patrick suppes philosophical essays - volume 2

PATRICK SUPPES: SCIENTIFICPHILOSOPHER

Volume2

SYNTHESELIBRARY

STUDIES INEPISTEMOLOGY,

LOGIC,METHODOLOGY,ANDPHILOSOPHY OFSCIENCE

ManagingEditor:

JAAKKOHINTIKKA,Boston University

Editors:

DIRKVANDALEN,UniversityofUtrecht, TheNetherlandsDONALD DAVIDSON,UniversityofCalifornia,Berkeley

THEOA.F. KUIPERS,UniversityofGroningen, TheNetherlandsPATRICK SUPPES, StanfordUniversity, California

JANWOLENSKI,JagiellonianUniversity,Krakow,Poland

VOLUME 234

PATRICK SUPPES, 1994

(Photo takenby Winston Scott Boyer andprintedhere withhis kindpermission)

PATRICK SUPPES:SCIENTIFICPHILOSOPHER

Volume 2.Philosophy ofPhysics, Theory Structure,andMeasurement Theory

Editedby

PAULHUMPHREYSCorcoranDepartmentofPhilosophy,

Universityof Virginia,Charlottesville, VA, U.S.A.

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TABLEOF CONTENTS

Volume 2: PhilosophyofPhysics, Theory Structure,andMeasurement Theory

PARTIII:PHILOSOPHYOFPHYSICS

BarryLoewer /Probability and QuantumTheory /Commentsby Patrick Suppes 3Arthur Fine /Schrodinger's Version of EPR, andItsProblems./ Commentsby Patrick Suppes 29

Jules Vuillemin /ClassicalFieldMagnitudes /Commentsby Patrick Suppes 45BrentMundy / Quantity,Representation andGeometry /Comments byPatrick Suppes 59

PaulHumphreys /Numerical Experimentation /Commentsby Patrick Suppes 103

PARTIV:THEORY STRUCTURE

RyszardWojcicki / Theories and TheoreticalModels /Commentsby Patrick Suppes 125

N.C.A.Da Costa andF.A.Doria /Suppes Predicatesand theConstruction of UnsolvableProblems in theAxiomatized Sciences /Commentsby Patrick Suppes 151JosephD.Sneed /Structural Explanation/Comments byPatrick Suppes 195

PARTY: MEASUREMENTTHEORY

R.DuncanLuce andLouis Narens /Fifteen ProblemsConcerning theRepresentationalTheoryofMeasurement /

vii

CommentsbyPatrick Suppes 219Fred S.Roberts andZangwillSamuelRosenbaum/The Meaningfulness of Ordinal Comparisons for GeneralOrderRelational Systems /Comments by Patrick Suppes 251C. UlisesMoulines andJose A.Diez / Theories asNets: The Caseof Combinatorial Measurement Theory /Comments byPatrick Suppes 275NameIndex 301

SubjectIndex 305

TableofContents to Volumes1and3 311

viii tableofcontents

PART III

PHILOSOPHY OFPHYSICS

BARRYLOEWER

PROBABILITY ANDQUANTUMTHEORY

My historical view of the situation is that ifprobability theory hadbeendeveloped to any-thing likeits current sophisticatedstate at thetimethebasicworkinquantummechanics wasdone in the twenties, thena verydifferent sort

of theory wouldhavebeenformulated. x

ABSTRACT.Patrick Suppes has writtenanumberofpapersin whichhe hasemphasizedthenon-standardnatureofprobabilityinquantum mechanics.Specifically, thequantumstateof asystemcannotbeunderstoodas characterizingaprobability measure over thespacegeneratedby thepossiblevaluesof observables.Hehassuggested thatarethinkingof quantum theory in whichprobabilityis treatedin a standard fashion wouldlikelysolve foundationalproblems inquantum theory. Iargue that thenon-standardness ofprobabilityis intimatelyrelated to themeasurementproblem. Once astand is takenonthat problemprobabilitycan be understoodclassically. Idiscuss two ways of doingthis; the orthodox account of quantum theory and Bohm's theory and argue for thesuperiority of thelatter.

Quantum theory plays a significant although ambivalent role inPatrickSuppes' philosophy of science. On the one hand, he appeals to itto support a number of his favorite themes - the fundamental role ofprobability in science, the inadequacy of Laplacian determinism, andthe incompleteness of scientific theory. On the other hand, he hasemphasized the peculiar natureof quantum mechanical probabilities.Heremarks thatProbabilityconcepts have a strangeand awkwardappearanceinquantum mechanics,as if theyhavebeenbrought withinthe frameworkof the theory only as anafterthoughtand withan apology for theirinclusion.2

Suppes is certainly correct about the awkwardness of probability con-cepts in quantum theory. A probabilistic theory is usually given byspecifyingaprobability space anda probability measure onthat space.Butin quantumtheoryprobabilitiesareintroducedbyBorn'srule which

3

P.Humphreys (cd.), PatrickSuppes: Scientific Philosopher,Vol. 2, 3-28.© 1994 Kluwer Academic Publishers. Printedin theNetherlands.

4 BARRYLOEWER

assigns themonly to the outcomesofmeasurements.Suppeshasempha-sized in a number of articles3 that this leads to probability assignmentswhich apparently fail to conform to the usual mathematical laws ofprobability. Specifically, if Q and Q* are non-commuting observablespertaining toa systemin state$ thenBorn's rule determines probabilitydistributions for each separately but not for their conjunction. Further,aseries of important results has shown that 'natural ways' ofextendingquantummechanical probability distributions to jointdistributions leadto absurdities.4 The most significant results, Gleason's theorem and itscorollary, the Kochen-Specker theorem, establish that it is impossibleto embed quantum mechanical probabilities into aclassical probabilityspace in such a way that these probabilities reflect the distribution ofvalues for observables.5 Inaddition to issues concerningthe mathemat-icsofprobability in quantum theory there is the questionofexactly howprobability in the theory should be interpreted. Are quantummechan-ical probabilities best understood as frequencies, degrees of belief, orobjective chances? These are the issues whichIwill address in mycontribution.

Suppesfinds the formalresults-especially thenon-existenceof jointprobability distributions- tobe at the heart of thepuzzles of quantumtheory. At one time he sought to adapt probability theory and logicto orthodox quantum theory. In fact he arguedfrom the non-existenceof joint probability distributions to a non-standard logic.6 But in morerecent writings he has reversed this line of thinking and instead hassought to adapt quantummechanics to classical probability theory andlogic. HesaysRelative to the historyof quantum mechanics over the past fifty years, it is aradicalthesis tomaintain thatorthodoxprobabilitytheorywill winout overorthodoxquantummechanics in providing a satisfactory theoretical framework. To a large extent, aphilosophicalviewpointthatderives from classicalrealismmotivatesmyownincreasingacceptance of theorthodoxprobabilisticviewpoint.Ifind it verymucheasier to thinkabout particles as havingcontinuous trajectories. The Copenhagen interpretation islacking intuitive persuasivenessand is too closely alliedto narrow positivistic viewsthatno longer seemplausible.7

Suppes's view hereseems tobe thatone shouldtry to formulate quantumtheory or a theory which accounts for quantumphenomena within theframework of classical logic and probability.Ivery much agree withthis approach. In factIwill argue that despite the results previouslycited it is not all that difficult to interpret quantum theory so that itslogic and probability theory are perfectly classical. But seeing this

5PROBABILITYANDQUANTUMTHEORY

requires areorientation of the way in which we are accustomed to thinkofquantum theory. Specifically,Iwill argue thatprior to understandingthe role of probability in quantum theorem the so-called measurementproblem or Schrodinger's Catparadox,must beresolved. Oncea standis taken onthemeasurementproblemthe theorycan be formulated withperfectly classical logic and probability.Iwill illustrate this claim bydiscussing twosolutions to the measurementproblem,theorthodox yon

Neumann accountand ahidden variable theory due to DavidBohm.Let us start by listing the principles of elementary non-relativistic

quantumtheory.1. Every isolated physical system is described by a quantum state

$ which is represented by a vector |$) in an appropriate Hilbertspace andeveryphysicalquantity (observable)Q isrepresentedbya Hermitian operator Q in that space. \Q= q) is a state in whichQ possesses the valueq.

2. Linearity: the quantum stateof a system evolves in accord with adeterministic linear law -Schrodinger's equation:

3. Eigenstate-eigenvaluerule: thequantum stateof asystemcontainsa complete specification of its physical state andan observableAhas avalue when and only when that state is an eigenstate of theobservable.

4. Born's rule: if S is in state $ then the probability that a perfectmeasurementofQon$willyield theresultQ = 6 is|($ | Q =b)\2.

Inaddition to theseprinciples elementaryquantum theorycontains avastamountofinformation concerning theconnection betweenphysicalquantitiesandoperators and thequantumstatesof realphysicalsystems.The whole theory is enormously well confirmed within non-relativisticdomains.

Prior to formulating the measurementproblem a few comments onthese principles are in order. First, it used to be, and to a lesser extentstill is,usual to try to understand quantum theory instrumentally -asmerely an algorithm for predicting theresults ofexperiments. But likeSuppesIfind 'classical realism' a more plausible philosophical view-point. If the theory isunderstood realistically then itsays that theentirephysical reality of an isolated system is characterized by its quantumstate. Second,Schrodinger'slaw is the soledynamical law of this theory

ot 2m

6 BARRYLOEWER

and it is deterministic and linear. These features will be very impor-tant to our discussion. Third, the eigenstate-eigenvaluerule is to beunderstood literally. Anobservablehas avalue when andonly when thestateis an eigenstate of that observable. Since no state is an eigenstateofnon-commuting observables suchobservables cannotsimultaneouslypossess values. Finally,Born's rule introduces probabilities into quan-tum theory but only for the outcomes of measurements. If Q and Q*are non-commuting then there is no state in which they are both welldefined; i.e. no statecorresponding to Q = qand Q* = q*. So, whileBorn'srule providesprobability distributions for Q andQ* separately,itgivesno wayofcalculating a jointdistribution of these twoobservables.

To keep matters simpleIwill suppose that Q is an observable ofsystem S with two values q\ and q2 and thatMis a measuring devicewhose pointer observable A has three values ao, a\, and ai (whichcorrespond to three pointer positions). The state \A = ao) is M'sready-to-measure state. A perfect non-destructive measurementof QonSis an interactionbetween S andMwhose Hamiltonian guaranteesthatif theinitial stateofMis the ready state and the initial state of Sis\Q =q{) then the final stateof S andMis\o =qi) \A =ai) (wherei=1,2).8 Since the interactionbetween Mandsisa physical interactionitconforms to Schrodinger'sequation. Thatequationpredicts that if theinitial stateof Sisnot aneigenstateof Qbut is ci |Q =q\ )+c 2\Q+q2)then it follows from the linearity ofSchrodingerdynamics thatthe finalstate ofM+Sis

Theproblem isobvious. MEAS is notan eigenstateofeither Qor A so,according to principle 3,neither Q nor A possesses a definite value inMEAS.Inother words,ifMEAS is the state ofM+S then the pointeris notpointing in any of its threepossible directions.

Schrodinger's famous cat paradox is a specialcase of this situation.If the Mis a cat and the pointer observable is the cat's being alive ordead then,according to 2 and 3,at the conclusion of the measurementthe cat isnot aliveandis notdeadbut isin asuperposition with both liveand deadcomponents.9 There is a moregeneral versionof the measure-ment problem which Philip Pearle calls 'the reality problem. Pearleobserves that in ordinary circumstances (even circumstances in whichno measurementsare being made) the states of physical systems willevolve so that they arerarely eigenstatesofusual physical quantities.10

MEAS c x \Q=qx) \A =ax ) +c2 \Q =?2) |>1 =a2>.

PROBABILITYandquantumtheory 7

So quantummechanical principles predict that most of the time mostof the properties which we think have values do not in fact have val-ues. Anyone who thinks of quantum mechanics as providing a truedescription of reality must come to terms with thisproblem.

The relationship between the measurementproblem andBorn's ruleshould be clear. According to Born's rule the probability of a mea-surement of Q yielding the result that Q=qi is equal to |ci|2. ButSchrodinger's law is completely deterministic. Rather than assigningthe quantummechanical probability it entails that this result will defi-nitely notoccur (since the final state is notan eigenstateof Q). So thereis a flat contradiction betweenBorn's rule, Schrodinger'sequation,andtheeigenstate-eigenvaluerule. The measurementproblem ismore fun-damental than the non-existence of joint probability distributions fornon-commuting observables since without a solution it does not evenmake sense to talk aboutprobability in quantum theory.11

Various solutions tothemeasurementproblemhavebeenproposed.Icannotsurvey them allhereandsowill,somewhat dogmatically,remarkthat as longas quantum theory is understoodnon-instrumentally thereare only two strategies that can possibly work. One is to deny the uni-versality of Schrodinger's equation(or replace it with anon-linear law)and the other is to deny the completeness of the quantum mechanicalstate. This strategy amounts to saying that a physical quantity maypossess adefinite value even when the state is not an eigenstate of theobservablecorresponding to thatquantity.12Iwill describe approacheswhich follow each strategy.

THEORTHODOX SOLUTION

The orthodox - that is, the usual text-book -way of dealing with themeasurementproblem, which was given its canonical formulation byyonNeumann13 replaces 2 with2. As long asnomeasurementisbeingmade onasystemSitsquantum

state evolves deterministically in conformity with Schrodinger'slaw

and adds5. The Collapse Postulate: ifa measurementis beingmade on Sthen

the quantum stateof M+ S does not evolve in accordance with

8 BARRYLOEWER

Schrodinger's law but rather jumps so that the probability that itevolves to \Q = qi) \A =ax) is equalto |q|2.

This changeof state is called the 'collapse' of thequantumstate.Itis clearhow yonNeumann'sproposal intends to solve themeasure-

mentproblem. Itentails that when an observable Q is beingmeasuredthe post-measurement state of the measuring apparatus and the quan-tum system is an eigenstate of that observable. So, at the end of themeasurement considered above, the state is not MEAS but is either\Q = qx) \A = ax) or \Q = q2) \A =a 2). The status ofBorn's rule isclear on yonNeumann's account. The ruleis entailedby 5 and 3 sincethe probability that the post-measurement state is an eigenstate of Q isthe onegivenby therule.Inow wantto show that if we take a realist view of orthodox quan-

tum theory then its logic and its probability theory can be understoodclassically. Letuslook first at the logic. According to the theory,phys-ical reality consists of aquantumstate function which usually evolvesdeterministically in conformity with Schrodinger'sequationbut which,every once in a while, when measurements are made, jumps in con-formity with the collapse postulate. We define 'an orthodox quantumpossible world' @ as a world history which so evolves. This defini-tion,of course, reflects the vaguenessof the 'measurement. A precisedefinition should include a precise physical characterization of thosequantumstates in which measurementsoccur.

Anorthodoxquantummechanical propositionisa setof suchworlds.A statement about a quantum mechanical observable, e.g. Qst = q,expresses theproposition that the state of S at t in @ is an eigenstateof Q with value q. Note that the quantum mechanical statement thatQ possessesno value alsoexpressesa quantum mechanicalpropositionsince it says that the state is one of those which is not an eigenstate ofQ. IfQt

— qandQis measured at t then the resultof themeasurementwill certainly be that Q = q but if Q possesses no value then themeasurementmay result in anypossible value of Q. Thelogic of thesestatements is ordinaryBoolean logic. Forexample,the statementQ = q& Q* — q* is perfectly well formed. IfQ and Q* are non-commutingthen theconjunctionis false since thereisnostate whichis aneigenstateof both operators.

The appearance that quantum mechanical statements obey a non-standard logic may arise from not taking seriously the position thaton this view the physical reality of a system is completely given by

PROBABILITYAND QUANTUMTHEORY 9

its quantum state. If we forget this then we may be tempted to thinkthat quantummechanical statements shouldbe related to each other incertain ways which suggest anon-standard logic. For example, let Qlbe an observable whose value is 1iff the stateassigns an amplitude of1 to the particle's being located in region L and Qr be an observablewhose value is 1iff the state assigns an amplitude of 1 to the particle'sbeing located in region R disjoint from L and Qrul an observablewhose value is 1 iff the state assignsan amplitude of1 to the particle'sbeinglocatedin theunionofRandL.Itis possible for Qlur — 1eventhoughQr 1and Qi 1.This has suggested to some that the logicof thesequantum mechanical statementsis non-standard since it seemsthat,according to quantum theorya disjunction - the particle is inLorthe particle is in R-can be true even thoughboth of its disjuncts arefalse. But from the point of view of a theory which takes the quantumstate as the real and complete physical state this is misleading. Theproposition that the quantumstate is an eigenstate of Qlur with value1is not thedisjunction of the propositions that the state isan eigenstateof Qr with value1and the stateisaneigenstateofQlwith value 1.Ofcourse, this doesnotmean that theorthodox view in which the quantumstate is real and complete is satisfactory. It entails that particles maylack definite location or move on definite trajectories and so on. Thatis quite puzzling. Mypoint is that the logic of propositions about thequantumstate isperfectly classical.14

Themathematical theoryofprobability for orthodoxquantumtheoryis alsoperfectly classical. Therelevantprobability spaceisan appropri-ate set ofsubsets of the setof possiblequantumhistories. The historiesevolve in accord with the dynamics described by 2 and 5. This givesrise to a branching tree structure where the branching points involvemeasurements.The probabilitiesof thesebranches atabranchpointaregivenby 5. For a givenstate $ at time t the probabilities of statementsof the form Qt = q areperfectly well defined. If$(£) is an eigenstateof Qt then P(Qt =q) is either 1or 0. IfQ and Q* are non-commutingobservables then the joint probability P(Q = q& Q* = q*)= 0 sinceno state is an eigenstate of both Q and Q*.

The probability at t that the system (or world) whose quantumstateis $f will evolve so that at Pt>(Q = q), where t' is after t, is givenbythe branching tree probabilities. It is the sum of the probabilities ofall the branches which are eigenstatesof Q at t'. Onceagain the jointprobability Qt> =q& Q*i =q*is0since the systemcannotevolve into

10 BARRY LOEWER

a state which is an eigenstateof both observables. There is noneed fornon-standardprobability mathematics.

According to Born's rule when a measurementof Q is made on asystem S in state $ which is not an eigenstate of Q the probabilitythat the result is Q=q is |(Q =q | $)|2. In the orthodox accountthis probability cannotbe an epistemic or ignorance probability. If itwere then there would be a matter of fact concerning the value of Q.But there is no matter of fact about the value of Q prior to its beingmeasured since the state is notan eigenstate of Q. For a similar reasontheprobabilities cannotbe understoodas frequencies. It is not thecasethat Sis one of an ensemble of systems in state $, \(Q = q | $)|2 ofwhichareonesin whichQ =q. If the stateofall these systems is$ then,on theorthodox account, for none of them does Q = q. Instead, theseprobabilities must be understood as objective chances. The quantummechanical probability is the objective chance of the system evolvinginto an eigenstateof Q with value q when Q is measured.

If whatIhave said so far is correct then the logic and probabilitytheoryof the orthodox accountis classical. Theaccountclearly recon-cilesBorn's rule with the dynamics andin doing so goessome distancetoward solving themeasurementproblem and the realityproblem. ButnowIwant topoint outsomeof the ways inwhich theorthodox accountis implausible and inadequate.

First,it should be noted that there is an inexplicitness in the ortho-dox view which up until now we have mostly ignored. It is that theview is not clearly specifieduntil aclear rule is given whichsays whenSchrodinger evolution occurs and when the collapse occurs and that,of course, involves saying exactly which physical processes are mea-surements. Defenders of the orthodox interpretation are rather casualconcerning this question since it turns out that as long as the collapsedoes not occur in interactions involving only isolated microscopic sys-tems (i.e. measurementsalways involve macroscopic systems) it willbe enormously difficult to empirically test whether or not a collapsehas occurred. For this reason the precise point at which Schrodinger'sequation gives way to the collapse isnot practically testable.

Second, itis difficult to believe that there are tworadically differentdynamical laws which govern interactions; the probabilistic collapsefor measurementsand Schrodinger's law for all the others. There isno smooth transition from one dynamic to the other. Theproposals forwhere to put a cut-off are not plausible; e.g. that Schrodinger's law

PROBABILITY AND QUANTUMTHEORY 11

holds for systems with fewer thannparticles (or degreesof freedom orwhatever)while thecollapseholds for systems with normore particles.

Third, the collapse of state is a physically peculiar process since itis non-local. Einstein,Podolsky, andRosenpointed this out long ago.David Bohm's version of their thought experiment involves a pair ofelectrons in the singlet state which are spatially separated. Orthodoxquantum theory predicts that if the 2>spin of one of the electrons ismeasured the statecollapses and instantaneouslyresults inan eigenstateof ;c-spin for both particles. In effect, this means that the physicalreality -whether ie-spin has a definite value or not -of one particle isinstantaneously affected bya measurementonthe other particle. Muchismadeof thepoint that thenon-locality involvedinthecollapseofstateis empirically compatible with special relativity since it is not possibleto transmit messagesor energyby the collapse process. This is correctbut still the collapse is a non-localprocess and it is no easy matter toformulate it without supposing thatapreferredreference frame exists.15

If it is so formulated in a preferred reference frame then that amountsto treating special relativistic claims about the structure of space-time(that there is no preferredreference frame) not completely realistically,orit involves treating the collapse of state notcomplete realistically.

Fourth, as we have already noted, the orthodox theory allows forstates which are not eigenstates of ordinary physical quantities, likeposition, momentum,energy,etc. and, in fact, doesnot allow for stateswhichare eigenstates ofbothposition and momentum.Inconsequenceparticles fail to possessdefinite trajectories.

Fifth, the orthodox view's attempted resolution of the measurementproblem is ultimately unsatisfactory. There are situations in which nomeasurementsare being made (on any reasonable construal or 'mea-surement') which therefore evolve linearly in such a way that,exceptinrare circumstances, theirquantumstates are noteigenstatesof ordinarymacro-properties. That is, the quantum states of cats (or systems ofwhichcatsareparts) typically evolveso that itisnot aneigenstateof thecat'sbeingalive or dead. Immediately after aposition measurementthewavefunctionbeings to spreadoutso that soonitis notan eigenstateofposition. It follows that the orthodox account fails to solve the realityproblem.16

12 BARRYLOEWER

BOHM'SHIDDEN VARIABLEVIEW

Inow want to consider a solution to the measurementproblem whichfollows the secondstrategyofdenying thatthe quantummechanical stateis the complete physical state. Such interpretations are misleadinglycalled 'hidden variable' theories.17 While there are anumber of hiddenvariable approaches, by far the most satisfactory was developed byDavidBohm (based on an idea originally due to de Broglie) and wasgivenan especially nice formulation by John Bell. It is a scandal thatBohm's theory has received so little attention from philosophers ofquantummechanics.18 The reason for this is probably the existence ofa number of 'no go' theorems due to yonNeumann,Gleason,and Bell,among others, whichhavebeen widely interpretedas demonstratingtheimpossibility of theories like Bohm's.

Suppes mentions John Bell's discussion of some of the 'no hiddenvariable theorems' (specifically yonNeumann's resultand the Kochen-Specker theorem)and seems toagreewithBell that theydo notdemon-strate the non-viability of hidden variable interpretations. But he goeson to say

In a beautiful series of papers beginning with Bell (1964 and 1966), a much morereasonable and intuitive treatment of hidden-variable theories has been given, andtheir impossibilityhas beendemonstratedexperimentally,at a rathersatisfactorylevel(Suppes, 1984,p. 25).

Itwouldbeuncharitable to interpretSuppes asendorsing the commonlyheld view thatBell's resultand Aspect'sexperiments literally show theimpossibility ofhidden variableresults. Bellhimselfwasanenthusiasticproponent of Bohm's hidden variable theory.19 In fact,he obtained hisfamous result by reflecting onBohm's theory and asking whether thenon-locality whichit manifests is anunavoidable feature of anyhiddenvariable theory capableof reproducing quantum mechanical statisticalresults.20Hisaffirmative answer doesnotshow that suchhiddenvariabletheories are impossible, but rather that they must be non-local. Onany such theory events which are related in a space-like way may becausally related. Perhaps this would count as an objection to hiddenvariable theories from the perspective of orthodox quantum theory ifthe latter were a local theory. Butit isnot. AsImentionedpreviously,Einstein,Rosen,andPodolsky hadalready established the non-localityof orthodox quantum theory and Bell himself presents a beautifully


lucid argumentthatanytheory-hidden variableornot-whichlawfullyreproduces quantummechanical statistics mustbe non-local.21

Whatever the reasons for its neglect, Bohm's theory should not beignored by philosophers of quantum theory. After briefly describingthe theoryIwill show that its logic andprobability theory is perfectlyclassical. Further,it provides a much more nearly adequate solution tothe measurementproblem than does the orthodox account. However,there are questions about exactlyhow probability shouldbeunderstoodwithin the theory andIwill close with some reflections onthis issue.

Bohm's theoryposits that,in addition to possessingaquantumstate,a system consists of a collection of particles (their number and naturedepending on the quantum state) which always occupy definite loca-tions. Itmodifies standard quantum theory by dropping 4 (since par-ticles have definite positions even when the wave function is not aneigenstate of position)and adding to 1, 2, and3 the following:

where q= (qx,...,an) specifies the positions of particles1through n.The velocity law says, in effect, that the quantumstate assigns to eachpoint in spacea velocity vector whichdetermines the particle's velocityif it occupies thatposition. If a systemconsistsof nparticles then thewavefunctionevolvesin3 — ndimensionalconfiguration spaceandthesystemof particles is located ataposition qin configuration space. Thevelocity law then describes the motion of the system in configurationspace andthis determines themotionofeachparticle inordinary 3-space.Notice that how a particle (or system of particles) moves dependsonlyon the value of the wave function where the particle is located (or thesystem is located in configuration space). Non-locality arises since,for example, the motions of a two particle system in 6-dimensionalconfiguration space may involvecorrelation of motions ofeach particlein ordinary 3-space.

The centralresultof Bohmian quantumtheoryis that theprobabilitydistribution given by 7 is preservedas the state of the system evolvesin conformity with Schrodinger's equation,andthe positionsof the par-ticles evolve in conformity with the velocity law. In fact, the velocitylaw is selectedprecisely because it has this feature. If a measurementinteraction is understood as correlating the position of something (say

do ,„ grad|s) , .6. Velocity Law: -^ =hImB , ' '

(q)at |$)

7. Probability Law: Pd(q) = |($ |$)|2

14 BARRY LOEWER

apointer) with an observablebeingmeasured then Bohm's theory vali-dates Born's rule for all observables. This means that, given the aboveassumption concerning measurements, Bohm's theory is empiricallyequivalent to standard quantum theory.22

It should be clear how Bohm's theory proposes to solve the mea-surement and reality problems. Consider a measurement like the onediscussed earlier in thepaper in which the stateofM+S evolves intothe stateMEAS. MEAS is a superposition of two components corre-sponding,say, topointeron the left andpointer on theright, whichareeffectively separatedin configuration space (this separation is requiredof a measurement). The pointer's particles will evolve in such a wayso that they are located in configuration space with one or the other ofthese components. In consequence the pointer will be pointing eitherto the left or to the right. Further, as long as these components do notsubsequently overlap, the movements of the particles will be directedonly by the component in which they are located. This has the samepractical effect of a collapse of the wave function although the othercomponent continues to evolve. Should the components subsequentlyoverlap the other component may havea physically detectable effect.23

Similarly, in Schrodinger's catexperiment,by theend of the measure-ment the 'live' and 'dead' components of the wave have separated inconfigurationspace and thecat's particlesare located inoneor theothercomponent. So the cat will be either alive or dead. More generally,by addingparticle positions to the ontology Bohm's theory is able tovalidate ordinaryproperties possessingdefinite valuesand sosolves therealityproblem.

Developinglogic andprobability theory for Bohm's accountis fairlystraightforward. Here is how we can develop the logic. A Bohmianpossible world is a world history of the evolution of the world's wavefunction in conformity with Schrodinger's law and the evolution ofparticle positions in accord with the velocity law. We represent itbyan orderedpair ($(0),g(0)), where 0 is some 'initial' time. Note thatthe part of the history representedby $ is different from an orthodoxhistory since it always conforms to Schrodinger's law i.e. there isno collapse of the wave function. Since the evolutions of both $(0)and q(0) are deterministic these serve to determine their values at alltimes. A Bohmian proposition is a set of Bohmian worlds. Ordinaryquantummechanical statements areunderstood asexpressingBohmianpropositions; e.g. Qs(t) =qexpressestheproposition thatthequantum

15PROBABILITY AND QUANTUMTHEORY

stateofSis aneigenstate of Q withvalue q. InBohm's theory therearealso statements like X(t) = x which specify the location of the systemof particles (in configuration space). Truth functional combinationsof these statements are always well defined and their logic is classicalproposition logic.

The probability theory for Bohm's accountis developedin this way.For each world quantumstate$(0) there is aclassical probability mea-surep(q{o)/$(0))overtheparticlepositions whichsatisfiesBorn's rule.Theremay alsobe aprobability distribution,r(s),over the set of quan-tum mechanical world histories,but this probability assignment is notpart of the theory (it represents subjective uncertainty about the truequantum state). The probability of a statement about the value of anobservable; e.g. p(Qt

= q/$o) is 1 if $o is an eigenstate of Qt and 0otherwise. Notice that theprobability (and the truth value) of a state-ment about the quantum mechanical position observable may differfrom the probability (and truthvalue) of a statementabout position. Sothe probability that the position observable has some value qmay be0 (or 1) even though the value ofp(q(t) = g/$o) differs from 0 (or1). Similarly, if e.g. momentumis defined in terms of rate of changeof q(t) then its value and probability given $(0) may differ from thevalue andprobability of the momentumobservable given $(0) (whichwill either be 0 or 1). Notice that joint probability distributions areperfectly well defined. Just as in orthodox quantum theory, if Q andQ* are non-commuting then P(Q = q & Q* = q*) = 0. If V(t) isthe velocity of the system at t definedin termsof rate ofchange of q(t)then P(q(t) =x & V(t) =y) is also well defined andcalculated fromthe Bohmianprobability distribution. The jointprobability ofobtainingparticular results for the simultaneous 'measurement' of Q and Q*, ofcourse,does notexistsince itisphysically impossible tosimultaneously'measure' non-commuting observables. Themoral is that as long as weareclear aboutexactly what propositionaBohmian statementexpressesthere isno reason to think thatprobability mathematics isnon-standard.

An interesting feature ofBohm's theoryis thatperfectmeasurementsof position within Bohm's theory are faithful in that they reveal pre-existing values of q(t). Suppose that the pre-measurement Bohmianstateof S+Mis

[SUMci|si(O))|/2),g5(O),0M(O)]

16 BARRYLOEWER

(where the $i are orthogonal position states). The measurementcor-relates the values of a position observable A of M with values of theposition observable of S. Suppose also that qs(0) is located in $fc(0)and that qm(0) is located in \R). Then the post-measurement state ofM+Sis

and it is guaranteed that qM.(t) is located within the region associatedwith \A = dk). That is,the pointer faithfully records theposition of theparticles. The situation contrasts with orthodox quantum theory sinceon that accountmeasurementsare faithful only when thesystem's stateis an eigenstate of the observablebeingmeasured.

The situation in Bohm's theory with respect to observables whosevaluesarenotdeterminedby thevalue of thepositionobservableisquitedifferent. Consider such a measurementof observable Q of system Sbyobservable A ofsystemM;i.e. the measurementcorrelates \Q =qi)with \A = a^. Suppose that the initial quantum mechanical state ofS+Mis |$(0)> = SUMCi\Q(o) =qi) \R(0)) and that gM+s(o) (theposition of particles in S and the measuring device) are distributed inaccord withBorn's rule. Then,as wehavenoted,thepost-measurementstate will be \s(t)) = SUM c% \Q(t) =qi) \A = a^ and the particlesin the pointer will still be distributed in conformity with Born's rule.This means that the probability of obtaining the 'result' Q = qi isgiven by Born's rule. We have to be careful in interpreting this asa 'measurement' since neither at the beginning nor at the end of themeasurementinteraction is the state an eigenstate of Q. So it would beas much of a mistake onBohm's theory as on the orthodox account tothink of the pointer asfaithfully recording aprior value of Q.

But the situation inBohm's theory is even more peculiar. Exactlywhich value of Q is recorded by the measuringdevicedependsnot juston5andits statebut also on the stateofM, including the positions ofits particles. This is easily seen by recalling that inBohm's theory thepost-measurement positions of the particles in M- where the pointeris pointing at the conclusion of the measurementis determined bythe pre-measurement Bohmian state of S+M (assuming that this isan isolated system) including the pre-measurement positions of theparticles in M. Finding that at the endof the measurement the pointeris in the position associated with A = a^ not only does not recorda pre- or post-measurement value for Q on 5; it does not record an

[S\JM\si(t))\A = ai)iqs(t),qM(t)]

17PROBABILITY ANDQUANTUMTHEORY

intrinsic propertyof S atall. If the original positions of theparticles inthe measuringdevice had been different the outcomemighthave beendifferent.

The Kochen-Specker theorem says that any hidden variable theo-ry in which the outcomes of measurementsare determined containsobservables which are contextual.24 Contextuality means that the out-comeof ameasurementof the observabledependson mattersother thanthe state (quantum andhidden variable) of the systembeingmeasured- e.g. on what other observables are being measured or the state ofthemeasuring apparatus. InBohm's theory contextuality is manifestedby the fact that the outcomeof a measurementofan observable (otherthanobservables which areafunction ofposition) dependsonthe initialpositionsof particles in M.

The inevitability of contextuality is often taken tobe an objection tohidden variable theories. But even the verystrongkindof contextualityexhibitedby Bohm's theory is not problematic given the ontology ofthe theory. Thatit seems so may arise from the habit of thinking of allquantum mechanical observables as representingproperties intrinsic toS andmeasurementsas passive interactions which reveal those values.It is natural to think thatif Q is really a propertyofS then its value,ifit has a value,should be counterfactually independentof the propertiesof M and, in particular, exactly how the measurement is carried out.But inBohm's theory the momentumobservable for example does notcharacterizethemomentum(definedinterms ofposition) of theparticlesbut rather characterizes thequantumstate. And 'measurements'of suchobservables do not record values but are interactions whichdeterminethe evolution of the Bohmian state of M+S. From this perspectivethere is nothingmysterious or objectionable in its being the case thatexactly how a 'measurement' is carried out may affect the outcomeofthemeasurement.

How shouldprobability be interpreted withinBohm's theory? Sincethe theoryis thoroughly deterministic,its probability statements cannotbe understood as referring to objective chances. It seems that theseprobabilities mustbeconstruedepistemically, somethinglikeprobabili-tiesin statistical mechanics. There is,however,an important differencebetween the two theories. Instatistical mechanics itisat least inprinci-plepossible to obtain complete information about the physicalstate ofa system. In contrast, it is in principle impossible to obtain any moreinformation about the Bohmian state than that allowed by orthodox

18 BARRYLOEWER

quantum theory. The uncertainty relations,for example,put a limit onhow much information one can have about the outcomes of measure-mentsof non-commuting observables. While itmaybepossible to learnthequantumstateof asystem the mostonecan know about thelocationsof its particles is givenby Born's rule. So the probabilities inBohm'stheory, while epistemic, have akind of objectivity that is grounded inthelaws of the theory.

Thelimits on obtaininginformation about theBohmian state and, infact,all theempiricalconsequencesof Bohm's theory(and so the theo-rem thatBohm's theory is empirically equivalent to orthodox quantumtheory),dependessentially onthe assumptionthat particlepositions areinitially distributed inconformity withBorn's rule. AsIsee it, the mainfoundational problem with Bohm's theoryconcerns how tounderstandthis assumption. The easiest way is to imagine that the universe hada beginning and at that initial moment some chance process -Godtossing particles into the original quantumstate? -distributed particlesin accord with Born's rule. These initial chances serve to ground theepistemic probabilities of Bohm's theory at all subsequent times. But,it is hard to take this myth seriously.25

Without objective chances togroundBohm'sepistemic probabilitiesit is difficult to understand their status. What is it about the world andour relation to it whichrenders thesetheappropriateprobabilities?Iamnot sure thatthis questionhas,or evenneeds,an answerbutIwould feelthatIunderstoodBohm's theorybetterifIknew some wayof groundingits probability claims.

Suppeshascalled for arethinkingofquantum theoryin which ortho-doxprobability theory wins outoverorthodox quantummechanics. Myargument in this paper has been that the rethinking should begin byconfronting the measurementproblem since, without a solution to thatproblem, we cannot even make sense of quantum mechanical proba-bilities. Bohm's theory provides a way of resolving the measurementproblem andalso provides a framework in which logic andprobabilityis perfectly classical. It even assigns particles continuous trajectories.For these reasons Suppesmay find it attractive. On theother hand,itsdeterminism andits focus onthepositionsof particles isreminiscent ofLaplace's world picture. These features are probably not congenialtoSuppes'sphilosophical views. Ido not know whether it is the kind oftheory that Suppeshad inmind when he wrotethat:


ifprobability theory hadbeendevelopedto anything likeits current sophisticated stateat the time thebasic workonquantum mechanics wasdone in the twentiesthen a verydifferentsort of theory wouldhavebeen formulated.26'27

DepartmentofPhilosophy,Rutgers University,NewBrunswick,NJ08903, U.S.A.

NOTES

1 Suppes (1979), p.12.2 Suppes(1962), p. 334.3 Suppes(1966) and (1984).4 Wigner (1932) andNelson(1967).5 Gleason (1957),Kochenand Specker (1967), andBell(1989).6 InSuppes (1966) he argues that every proposition or event should be assigned aprobability and concludes from the non-existenceof joint probabilities that arbitraryintersectionsof propositions arenot propositions. Suppes does not endorse the argu-ment andinlater writings seems quiteskeptical of it.7 Suppes inBogdan (1979), pp. 210-211.8 This is anecessaryconditionfor aperfectmeasurement. Givingasufficient conditionis a muchtrickiermatter.9 1amassuming thatalivenesscorrespondstoaQMobservable.This seemsreasonableif we think that the cat is a physical system and its state of health supervenes on thevaluesof physical quantities ofits particles.10 The same point ismade by AlbertandLoewer (1991).11 The habit of talking about probabilitiesprior to resolving the measurement prob-lemis very misleadingand has caused some writers- e.g. Prosperi andLongeri-

to think that the measurement problem is not a genuine problem or is solvedbynoting that the statisticalpredictionsof asuperposed state likeMEAS are forallprac-tical purposes indistinguishable from the statisticalprediction of a mixture of its twocomponents (which allegedlyis non-problematic). But the problemis that given theeigenstate-eigenvaluelink,states likeMEAS areones in whichthestatisticaloutcomesare non-existent.12 This claimcanbe appreciatedby noting that linearity leads to the stateMEAS andtheeigenstate-eigenvaluelinkentails thatin this state thepointerobservableisnot welldefined.13 YonNeumann (1955).14 Thereis atraditionofrespondingto the foundationalproblemsofquantum theorybyarguing for a non-classical logic with a structure that mirrors the algebraof quantummechanicaloperators.Iknow of no such proposalthat actuallysucceeds in resolvingthemeasurementproblem.15 SeeMaudlin (1993).16 Thereis another way of pursuing the strategy of modifying linearity, which does a

20 BARRYLOEWER

much better job of dealing with the measurement problem,due toGhirhradi, Rimini,andWeber anddevelopedbyPearleandBell. According tothis strategy thereis asinglenon-lineardynamic whichgoverns theevolutionof thequantum state. The basicideaisthatasystem's wavefunction usually evolvesinconformitywithSchrodinger's lawbuteveryonceina while,withaspecificprobabilitythatdepends onthenumber ofparticlesin the system, the wave function is multiplied by anarrow Gaussian. The probabilitythattheGaussianis centeredat apointxdependson theamplitudeof thewavefunctionat x. Since the chance of a collapse (multiplicationby the Gaussian) is very smallfor systems with few particles this theoryispractically indistinguishablefromstandardquantum theory for such systems. When a quantum systembecomes correlatedwitha macro-system the chanceof a collapse is very largeso states like MEAS are veryunstable and soonprobabilisticallyjump into states which(approximate)states whichare eigenstatesof pointer position. For discussions of GRW seeJohnBell(1989) andAlbert andLoewer (1992).17 Although, as Bell observes, the title 'hidden variable' is not apt since the hiddenvariables areusually whatis directly observed whilethequantum statemanifests itselfonly in thestatisticsof thehidden variables.18 It receives no mention inRedhead (1987), Hughes (1989), or vanFraassen (1991).Many oftheclaims foundin thesebooks arerefutedby theexistenceofBohm's theory.See Albert andLoewer(1989).19 Bell,J. (1989), The ImpossiblePilotWave.20 Bell(1989).21 Bell(1989). The situationis a bitmorecomplicated.First, ithas becomepopular toargue that thenon-locality inorthodox quantum theoryisnot asbadas thenon-localityof hiddenvariabletheories. Sometimesit is claimedthat thelatterallows for superlu-minalsignalingand sois incompatible(or more incompatible?) withspecialrelativity.But it can be shown that no theory which reproduces quantum mechanical statisticscanallow for superluminalsignallingand, for example,Bohm's theory does not. Theothercomplicationis that there is, infact,one interpretationof quantum theory whichsolves the measurement problem and is thoroughly local. This is the 'many mindsinterpretation'due to David Albert and myself (Albert and Loewer, 1988). There isnoconflict withBell's argument since this interpretationdoes not validateoneof theassumptions whichBellmakes.22 This claimisnot exactlycorrect. Anycollapse theorydiffers inprinciplefromanon-collapsetheory likeBohm's,but in orthodoxquantum theorycollapsesareassumedtooccur only when systems involve many particles andfor such systems it is not practi-cally possibleto empiricallytestwhethera collapsehas occurred. See Albert(1992).23 Noticeableoverlap formany-particle systems is enormously difficult to bringaboutexperimentallyandprobably seldomoccurs for any length oftime.But in few particlesystemsit is whatis responsibleforcharacteristically strangequantumphenomena.Forexample,in the famous two-slitexperiment the wave functionof aparticle is literallysplit,with components going througheachhole. The particle travels withone or theothercomponent. The apparatusis setup in such a way that thecomponentsoverlapatthe screen. This overlap is what accounts for the interferenceeffects. SeeBohmandHiley (1992) for adetailedanalysis.24 A physicalquantity is said tobe 'contextual'iff theresultofmeasuringitdepends notonly on its value, if any,buton othercontextual featuresof themeasurement situation,


e.g. whatotherquantitiesarebeingmeasured.25 Thereare stochastic theories similar toBohm's. Aninterestingdiscussion of proba-bility inBohm's theory is foundinGoldsteinetal. (1992).26 Suppes (1972),p. 12.27 Thanks to DavidAlbert.

REFERENCES

Aharanov,V and Albert,D.: 1981, 'Can We Make Sense of theMeasurementProcessinRelativisticQuantumMechanics?', Phys.Rev. D., 21(2), 359-370.

Albert,D.: 1992,QuantumMechanicsandExperience,Cambridge:HarvardUniversityPress.

Albert, D. and Loewer,B.: 1988, 'Interpreting theMany Worlds Interpretation',Syn-these,77, 195-213.

Albert,D. andLoewer,B.:1989, 'TwoNo-ColpaseInterpretationsofQuantumMechan-ics',NOUS,23, 169-186.

Albert,D. andLoewer,B.: 1990, 'Wanted Dead or Alive: Schrodinger's Cat', in: A.Fine, M.Forbes,andL. Wessels (Eds.), Proceedings ofScience Association1990,VolumeI,East Lansing: Philosophy of Science Association.

Albert, D. and Loewer, B.: 1991, 'The Measurement Problem: Some Solutions',Synthese, 81, 140-152.

Albert,D. andLoewer,B.: 1992, 'Tails of Schrodinger'sCat' (unpublished).Bell, J.: 1989, The Speakableand Unspeakablein QuantumMechanics, Cambridge:

Cambridge Univ. Press.Bogdan,R. (Ed.): 1979,Patrick Suppes,Dordrecht: Reidel.Bohm, D.: 1952, 'A Suggested Interpretationof QuantumTheory inTerms ofHidden

Variables: PartI',Phys. Rev.,85, 166-179.Bohm,D. andHiley, D.: 1992, The UndividedUniverse,London: Routledge.Daneri,A.,Longer,A., andProsperi,G. M.: 1962, 'QuantumTheory ofMeasurement

of Ergodicity Conditions',NuclearPhysics, 33,297-319.Einstein,A.,Poldosky,8., andRosen,N.: 1935, 'Can Quantum MechanicalDescrip-

tionsofPhysicalRealityBeConsideredComplete?',PhysicalReview,47,777-780.Gleason,A. M.: 1957, 'Measureson theClosedSubspaces ofaHilbert Space',Journal

ofMathematicsandMechanics, 6, 885-893.Goldstein, S.,Detles,R., Durr,D.,andZangi,N.: 1992, 'QuantumEquilibriumandthe

Origin of AbsoluteUncertainty',JournalofStatisticalPhysics, 67, 843-907.Hughes, R. I.G.: 1989, The Structure and Interpretation of Quantum Mechanics,

Cambridge: Harvard University Press.Kochen, S. and Specker, E.: 1967, 'The Problem ofHidden Variables inQuantum

Mechanics',JournalofMathematicsandMechanics,17, 59-87.Maudlin,T: 1993, QuantumMechanics andRelativity, Oxford:BasilBlackwell.Nelson, E.: 1967, DynamicalTheories ofBrownianMotion,Princeton,NJ: Princeton

University Press.Pearle,P.:1992,Talk atConferenceon theFoundationsofQuantumTheoryat Columbia

University (unpublished).

22 BARRY LOEWER

Redhead,M.: 1987, Incompleteness,Nonlocality,andRealism:A Prolegomenonto thePhilosophy of QuantumMetaphysics,Oxford: ClarendonPress.

Suppes,P.: 1961, 'ProbabilityConcepts inQuantum Mechanics',Phil, ofScience, 28,278-289.

Suppes, P.: 1962, 'The RoleofProbabilityinQuantum Mechanics', in: B.Baumrin(Ed.), Philosophy ofScience, the Delaware Seminar, Vol. 2, New York: Wiley,pp. 319-337.

Suppes, P.: 1966, 'TheProbabilisticArgument for a Non-Classical Logic of QuantumMechanics',Philosophy ofScience, 33, 14-21.

Suppes,P.: 1979, 'Self-Profile',in: R.J.Bogdan(Ed.),PatrickSuppes, Boston: Reidel.Suppes,P.: 1984, ProbabilisticMetaphysics, Oxford:BasilBlackwell.vanFraassen, B.: 1991, QuantumMechanics: An Empiricist View,Oxford: Clarendon

Press.yonNeumann,J.: 1955, TheMathematicalFoundationsofQuantumTheory,Princeton,

NJ:PrincetonUniversity Press.Wigner, E. P.: 1932, 'On the Quantum Correction for Thermodynamic Equilibrium',

Phys. Rev.,749-759.

COMMENTS BY PATRICK SUPPES

Thereis muchthatBarry andIare in agreementabout concerningquan-tummechanics,especially in our critical view of the weaknessesof theorthodox interpretation of quantum mechanics and in our skepticismtoward the need for any nonclassical logic or probability -a matteronwhichIhave changedmy position, as he indicates. Moreover, wearealsoin agreementabout staying within the generalconfines of classicalrealism. For reasons that Iwill state, Ido not share his enthusiasmfor Bohm's theory, althoughIcertainly do agree that it is one of theimportant hidden-variable theories andhas undoubtedlybeenmuch tooneglectedby philosophers of quantummechanics. There is much to besaidabout almost allof theissues hediscusses,at least,much to be saidfrommy own viewpoint toward the foundations ofquantummechanics.However,Iwillrestrictmyself to four generaltopics because theyrepre-sent thematters on whichheandIdiffer themostand therefore it willbemostuseful to discussin detail.Idohave the feeling that with sufficienttime for analysis we would come to have positions that are relativelyclose, both of us probably modifying some of our current views. ThereasonIsay this is thatLoewer andIbegin with such an agreed-upongeneralframework for quantummechanics. Thepredictionisalsobaseduponmy experience withhimas agraduatestudentmany yearsago. Hehas always been prepared todefend in detail unorthodox or unpopular


views,buthisunderlyingoutlook isreasonable andamenable to makingrevisions tomeetdetailed criticism.Ihope theseare characteristics thatIshare, at least in part.

Difficulties ofBohm's Theory. There are at least two majoraspectsofBohm's theory that disturbme. One is the existence of n-dimensionalwave functions (for large n in many-particle systems) that never col-lapse. Both the physical interpretation of such waveseven for a singleparticle, andeven more the problem of their notcollapsing, aremattersthatIfinddifficult to haveany sort of feeling for. Unlike Barry,Ireallydo think of the wave functions that satisfy Schrodinger's equation asbeingwavefunctions that are probability amplitudes-the exactcharac-terof these distributions isdiscussedmore thoroughlybelow. Thepointis thatphysicalexistence,exceptas probabilitydistributions,especiallyin n dimensions for many particles, is not something thatIhave anyphysicalunderstandingof at all.

Second,Bohm's introduction ofa quantumpotential which leads toexpressions for the velocities of particles, is also something thatIfindhard to have a feelingfor. It seems too much like a deus exmachina.How are we to think of thispotential, whichinmanyofitscharacteristicsis like a classical potential? In the classical case, however, we have agood feeling for the physical source of the potential. In the quantummechanical case,however,Iat least do not.

Third,Iam skeptical as to how the Bohm theory can be extended toaccountfor thebehavior ofphotons andelectrons intheirusualquantumelectrodynamical framework. More generally,it is notclear to me howtheBohm theory canbe successfullyextended to arelativistic setting.Ido want to emphasize that Iam not in principle against Bohm's

theoryandmaybe ifIunderstood itbetterIwouldbe more sympathetic.

QuantumMechanicalProbabilities as Averages. AsIhave empha-sized on several occasions,but most recently in Suppes (1990),Ithinkof the probability distributions that arise from the Schrodinger equa-tion as average ormean distributions. Consequently, the application ofthe results to the behavior of single particles is to be treatedcarefully,thoughit is not inconsistent todo so. Thepoint is that the informationcontained in the distribution is very weak. We cannot, for example,from the probability distributions arising from the Schrodinger equa-

24 BARRYLOEWER

tion, compute autocorrelations of positions for a givenparticle. Whatwe really getis for each time t a mean distribution for particles satisfy-ing the givenphysical conditions;for example,that ofa freeparticle, aparticle that is a harmonic oscillator,a particle in a Coulomb field,etc.Consider,for example,the freeparticle as thesimplest case. Letusspec-ify some distributions at time to,and we thenget from the Schrodingerequation themeandistribution at anyother time t,pastor present. Thismean distribution just tells us the distribution at time t. Itdoes not tellus about how the distributions across time are related. This would gobeyondthe formalism of standard quantummechanics. What is impor-tantabout thesemeanor average distributions is that they arecapaciousin the sense that theycan accommodate many different hidden-variabletheories. This is a familiar fact about theories that are formulated onlyin terms of meandistributions. Manydifferent detailed theories can bedevelopedin a way such that the amplification is consistent with theoriginal givenmean distributions.

Also notice that in the relativistic theory of quantum mechanicsthis mean view does not really hold up too long, for something likeFeynman's propagators arise naturally and are constantly used. Buta propagator is just a way of studying aspects of the trajectory of aparticle. Note also that in standard quantummechanics what amountsto the kindof conditionalization that takes place in a propagatoris notreally admitted. We cannotmeaningfully conditionalize on the meandistributions that are solutions of the Schrodinger equation. It is goodthat we cannot, for conditionalizing on mean distributions in generaldoes not yield a meaningful result,but any more detailed theory, i.e.any hidden-variable theory, is crying out for such conditionalizationas a legitimate and appropriate method for thorough investigation oftheoretical details. If we can say something about where a particlehasbeen, we can say a lot more about where it will be in the future. It isperhaps theaspectof quantum mechanics that seems the mostpeculiarfrom the viewpoint ofprobability andclassicalphysics,but if we thinkof it in terms of mean probabilities it is not peculiar at all. It is just aweak theory that does not have anything to say about trajectories but,contra Bohm in his more extremephilosophical moments, this doesnotmean at all that such trajectories do not exist. Itonly means thatclassicalquantummechanics can have little to say about them. In fact,itdoesnot take muchexperiencewith the talk ofphysicists toknow that

25PROBABILITYAND QUANTUMTHEORY

the concept of particles having trajectories is uneliminable from theirintuitive andexperimental discourse.Iemphasize then that thecomplex wavefunctions that arise in clas-

sical quantum mechanics require no peculiar interpretation. Whensquared, they are just mean distributions at a given time. Irealizealso that this needs detailed development for cases not involving thedistribution of position,orinvolving particles that must be givenarela-tivistic treatment,as in thecase ofphotons. But it is notreally possibletopresent further details in thepresent context.

Other Hidden-Variable Theories. Although it is by no means myfavorite,asIhavealreadyindicated,IconsiderBohm'stheory animpor-tant one. It must be taken account ofby anyone who is serious abouthidden-variable theories. Iwant tonow turntosome ofthealternatives.

One that Iwas devoted to for some time is stochastic mechanics,developedespecially by EdwardNelson (1967,1990). Nelson wantstostart with classical mechanics andadd Brownian motion, sohe derivesin a natural way a diffusion for particles and from this he is able todevelopa goodpartof classicalquantum mechanics.

On the other hand, critical parts are not really satisfactory. Forexample, the analysis of interference is not persuasive, and more gen-erally the account he gives of wave phenomena is not worked out indetail, and only the virtuosity of some of his explanations gives it anair of plausibility at all. Moreover,it is even less clear how stochasticmechanics canbe extended to therelativistic case andprovideanythinglike aproper treatmentofphotons and their interaction with electrons.

Albert and Loewer (1989, 1991) introduce a novel 'many minds'hidden-variable theory which takes off from Everett's 'many worlds'interpretationof quantum mechanics. Certainly this isperhaps the mostnovelof any of the hidden-variable theories to be seriously discussed,but it is too new and too strange for me to yetcommenton seriously.Myown taste is toomuch for trying toaccountfor quantumphenomenain a much more austere and more standard physical setup. It would,above all, be useful to understand how Albert andLoewer propose toextend their theory to the standard relativistic phenomenaof photonsand electrons.

Finally,Icome to my own current favorite view. Ihave had aconversion late in life,away from stochastic mechanics and the use ofdiffusion,to theuse of random walks from which onecanderive wave

26 BARRY LOEWER

equations rather than diffusion equations. Given the importance bothhistorically andconceptuallyof wavephenomenainquantummechanicsit seems to me that this is essential for anyone like myself who wantsto begin from classical probability and as much classical physics aspossible. This particular approach is as yet not sufficiently developedto permit a really detailed evaluation of its strengths and weaknesses.My own work on itis being done inconjunction with ayoungBrazilianphysicist, J. Acacio de Barros, and our first paper is about to appear(Suppes and de Barros, in press),but this first qualitative analysis hasalready been modified along the following lines.

HereIwill just briefly sketch our approach. We take as a criticalcase diffraction andinterference phenomena,especially asexhibitedbyphotons. The four fundamental qualitative assumptions are these:

I.Photons have velocity c.11. Photons movein straight lines following classicalpaths inabsence

of charge.111. In thenear presenceofacharge,aphotonhas apositiveprobability

of being scatteredat some angle.IV. Photons are emitted uniformly in alldirections by a harmonically

oscillating source.Whentheseassumptions are spelledoutinmathematical detail,stan-

dard resultscan bederived,but now the wavepropertiesare themselvesderived as properties of mean probability distributions p(x,y,z,t),which, in the absence of charge,satisfy Huygens' wave equation and,with thepresenceof charge,satisfy Maxwell's equations.

One thing that has motivated us in constructing such a hidden-variable theoryatthis stage is thatweshouldtake accountofthe standardrelativistic behaviorofphotonsandelectrons. Itis toomuchatthis stagetoclaim that wecan carry theday with this theory,but we do think thatits development, even if conceptual modifications will be needed, isvery much worth pursuing. Notice that the ideas are very differentfrom Bohm's and very much driven from the standpoint of standardprobabilistic ideas. In my view there is much to be said for a theorythat assumes particles have trajectories but not necessarily the randomwalks ofclassical diffusion.

Determinism Is Transcendental. AgainIwant to confess a changein mind that has been underway for several years but now has firmly

27PROBABILITY ANDQUANTUMTHEORY

crystallized. It is a change from the view thatIheld in ProbabilisticMetaphysics (1984) andis thebelief that the choicebetween determin-ismor indeterminism is transcendental,inKant's sense of transcendingexperience. Barry rightly points out that in my former viewsIwouldhave been against the determinism of Bohm's theory and might wellhave leaped upon the necessity of objective chance, as he describesit,as a refutation of the theory. Now Idonot feel that way at all. Iam comfortable with all kinds of probability, includingobjective prob-ability, within a deterministic framework. Two kinds of fundamentalresults havepersuadedme,bothof whichIhaveexpoundedon severaloccasions, but have particularly summarized in a recent paper on thetranscendental character ofdeterminism (Suppes,1993).

It is in this respect that my ideas really do differ from Loewer'sconcept of separation of determinism and probability. Deterministicsystems do nothave to be content only with epistemic probabilities.A good example is the generation of arbitrary random sequences bythe motion of selected deterministic systems of three bodies operatingonly under gravitational forces. Idiscuss such an example in somedetailin Suppes (1987). The secondkind of theorems are those provedby DonaldOrnstein and hiscolleagues concerning the isomorphism ofindeterministic systems and mechanical systems of given complexity,for example,billiard tables withconvexobstacles. Theseisomorphismsor the randomness in three-body problems do not depend at all uponignorance of initial conditions or the many other kinds of epistemicprobabilities introducedeither intostatistical mechanics or inPoincare'smethod of arbitrary functions for analyzing roulette wheels or throwsof dice. These objective probabilities, orpropensities if you will, livenicely and happily within systems that are as deterministic as any weordinarily countas such.

Of course, this does notmean thatdeterminism is true. Itjustmeansthat we have a transcendental choice for a wide variety of phenomenabetween determinismor indeterminism. Bohm'shidden-variable theoryis an opportunity tomake a deterministic choice. My transcendentalmetaphysicalprejudices run the other way, andIsupposeIwould optfor indeterminism just because of its less exotic and more minimalistfeatures. But that is not important from a scientific standpoint. Whatis important is to provide a proper probabilistic formulation, whetherthe underlyingprobabilities arise from deterministic or indeterministic

28 BARRY LOEWER

processes,a matter we shallnot be able tochoosebetween on thebasisof empirical data.

So, inmy own view,thedemonof determinismin quantumorhumanaffairs andits twin demonofindeterminism havefinally beenput tobed,tolie inpeace forever covered by aneo-Kantian quilt thatcanneverbepenetratedby experience.

REFERENCES

Albert,D. andLoewer,B.: 1989, 'TwoNo-CollapseInterpretationsof QuantumTheo-ry',Nous,23, 169-186.

Albert,D. and Loewer, B.: 1991, 'The MeasurementProblem: Some "Solutions'",Synthese,86, 87-98.

Nelson,E.: 1967, DynamicalTheories ofBrownianMotion,Princeton,NJ: PrincetonUniversity Press.

Nelson,E.: 1990, QuantumFluctuations,Princeton,NJ:PrincetonUniversityPress.Suppes,P.: 1984,ProbabilisticMetaphysics,Oxford:BasilBlackwell.Suppes,P.: 1987, 'PropensityRepresentationsofProbability',Erkenntnis,26,335-358.Suppes,P.: 1990, 'ProbabilisticCausalityinQuantumMechanics',JournalofStatistical

Planning andInference, 25,293-302.Suppes,P.: 1993, 'The TranscendentalCharacterofDeterminism',MidwesternStudies

inPhilosophyXVIII, 242-257.Suppes,P. and de Barros, A.: inpress, 'A Random Walk Approach toInterference',

InternationalJournalofTheoreticalPhysics.

ARTHUR FINE

SCHRODINGER'S VERSIONOFEPR,ANDITSPROBLEMS*

[TJhere is no probabilisticcausality in quan-tum mechanics, for the essence of causality isto relatebehavior at one time to behavioratanothertime(Suppes, 1990).

ABSTRACT. Schrodinger'sversionofEPR involvesmeasuringadifferentobservableon each of the two component systems and using the strictquantum correlations toinfer valuesof theunmeasuredobservables. This procedure assigns values to all fourobservables. We explainhow a recent resultby Peres shows thatthis cannot bedonewithoutviolatingtheproduct rule forcommutingobservables. Weconnect this,briefly,withspeculations aboutrestoringsome sort of causality in thequantum theory.

0. PRELIMINARYREMARKS

Canbehavior at different times be related? As Suppes notes, the quan-tum theory calls that into question, even for something so simple asthe location of a systemat two different times. Position operators fordifferent times do not commute. There are no joint distributions forlocations at different times. Still,can we not somehow hold the loca-tions together,at least in thought, soas tocompare them? Conventionalwisdomanswers, 'No.This essay describes the fate of a tempting wayof trying to getbeyondthis conventional wisdom.

1. SCHRODINGER'S VERSIONOF EPR

Almost immediately after the publication of the EPR paper inMay of1935 (Einsteinet al,1935),Schrodingerbegansoliciting reactions toitfrom many of the leadingquantumtheorists. He shared hisamusementover their sometimes contradictory responses withEinstein, writing tohim on July 13,[I]t worksas wellas apikeinagoldfishpond andhas stirredeveryoneup....Itis asifoneperson said, "It is bittercoldin Chicago;" andanotheranswered, "That is fallacy,it is veryhot inFlorida."(QuotedinFine,1986, p.74.)

29

P. Humphreys (cd.),Patrick Suppes: Scientific Philosopher, Vol. 2, 29-43.© 1994 Kluwer AcademicPublishers. Printedin theNetherlands.

30 ARTHUR FINE

During thatsummerSchrodingerhimself wrotetwopapers(Schrodinger,1935a,b) inspiredbyEPR whereheanalyzedandrefined theanalysisofwhat he calledquantum 'entanglement' (Verschrankung),and he laterwrotea third paper onthe same theme (Schrodinger, 1936). Inboth ofthese 1935 discussions (the substance of his (1935a) analysis appearsagainin (1935b),Sections 12and 13)Schrodingernotes that in theEPRsituation we could determine simultaneous positions andmomentaforeach of the entangledsystems, "one bydirect observation,the otherbyinference from an observation on the other system" (1935a, p. 561).That is, if we measure position on one system and momentum on theother then we can use the quantum correlations to infer, respectively,momentumonthe one systemand position onthe other,soas to assignvalues toall four observables. Irefer to this variantof EPR, involvingmeasurementsonboth systemsand inferences across them toyield val-ues for allfour observables,asSchrodinger'sversion. (SeealsoPopper,1959.)

Using the analogyof a student in an examination who respondscor-rectly toany questionposed,Schrodingerconcludes that inhis versionofEPR allfour observables really dohave definite simultaneous values,just as we would naturally assume our student to know all the answers.This only deepens the puzzlesof EPR, however,because these valuesdonot relatein theexpected ways. For instance, for anynumber b > 0,Schrodingernotes that theeigenvaluesof theoperator(p2/b+bx2) areodd integral multiples of Planck's constant h. But ifpx and xx were,respectively,the valuesofpi andxiononesystem,it issimply notpos-siblefor the valueof the correspondingexpression(p2/b+bx\) to beanodd integral multiple ofhfor everypositive b. Thus the valuesassignedto non-commuting operators cannot be combined in a way that corre-sponds to therelations between the operators themselves. Inparticular,the correspondence fails for combinations of sums andsquares.

The situation,Schrodinger argues, is even more paradoxical. For,despite this failure of correspondence,the valuesof functions definedonthe separatesystemsrelate justasif'theassignedmomentaandpositionsdid combine inthe expectedmanner. Thuslook at the operators

x=xi- x 2 and p=pi+p2,

where the subscripts refer to the two entangledsystems. Operators xand p commute, so they have a simultaneous ip in which they takeeigenvalues,say, x andp. Now consider the operators A = /(xi,pi)

SCHRODINGER'S VERSIONOFEPR 31

and B = f(x2 + %,P— P2), where /(",") is any 'nice' two-place

function. Schrodinger shows that (A — B)ip =0andhe concludes thatin the stateip the valueof Aisequal to the value ofB.Hence the valuesof these functions areconnected justasif'the valuepofp wereidenticalto the sum of the value ofpiplus the value ofp2,andsimilarly for thevaluex of x-although weknow that theseconnectionscannot actuallyobtain in general.

Let me draw these various elements together. (1) Schrodingerpro-poses avarianton theEPR situation wherewemeasure different observ-ables on each systemandinfer values for thecorrelated observables ontheother system. He concludes that allfour observables that enterintoEPR (the positions and momenta for both systems) can be assignedsimultaneous values. (2) He proves that the following rules relatingtheseassigned values cannotboth hold in general,

where the observables A and B do not commute. (3) He shows that,paradoxically,values of observables defined on separate systems,nev-ertheless,do relate as though the sumruledid hold. Indrawing this lastconclusionhe makes use of the following inference. In the case whereobservables A and B commute, infer from the fact that the value of(A — B) is zero, that the value of A is equal to the value ofB. Notethat this is,roughly, to apply the sumrule tocommuting observables.

Schrodinger's discussion may call tomind yonNeumann's famousno-hidden-variables proof (yon Neumann, 1955, 1V.2). The similari-ties are striking. In his no-go theorem, yon Neumann shows that thesumrule (fornon-commuting A and B) is inconsistent with the squarerule (which is the principle that, in yonNeumann's terms, the ensem-ble is dispersion-free). Schrodinger's example for proving (2) aboveis actually a simple (and improved) way of demonstrating exactly thesame result. In the July 13 letter to Einstein, written while he waspreparing these 1935 papers, Schrodinger calls Einstein's attention toyon Neumann's book, which he says he has been studying. More-over, Schrodinger's (1936) uses results onmixtures that yonNeumanndeveloped in the book in tandem with his discussion there of hiddenvariables.lItis,therefore,noaccident that inexploring the issuesraisedby EPR, Schrodinger makes use of these functional rules featured byyon Neumann. In trying to deepen the paradox of EPR, however,

val(A +B) = val(A) +val(B) (SUMRULE)val(A2) = [val(A)]2 (SQUARERULE)

32 ARTHURFINE

Schrodingergoes one step further andconsiders the implications of thesumrule applied only to commuting observables. He is on the vergehere of an important result, although one that casts into doubt his ownunderstandingof the Schrodinger version of EPR.

2. ALGEBRAIC CONSTRAINTS ONHIDDEN VARIABLES

Withhindsightitiseasy to see,bymeansofexampleslikeSchrodinger's,that theyonNeumannsumruleisan arbitrary andunreasonable require-ment toplace on the assignment of values to non-commuting observ-ables. Itmight, nevertheless,seemlike areasonable condition for com-muting observables,as Schrodinger apparently took it tobe. Resultsachieved independentlybyBell(1966)andKochenandSpecker(1967),however,can beused to show that if we restrict the possible values ofan observable to the spectrum of the correspondingoperator, then thereis no way to assign values to a certain finite set of pairwise commut-ing observables in satisfaction of the sum rule (Fine and Teller,1978,Proposition 4). Itisan easyconsequenceof this result onsums to provethat the followingproductrule for commuting observables A and B isalsoinconsistent.

(PRODUCTRULE)

For suppose that valuessatisfying thisrule arealready assignedaccord-ing to some function val(-) from the spectrum of operators. Thenwe can define a new assignment of values, VAL(-) by requiring thatVAL(X) = log2[val(2x)]. Since val(2x ) is in the spectrumof 2X byassumption, log2[val(2x)] will be in the spectrum of X as required.Moreover, VAL(-) satisfies the sum rule since the val(-) function isassumed to satisfy the productrule. Thus the inconsistencyof the prod-uct rule follows from the inconsistencyof the sumrule (FineandTeller,1978,Proposition5).

This simple demonstration of the inconsistency of the product rulepiggybacks on the considerably more complex proof of the Bell-Kochen-Speckerresult.2 Inaremarkable paper AsherPeres shows theinconsistencyof theproductruledirectly andmoreperspicuously (Peres,1990). The starting point for Peres' proof is a Schrodinger version oftheBohm spin-^ realizationof theEPRexperimentschematizedbelow,where the crs are the spin-component operators (or, ambiguously, the

val(A " B) = val(A) " val(B).


spin observables) defined on the two subsystems (System 1 and Sys-tem 2).

The state ip of the two-particle composite systemis the 'singlet state',which canbe written as .

where a maybeany direction (including x,y,or z)and where

cr^ = the spincomponent in direction aon System1

J +1 in state 0]*"(a)\ — 1 in state.0f(a)

and

cr2 = thespin component in direction aonSystem 2j+1 in state\ — 1 in state

In state -0 the total spin in any directionis zero; i.e.,if the spinis foundto be 'up' (+1) in direction aon one system it will be -down' (-1) indirection a on the other system, and vice versa. (Put otherwise, theproduct of the spin components on the separate systems in the samedirectionis always -1.) Suppose that weconsider aSchrodingerversionof the experiment and make a measurementof al

x on System 1 and(simultaneously) of <x 2 on System 2,finding,respectively,values aandb. We can then cross-infer values for cr2 and for <x*,to get the resultsas follows.

Since the only possible values for a and b are ±1, if we multiply themtogether theproduct (ab) will alsobe ±1, and its square (ab)2 = 1. So,

0=-frttt^ ® («)] "775^1 («) ® </>2»l

Observables: trlx, cr2y, <r2, a\

Values: a, 6, —a, —b

34 ARTHUR FINE

wehave

Hence,

However,byusingstandardrelationshipsbetween the variousspinoper-ators we can calculate the value of the product $<rx " cr2)(o~y

-2)] in asecond way.Firstof all,thespin operators in orthogonaldirections (like*x* and V above) on a single system (say, System 2) anti-commute;that is,

Also, spin in the z-direction is related to spin in the x-andby the equations

where i2 =— 1. Using [cr2x

" a 2) = —{cr 2 " cr2) we can rewrite theexpression for cr^,

Using the fact that (<r^ " cr2 ) = (cr2 " ay ) we can readily calculate[cr\ " cr2,) from theabove relations as follows.

val (<rx " <rl)(crl - <rl)=val (<r\ ■ (T2

y ) " val (<rly " cr2 )= val (<ri) " val (cr2) " val (<rj) " val (<r2 )=a- &"(-&) -(-a)=(ab)2.

(1) Val [(0"i-cr2)(crl.cr2)] =1

(a2x

■ a2y ) = -(a\ ■a2

x).

<T\ = Z(cr^ " (Ty )2 "/ 2 2\(Tz

= l((Tx" (Ty )

°\ = ~i(<rl " °i)

(al " a2z) = i(a\ ■a$ ■ (~i)(a2

y ■ a2x)

= ~(f)(a\ ■ a\)(a2y ■ a2

x)= (al-a2

y)(a'y -a2x).

SCHRODINGER'SVERSION OFEPR 35

Thus,reading from far left tofar right.

Now we know that for direction z,like anyother, the spin valuesonthe two systemsare opposite to oneanother, i.e., that

(Indeed, thesinglet stateip is an eigenstateof (alz " <x 2) with eigenvalue-1, so the only possible value for (cr* " cr2) is -1.)

So from (3), LHS (2) = -1. Since -1 1 (!), it follows that wecannotconsistentlyassign values as above.

Anexamination of thecalculations shows that theprinciples weusedto assign values are just these:(SPECTRUMRULE) The only possible values for an observable A of

a system in a state ip are the eigenvaluesof A that have non-zeroprobability in ip.

(PRODUCTRULE) IfAandB commute, then val(A >B) = val (A) "

val (B).Thus Peres provides a proof of the following no-go theorem for aSchrodinger version ofEPR:

Thereis noassignmentof exactvalues to the quantumobservables thatsatisfies the spectrum and theproduct rules.

3. DISCUSSION

Schrodinger's version of an EPR experiment yields a simultaneousassignment of values to all four observables, two by direct observa-tion and the other two by inference. Peres' proof shows that this isnot feasible provided, like Schrodinger, we alsosuppose that the valuesso assigned tocommuting operators follow the same algebraic rules asdo the associated operators. Following Bell (1966), Peres interpretsthis as showing the inconsistency of the assumption that the result ofameasurementof an observabledependsonly on the observable itself,together with the stateof the measured system. Thus Peres interpretsthe proof as demonstrating contextualism,in the sense that measured

(2) val(alz " cr2 ) = val [(alx ■ a2y)(a\ ■ a2

x ) .According to (1),RHS (2) =+1.

(3) val(alz " cr2) = val(alz) ■ val (cr2) = -1.

36 ARTHUR FINE

values do depend on the choice of other simultaneous measurements(maybeeven non-local ones).

While a number of investigators have found some sort of contextu-alism attractive when interpreting measurementresults in the quantumtheory, it is important to recognize that contextualism does not actual-ly follow from the no-go-theorem. All that follows is that the valuesassigned in Schrodinger's version of EPR fail the product rule. Thatrule, like the sumrule and other algebraic constraints on assigned val-ues,is not containedin the quantum theoryproper. Forquantum theoryhas nothing to say about interpolated, unmeasured values. The theo-ry,at least as it is usually understood,only concerns relations betweenmeasured values. For example, consider the joint probability formu-la for commuting observables A, B, their product (A " B); and theirrespective values x,y,andz in any state ip:

This assertion, whichholds for all ip,may seem to say that the value ofaproduct cannotbe different from theproductof the values. Soit does,provided that by 'value' we mean 'simultaneously measured value.More precisely, the rule is that (withprobability 1) in a simultaneousmeasurementofall three observables A,B,and(A " B)ona systeminstateip, themeasured value of theproduct willalways be the productofthe simultaneously measured values of the factors. This is certainly aprincipleof thequantum theory,but itis not the productruleused intheno-go theorems. In thePeresproof above,for example, the rule appliestoproducts like (ay

" cr2 ) where the values assigned to the factors areinferred andunmeasured. There is nothing in the quantum theory itselfthat covers suchcases. So what the Peresproof and the other algebraicno-go theorems show is that thequantumrules for simultaneously mea-sured values cannotbe extended consistently to unmeasured,assignedvalues. That is important toknow for, like the earlier yon Neumannand Schrodingerproofs, it rules out certain extensions of the quantumtheory. It is alsoimportant to realize,however, that (pace Bell,Peres,and others) theseno-go theorems do not single outanyparticular inter-pretationofmeasured valuesin theunextendedquantum theory,neithercontextual nornon-contextual.

In his letter to Einstein of August 19, 1935, Schrodinger remarksthat toviolate whatIhave calledabove the sum andsquare rules wouldbe "altering theconnections with the concepts of ordinary mechanics"

P^(A =x,B=y,(A-B) =z)= 0, if z^x-y.


(Fine, 1986, p.79). Clearly Schrodinger does not regard this as desir-able,evenfor sums ofnon-commuting operators. Inthese terms,Peres'proof shows that it is possible to assign values in a Schrodinger ver-sion of EPR only by making an even less desirable alteration of thoseconnections.

4. BACKTO BASICS

A fundamental question about the quantum theory,perhaps the funda-mental question, is how to understand superpositions. If the state ofthe system is not an eigenstate of an observable (nor aproper mixtureof such states), does the observable have a value in that state? Theacceptedanswer to thisquestionis 'No. Thatanswer isbased onacon-sequentialistanalysis. That is,onetries toextend thequantum theorybyassigningvalues in superposed states and one shows that the proposedextension leads to difficulties. Perhaps the best knownanalyses of thistype are Bohr's repeatedtreatmentsof the double slit experiment. TheBell theorem toocanbe looked at as an analysis of this sort. There wesuppose definite premeasurement values for a small set of observables,values that satisfy further constraints, including a principle of locality,and we show that the quantumstatistics rule this out.3 Similarly, thePeres theoremand theother algebraicno-go results alsopoint to aneg-ativeanswer. Butall theseresults candois topoint,andnot logically tocompel. For everyproof and every experimentaldemonstration makesuse of assumptions beyond the assignment of values. So in the endwe are faced with a choice of whether to give up the assumption ofdefinite values,or some others. These are precisely the circumstancethatcall for goodhuman judgment(Duhem'sbon sens) sinceneither allour theory norall our practicecan determine which choices to make.

Finally, to return to the initial theme, what then of relating behav-ior over time as required for causality? To do so involves programs(like Bohm's, or perhaps a stochastic mechanics) that assign simulta-neous values to non-commuting observables, thus allowing values insuperposed states. In thisconnection Suppes writes,"Butall the possi-bilities are notlost. If [the constructionof a 'non-Markovian stochasticmechanics'] can be carried through,probabilistic causality is restoredforquantum phenomena" (Suppes, 1990,p. 301). These programsruncounterto the direction of conventional wisdom,which is the direction

38 ARTHUR FINE

indicated by the no-go theorems. We will have to march the oppositeway if we insist on causality in the quantum theory. As just discussed,it isnot clear whether it wouldbe wise toperform such acountermarch.More fundamentally, it is not clear whether judgment that insists oncausality in thequantum theory,even aprobabilistic causality, is goodjudgment.

DepartmentofPhilosophy,Northwestern University,Evanston,IL 60208, U.S.A.

NOTES

* This essay was inspiredby a comment of Patrick Suppes. In the spring of 1980 Icirculateda series of notes in oneof whichIproved thatif there were a factorizable,stochastichidden-variablestheory, there wouldalsobeadeterministicone.Respondingto that note, Suppes remarked thatIseemed to be a 'crypto-determinist'. IneverunderstoodPat'scomment,butIthink this littleessay is areflectionon it.1 Thereis amoredetaileddiscussion of the Schrodinger-EinsteincorrespondenceoverEPRin (Fine, 1986,Ch. 5).2 See (Peres, 1991) for elegant and relativelysimple proofs of the Kochen-Speckertheorem. Inhis introductionPeres remarksthat theBellproofofthis resultinvolves acontinuum of vector directions. While this is true ofBell'sexposition,his actualproofonly requires finitely many directions,as shown in(Fine andTeller,1978).3 Indeed(Fine, 1974) showshow to interpret theBell theoremas a directproofof theinconsistency of theproduct andspectrum rules.

REFERENCES

Bell,J. S.: 1966, 'On theProblemofHiddenVariablesinQuantumMechanics',ReviewsofModern Physics, 38,447-452.

Einstein,A.,Podolsky,8., andRosen,N.: 1935, 'CanQuantumMechanicalDescriptionofPhysical RealityBeConsideredComplete?',PhysicalReview, 47,777-780.

Fine, A.: 1974, 'On theCompleteness ofQuantum Theory',Synthese, 29,257-289.Fine,A.: 1986, The Shaky Game:Einstein,RealismandtheQuantumTheory,Chicago:

University of Chicago Press.Fine,A.andTeller,P.: 1978, 'AlgebraicConstraintsonHiddenVariables',Foundations

ofPhysics, 8, 629-636.Kochen,S. and Specker,E.P.: 1967, 'The ProblemofHidden Variables inQuantum

Mechanics',JournalofMathematics andMechanics,17, 59-87.Peres,A.: 1990, 'IncompatibleResultsof QuantumMeasurements', Physics LettersA,

151, 107-108.

39SCHRODINGER'S VERSIONOF EPR

Peres, A.: 1991, 'Two Simple Proofs of the Kochen-Specker Theorem', JournalofPhysics A,24, L175-Ll7B.

Popper, X.: 1959, TheLogicofScientific Discovery, London: Hutchinson.Schrodinger, E.: 1935a, 'Discussionof ProbabilityRelations betweenSeparatedSys-

tems', Proceedingsofthe CambridgePhilosophicalSociety, 31,555-563.Schrodinger,E.: 1935b, 'Die gegenwartige Situation in der Quantenmechanik',Die

Naturwissenschaften,23, 807-812, 824-828, 844-849.Schrodinger,E.: 1936, 'ProbabilityRelationsbetweenSeparatedSystems',Proceedings

ofthe CambridgePhilosophicalSociety, 32,446-452.Suppes,P.:1990, 'ProbabilisticCausalityinQuantumMechanics',JournalofStatistical

Planning andInference,25,293-302.yonNeumann,J.: 1955,MathematicalFoundationsofQuantumMechanics, Princeton:

PrincetonUniversity Press.


Ihave learned a lot from Arthur Fine's various papers on quantummechanics and the presentoneis noexception.Ihaveprobably learnedeven more from our many conversations over the years. What seemsappropriatehereis toengageinyetanother conversation withArthur. Heprettywell throws down thechallenge toshow thatprobabilistic causal-ity should be taken seriously in a fully satisfactory theory of quantumphenomena. It is obvious enough that standard quantum mechanicsdoes not provide such causal analysis, and on this point Arthur andIagree.

It is important to mention one standard move that is often made.It is to claim that standard quantummechanics is causal because thesolutions of theSchrodingerequationaredeterministic,in the sense thatthe state at one time uniquelydetermines the state at another time,butthese statesdo notgive anything likea standard view of causalanalysisin physics, that is, an analysis in terms of forces acting on particlesand thereby determining individual trajectories. There is, it must beadmitted, something to this particular view, for as the Hamiltoniansused in the Schrodinger equation vary, so do the solutions, and theHamiltonians represent the waysin which the ideaof force is importedintoquantum mechanics via the Schrodingerequation. But still,all wegetare distributions and nota causal analysis of individual trajectories.

AsIhaveemphasizedinother commentsonpapersin these volumes,what is really missing in standard quantum mechanics is the conceptof a trajectory for a particle. It is around this concept that we would

40 ARTHUR FINE

expect to find the fulldevelopment ofcausalideas. This does notmeanthatevery aspectofa trajectory is givena causalanalysis. Itjust meansthat changes in trajectories away from uniform motion are explaineddirectly by the impact of forces. Ichoose the word impact deliberatelyfor one radical version of causal analysis inphysics,a Cartesian view,is that the only forces that reallyexist are very short-rangeones.

Arthur says at the end of his paper, in response to my remark aboutnon-Markovian stochastic mechanics beingapossible way ofprovidingaprobabilistic causalanalysis of quantumphenomena, "theseprogramsruncounter to the direction ofconventional wisdom, whichis the direc-tion indicated by the no-go theorems. We will have to march theopposite way if we insist on causality in the quantum theory." It isevident enough,however,that the consideration just ofno-go theoremswillnot in any sensesettle the issueof whether we canexpect tohaveaprobabilistic causal analysis of quantumphenomena. As Arthur headsthe final section of his paper,it is indeed a questionof back-to-basics.The fundamental considerations here go much more to the heart ofquantumphenomena than the no-go theorems by themselves. WhatIhavein mind especially are the elaborate methods devised in quantumelectrodynamics for studying the interactions ofelectrons and photons.Especially as conceptualized by Feynman, in general terms we have averynaturalprobabilistic causalanalysis. Evenif theprobabilitieshavea somewhat strange feature, the causality has a relatively intuitivelystraightforward feeling. This includes everything from path integralsto the derivation of propagators which furnish a version of trajectoryanalysis in quantumelectrodynamics. Iam not suggesting that allIwould like to see is to be found in quantum electrodynamics. Forexample, there are no assumptions of definite trajectories for photonsor electronsbut there areclearly assumptions about causal interactionsbetween photons andelectrons.

Thepoint thatInow want to make invarious detailed waysis thatthefurtherrelativistic developmentof the interactionbetween electrons andphotons,asubjectnothandledin anysatisfactory wayatallbyclassical,nonrelativistic quantum mechanics,inevitably and necessarily deals incausal concepts. Tosay this is not to say thata clear classicalconceptof trajectory is used,but rather that the classical, in fact preclassical,conceptof causalinteraction between objects isused repeatedly.

Tobeginlet me quote some sentencesfrom the abstract of afamous1949 paperby Feynman.


Electrodynamics is modifiedby alteringtheinteractionof electronsat shortdistances.Allmatrix elements are now finite, with the exceptionof those relating to problemsof vacuumpolarization. The latterare evaluatedin a mannersuggested byPauli andBethe, whichgives finiteresults for thesematrices also....The results thenagreewiththose of Schwinger. A complete, unambiguous, andpresumably consistent,methodisthereforeavailablefor thecalculationof allprocesses involvingelectronsandphotons(p. 769).

As another example that does not require a quotation because of itswidespreaduseinall detailed discussionsof quantumelectrodynamics,consider thetalk thatphotonsare absorbedor emittedin agivenprocessor in a given region. This talk of absorption or emission, and thedetailed analysis that accompanies it is inevitably causal in characterand is ordinarily recognized as such in informal discussions of theformalism. But still to give one particular reference,notethe way thatKallen (1972) describes apropagator function.In this waythe function ...actually gives a 'causal' descriptionof the collision. It isoftenreferredto in the literatureas the 'causal propagatorof thephoton',or simply asthe 'photonpropagator"(p. 106).

Againnotetheuseof theconceptofcollision,acausal conceptifeverthere was one in physics. Note this typical passage in the well-knownbookonnon-relativistic mechanics byLandau andLifshitz (1958)Born's formula...can beappliednotonly tocollisionsof two elementaryparticles,butalsotoan elasticcollisionbetween,say, anelectronandanatom, ifthepotentialenergyU(r) is suitably defined. The condition for the Bornapproximationto be applicableto such acollision requires that the velocityof theincidentelectron should be large incomparison with thoseof theatomic electrons(pp. 426-427).

This passage,asIhave already mentioned,occurs in a famous text-bookon classicalquantummechanics andis in the chapteronthe theoryof elastic collisions,which contains also a discussion of the generaltheory of scattering. Notice what is going on in this case and whyprobabilistic causality is more fundamental in the analysis of physicalphenomenaeven than the existence of detailed trajectories. Iin factlay down the challenge to Arthur that it is not even a part of anyone'sconventional wisdom to thinkin anyother thancausal termsabout thesecollisions,collisions of the kind that dominate the theory of scattering.Of course we can,as inclassicalmechanics,distinguishbetween elasticcollisions andthose whicharenot. Here thatdifference isnot important,although the mathematical analyses are quite different. What is quiteimportant for my point is that the thinking about interacting particlesis inevitablycausal incharacter. This does not mean that every aspect

42 ARTHUR FINE

of causality is brought over from classical physics. It does mean thatcausal concepts, the rock-bottom basics of collisions or interactions,cannotbe avoided.

A nice relativistic example of scattering and therefore of causalprocessesis photon-photonscattering. In this case the scattering ofonephotonby another is a quantumelectrodynamicalphenomenon whichdoes not occur inMaxwell's classical electrodynamicsdue to the factthat Maxwell's equations are linear. The typical diagrams for theseinteractions,as in the case of other fundamental particle interactions,are nothing ifnot causal in their conception and in physicists' waysof thinking about them. Of course, this is even more obvious in themoreaccessiblephenomenaofelectron-photoninteractions. Bothkindsof interaction exhibit well enough for my purposes natural notions ofprobabilistic causality.

Finally,inmy continuingdialogue withArthur about the foundationsof quantum mechanics,Iwant to throw back at him his conventionalwisdomandsay itisdatedandappliesonly to thefoundationsofclassicalquantummechanics,and not even there in its extensions to scattering.Probabilistic causality is alive and well in the actual physics used toanalyze quantumprocesses of interaction,andespecially inFeynman'sparticleconceptionofquantumelectrodynamics.

Itis obvious that in this discussionIhavegone wellbeyondclassicalquantummechanics in its standard formulation,becauseIthink this isthe direction in which the philosophy of quantum mechanics shouldmove. Any discussion of basics must look at the wider context ofquantum electrodynamics and quantum field theory.Ido not mean tosuggest also that my views are entirely accepted by everyone. Iamjust making the case that the robust discussion of causal processes isaliveand wellincontemporaryphysicsandisuneliminable from almostanyone'sdiscourseabout thesematters.Itisundoubtedlydeepintuitionsabout these matters, arising long ago in preclassical physics, that ledto Feynman's valiant attempts to develop a space-time approach andtherebyhispath-integral approach to quantummechanics. Thedefect isclassicalquantummechanics,not the wide-rangingintuitive conceptsofcollisions andinteractions usedbyphysicists. The theoryof thefuture,Iwouldpredict, willmove inthe directionof amoreexplicit treatmentofcausalprocessesandnotin thedirectionofclassicalquantummechanics.

SCHRODINGER'S VERSIONOF EPR 43

REFERENCES

Feynman, R.: 1949, 'Space-Time Approach to Quantum Electrodynamics',PhysicalReview,76, 749,769-789.

Kall6n,G.: 1972, QuantumElectrodynamics,trans,byC.K.IddingsandM.Mizushima,New York: Springer Verlag.

Landau,L.D.andLifshitz,E.M.: 1958, QuantumMechanics:Non-RelativisticTheory,Vol. 3,Courseof TheoreticalPhysics, trans, by J. B.Sykes andJ. S.Bell,Reading,MA: Addison-Wesley.

JULES VUILLEMIN

CLASSICAL FIELDMAGNITUDES

ABSTRACT. Difficulties in the measurementof physical quantities arisein classicalphysics as soon as classicalphysics has to deal with wavesin aconcrete sense. Theconcept of a monochromaticplane waveis completelyand precisely defined, but itremainsabstract. Concrete wavesas collective objects, on thecontrary, areburdenedwithspectralinequalities,characteristic ofclassicalfield magnitudes.

Quantummechanics defines physicalmagnitudesasoperators actingonavector,the vectorof stateof agiven system. Classical mechanics,forexampleLagrange'sanalytical mechanics,is completely foreign to thisconception. The evolution of a mechanism is completely determinedwhen the valuesofits generalizedcoordinates qi andof their derivativeswith respect to time q\ are given at any time. The state of such amechanism is nothing less than the assignation of these geometricalmagnitudes (distances and angles) and these kinematical magnitudes(generalizedvelocities).

Itis true to say thatquantummechanics hascompletely transformedboth of the concepts of physical magnitude and state. Nevertheless,we must not too hastily give quantum mechanics what belongs to theclassical theory of fields.

The proper concepts of physical magnitudes as operators and ofstates as vectors are relevant to such a theory. Quantum mechanicssteps in whenEinstein's andde Broglie'sequationsassociate propertiesof particles with properties of waves.

My aimhere is to analyze the conceptualchange of the concept ofphysical magnitude within the limits of classical physics. Ishall firstdescribe the main characteristics of the field magnitudes. This sketchwill then suggest some philosophical conclusions.

I

The celebrated representations of Nature by the atomsand their com-position in Greece and the less well-known analysis of the tides by

45

P.Humphreys (cd.), Patrick Suppes: Scientific Philosopher, Vol. 2, 45-57.© 1994 KluwerAcademic Publishers. Printedin the Netherlands.

46 JULESVUILLEMIN

Posidonius theStoic testify that ancient physicshadalready recognizedthe necessity of using two conflicting conceptualizations in order todescribe matter.

On the other hand reality is conceived as made of discontinuous,localized, exclusive bits which move by transportation according towell-defined trajectories. Physics has essentially to state the rules ofconservation thatgovernthecollisions oftheatomsand their aggregates.Collisions require laws ofconservation relative to physicalmagnitudeswhich may be locally defined. But the forces which finally yield theircollisions andsecureunity for a worldact atadistance. Newton showedhow to work with this hypothesisof separatedcentres ofgravity. WhenNewton's laws are rewritten in terms of gravitational fields, this intro-duction of the field conceptualization means that the atomist uses aformal trick rather than thathe makes concessions.

On the other hand there are genuine phenomena of propagationwithout transportation, where the magnitudes involved are continuous,defined at every point of space, and, therefore, everywhere present,and,moreover, intersect theirrespective paths without disturbing them.Because it is continuous, this conceptualization is more akin than itsrival to therequisites of geometry.This is thereason why the followersof Descartes and Huygens did not give ground despite the successesof Newtonianism inmechanics until their apparently final victory withYoung and Fresnel. Compare, for example, the laws of impact of onebody against another one with the law of thebeatof two sound wavesin the air. In the first case, we deduce the laws from the principle ofconservation of the quantity of motion, withoutbeingable to describewhat happens at the instant of impact. By contrast, given two soundsources with slightly different frequencies, we have only tomake analgebraic addition at each point in order to construct the figure of theinterference in time with allits particulars.

Geometry shows,however, thatalot ofsurprises maybe stored in acompletely intuitive construction. Letus review two of them under thehead of thecategoriesof causality andsubstance.

When from a law of conservation aphysicist predicts how twobil-liard balls will rebound, he simply equates two accounts before andafter the impact. Thereis succession,there is not causality. Predictionis still possible, but one shouldspeak ofglobaldeterminism rather thanof strict causal laws, since the balance of observablequantities beforeand after an event does not imply that the event itself be subjected to

CLASSICAL FIELDMAGNITUDES 47

a detailed and continuous computation. Hume's doubt is to the point.But add twowaves. Each development is entirely determined once theamplitude, the rate of temporal evolution (pulsation) and the rate ofspatialprogression (wave number) are fixed. And their superpositionobeys the same complete determination. We follow it as we followthem. Causality is here the product of algebra. The surprise comes,nevertheless,if we draw the physicalconsequences from the superpo-sition,namely that two vibrations, when added,mutually reinforce orextinguish themselves,according as theadditionis made at two troughsor at twocrests on theone hand,or at a troughand acrest onthe otherhand. The astonishing phenomena of interference are thus explainedaway without introducingKant's so-callednegativemagnitudes.1With-out forgetting thatmonochromatic wavesbelongto abstractphysics, nottoconcretephysics,wemightsay, figuratively, that,putting togethertwoilluminations may produce obscurity. In the same way, it will be seenthatall the disturbances bound with asuperposition of waves infinitelyextendedoverspace may be cancelled everywhereexceptover the tinyextensionof apacket'scentre.

As to thecategoryofsubstance,thesurprisecameoff in twoepisodes.First, in his Analytical Theory ofHeat, Fourier analyzed the diffusionof heat without taking sides with the upholders either of the caloricor of the kinetic theory. Whereas the physicistshad to understand themechanisms according to which columns of air are displaced by thepropagationof soundin order to state and tosolve the waveequationofsound, the stationary distribution of the temperatures that results fromthe diffusion of heat in a wall or in a ring is obtained thoughno mod-el in terms of fluid ormolecules is offered to support the distribution.Auguste Comte, the founder of positivism, well understood the impli-cations of Fourier's method: physicalmagnitudes can be measuredandknown, while the substances that theymake manifest are keptin com-plete ignorance. Therefore phenomenalmagnitudes,not substances arerelevant tomathematical physics.

The second episode is more telling still. In Fourier's perspective,physical magnitudes were abstracted from their substances,but theirsubstances, whatever they might be, were not denied a physical exis-tence. Onthe contrary,when themechanical theoryoflight,asconstruedbyFresnel (who assimilated the vibrations of light with the transverseelastic vibrations of solids and gave the ether, which was supposed tosupport them, theparadoxicalproperties of absolute incompressibility

48 JULES VUILLEMIN

and solidity) was forsaken and Maxwell's electromagnetic theory wasadopted, it was gradually realized that field magnitudes need no sub-stantial medium to which they should adhere. The last tie betweenmagnitudes andsubstances wasbroken.

Field magnitudes are superposable and may live an autonomousexistenceindependentofamaterialmedium. Theseunexpectedphysicalphenomenaresult from the geometricalconstraint ofcontinuity. Apureanalytical surprise, ifIdare say such, following Wittgenstein's ukase,still waitsforus,whenboundaryconditions areplacedupon the solutionsof the wave equation, asis requiredby thepossibility ofexperience,forexample,a plucked vibratingstring, avibrating membrane, a systemoftwo coupled pendula, or the evolution of aparticle in a potential well.Fourier's trigonometric seriesdecomposesanyperiodic function whichis regular enough into an infinite sum of harmonic functions affectedby suitablecoefficients that determine therelative participation of eachfunction in the whole. Twoanalyticalphenomenarelative tosuch seriesclaim our attention: the emergenceof eigenfunctions and eigenvaluesand thecouplingof the magnitudes.

Owing toboundary conditions,discontinuity, so to speak, emergesfrom continuity. When the vibrating string is fixed at two points, thespace partof the wave function,whichis an ordinary differential equa-tion,is constrained to vibrate atcertainnaturalmodes.2 The wave num-ber, and with it the frequency,is allowed to take as values only integralmultiples of a constant function of the period. These valuesare calledeigenvalues. To each of them there corresponds an eigenfunction. Theeigenfunctionshavetwo importantproperties. Theyareorthogonalandcomplete. This means that they define a generalized vector space ofan infinite number of dimensions and thusafford aconvenient basis fordevelopinganyregular enoughfunction thatsatisfies the sameboundaryconditions.

Understood as a theorem of generalized vector analysis, where theelements of the vector space are the real, continuous functions of areal variable defined over an interval,Fourier's decomposition has far-reachingconsequencesfor the conceptof physical magnitude. Inordi-nary geometry a length is defined by the coefficients which multiplyits projections upona Cartesian orthonormed basis. In the same waylet us suppose that the state of a system is characterized by a givenperiodic function of a continuous variable. Then, owing toFourier'sanalysis, the spectrum of theFourier coefficients oreigenvalues, if the

CLASSICAL FIELDMAGNITUDES 49

eigenfunctionshavebeenpreviously normalized,willdefine thepartsofaphysicalmagnitude. As the spectra ofeigenvaluesandeigenfunctionsare producedby the action of certainoperators upon the periodic func-tion,physicalmagnitudes become associated with operators. A simpleexample is given by theenergy theorem. The energy of a wave over aperiod isproportional to thesquareof itsamplitude over theperiod. Theoperator, or rather the functional - since the transformation is from avectorto anumber- actingupon the functionof time ishere the definiteintegral over the period. According to the theorem, the total energy ofa wave is simply the sum of the energies of allFourier's components.This is preciselyPythagoras's relation.

Fourier's analysismay beextendedtocomplex functions and tonon-periodic functions. Inthe first caseprovisions are made for getting realeigenvalues: theoperators must beHermitian. In the second case, con-tinuous spectra, integrals and Fourier transforms, respectively, replacediscrete spectra, series and Fourier components. The integrals whichoccur inFourier transforms are reciprocal functions of continuous vari-ables: the wave vector and the position (since we took the space formof the wave equation). Analytical considerations exhibit a classicalrelation between the widths of two Fourier transforms. Theirproducthas an inferior bound.

Let AA; be the width of the scale of undulation, the inverse of thewavelength,and letAxbe thecharacteristic heightor spreadof theone-dimensional position, i.e., the space extension containing the centreofthe wave packet. The function ip(x,o) is obtained by integrating itsFourier transform over k. If Ax is greater than l/Ak, the transformoscillates several times within the interval Ak and the integral overk takes a negligible value: there is destructive interference. If x =xo, Ax being less than 1/AA;, the function which is integrated doesnot practically oscillate and its integral over k takes an appreciablevalue; the wave packet centre, where the amplitude (x,0) is maximal,is situated at x = xq. Therefore, to form a wave packet with a limitedextension - to simplify let us consider an abstract wave packet byreplacing the concrete superposition of an infinity of plane waves by adiscrete interference of such waves,finite in number- we needseveralharmonic componentsof wavelengths, the range of which is the moreextended the more the wave packetis localized.3

The spatialspectralinequality couples the two physicalmagnitudes:wave vector and position. But since physical magnitudes have been

50 JULESVUILLEMIN

associated with operators, the question arises: what operators are asso-ciated with wave vectorsandpositions, and whatproperties ofoperatorsareresponsible for the couplingof thesemagnitudes? Therelevant oper-ators are shown, respectively, to involve a first derivative with respectto space and a multiplication by the space coordinate. Each of themis characterized by a spectrum of eigenfunctions and eigenvalues. Thebasisaffordedby the system of theeigenfunctionsofoneof themallowsthe development of the other. But coupled operators do not admit thesame eigenvectorsand do not commute. The magnitudes with whichtheyare associatedare therefore notsimultaneouslymeasurable with anaccuracy superior to afinite givenquantity.4

Analyzing the temporal form of the wave equation would producesimilar conclusions. A temporal spectral inequality should take theplaceof its spatialcounterpart. Insteadofarelation between the widthsof the waveand theposition vectors, therelation wouldbe between thewidthof thepulsation spectrumand the widthof thetemporalextension.

These limitations havenothing todo with quanta.They obtain with-in theboundsofclassicalphysicsandeven within thebounds of thepartofclassicalphysics thatprecisely deals with continuous magnitudesparexcellence. Specific laws rulethepropagationofcollectivephenomena,the form of whichmay beconfined while the superposed waves fill thewhole space. The interferences act constructively only within a thinvolume around the centre of the wave packet and destructively every-whereelse. As thegroup of waves runs towards either circle that limitsthe perturbation caused by a stone thrown in a pond, this form mayget a proper velocity, different from the phase velocity. The velocityof the maximum of the wave packet is not the mean phase veloci-ty, since the component waves in a dispersive medium have differentvelocities becauseof their different wavelengthsand the interferencesslowly change the determination of thismaximum. This group velocitydependson what spectralextensions the pulsation and the wave vectorof the wave packet have.5

Thereforeitisno wonder thatcommunication theory,which isentire-ly groundedon classicalprinciples, presentsus with the new conceptofphysicalmagnitudes associated with operators, when pulsations andgains are measured bylinear transmission of messages.6

The mutual relationbetween signaland transmission channels givesrise to the concept of impedance,a physicalmagnitude whichis gener-ally representedby an operatorandis measurable when the signal isan

51CLASSICAL FIELDMAGNITUDES

eigenfunctionof this operator, the resultof the measurementbeing theassociatedeigenvalue.7

II

Both of the basic conceptsof classicalphysics raisedifficulties that arebound up with conciliating discontinuity and continuity. A materialpoint means afinite quantity ofmatter,but without extensionand there-fore infinitely concentrated. In so far as a wave is infinitely extendedand everywheredefined by finite magnitudes, it is not burdened witha problem of singularity. The application of the laws of science isnowhere prohibited. Huygens' light waves, which progress throughthe void,or rather through the ether, with the same velocity may evenbe said to be unproblematic. The paradoxes that affect the classicalconceptofphysicalmagnitudesbeginwith Young'sorFresnel's waves,with dispersion.

Reading an equation - or rather into an equationis not an easytask. The solutions of the wave function are circular functions of thedifference between two products: of the wave vector into the spacecoordinate,andof the pulsation into the time coordinate. At the begin-ning thephysicistssupposedthat thepulsation whichis afunctionof thewave vectoralways expressesthe productof this vector with aconstantphase velocity. Abandoning this supposition led to important progressinphysical field theory during thenineteenth century.

This progress in a sense answers, or in another sense, questionsphilosophicalpositions which goback to Descartes andHuygenson theonehand,and to Kanton the other.

Eager toavoidNewton's singularitiesKant introduced,besides ordi-nary or extensive magnitudes corresponding to quantities and to theaxiomsof intuition,intended specific magnitudes,calledintensivemag-nitudes andcorrespondingtoqualities and to the anticipationsofpercep-tion. Whathehad inmindisexplainedin abookbyhim which hasbeen'undulyneglected'(exceptbyPatrick Suppes8),theMetaphysicalFoun-dationsofNaturalScience, where,in the division of scienceof motionbetween kinematics and mechanics,he inserts dynamics. Dynamicsrelies on the opposition of two forces- attraction and repulsion - thebalance of whichaccountsfor the size of the objectsin the universe. Letus remark that quantum mechanics will answer the same question by

52 JULES VUILLEMIN

putting forward Heisenberg'sinequality that results from the couplingof themagnitudes occurring in the wave equation, once itis interpretedin terms of quanta,and by postulating for identical particles a specificbehaviour which,in thecase of antisymmetry, obeys Fermi's principleofexclusion.

This remark and Kant's own weaknesses of construction advise usto replace Kant's so-called intensive magnitudes by the specifically'dynamical' concept of a wave, a substitution hinted by Kant himselfwith respect to colors and musical sounds.9 This move would giveKant's dynamics its proper scope. All the physicalmagnitudes shouldbe conceived of as field magnitudes, what remains of corpuscular ideashaving to be explained away as figures of speech. Kant often insistson the fact thatcontinuity is built into thepossibility of experience: anobject, he says, could not be known, were its properties not givenasphysical magnitudes. But it is precisely those properties that shouldnot meet the requirements of continuity imposed by space and timeas forms of possible experience that could not be given as physicalmagnitudes. The supreme transcendental principle goes so far as toidentify thepossibilityof experienceand thepossibilityof the objectofexperience.

Inother words, waves and their magnitudes do define anypossiblephenomenalreality.

In this view, when it is said that matter is opposed to radiation ascorpuscles are opposed to waves, this opposition results from a sheerappearance,since everyphysicaldescriptionmust fit in with the condi-tionsofpossibleexperience. Maybeatomistic hypotheseshaveaheuris-tic value. They are,however,deprived of ontological import. Phenom-ena are continuous. The question whether reality in itself is atomicor continuous,or whether theuniverse is finite or infinite bypasses thepower ofour knowledge andinvolves reasonin dialectical antinomies.

Suchaview wasgenerallyaccepteduntil thebeginningof this centu-ry. Duhem,for example,and withhim allthe followers ofenergeticism,thoughthey didnot professeither that space and time be the subjectiveforms of our intuition or that pure reason be affected with insupera-ble contradictions,agreed that physical theories should only 'save thephenomena' without speculatingupon the so-called realityof atoms.

Recalcitrant phenomena compelled the physicists to renounce thiskind ofphenomenalism, toadmit the wave-particle dualism within thefacts and therefore thepossibility ofexperience.Itisremarkable that the

CLASSICALHELD MAGNITUDES 53

revolution broughtaboutinournotionsofphysicalmagnitude and stateoriginated from the verydomain ofradiation where the classicaldevel-opments anddifficulties that havebeendescribed hadarisen. Quantummechanics may evenbe constructed frommerging the conceptsof func-tion and vector, as is requiredby Fourier's generalizedanalysis,on theconditionthatpulsationand wave vectorbe respectivelyassociated withenergyand momentum throughPlanck's constant.

Butletus staywithin theboundsofclassicalphysicsandask about themeaning of classical field magnitudes. It is here that adistinction usedbyDescartesandby all theCartesians,Malebranche,Huygens,Spinozaand Leibniz will be relevant,namely the distinction between abstractphysics and concretephysics. Its best and ideal illustration (since themathematical andphysical developmentscame later) isprobably givenby analyzing the conceptof wave.

A simple wave is an elementary harmonic wave, i.e.,a sinusoidalfunctionwhichmovesallinonepiece with aphase velocity thatdependsonthematerialmedium. Whatdefines theabstract individuality of theseelementary waves are the quantities by which the different periodici-ties are determined (period and pulsation for time, wavelength andwavenumber for space). A successive section shows equal amplitudeseach time the variable t has growna period and an instantaneous sec-tion shows equal amplitudes each time the space variable has grown awavelength. Such waves belong to abstract physics. According to thetemporal spectral inequality, the temporal extension of the sinusoidalwave is infinite with a unique pulsation. But a monochromatic planewave has no physical reality. Only superpositions of harmonic wavesmay be realized.

Concretephysics thendeals withcomposedor collective objects.Butwhileabstractindividual objectswere welldefined,difficulties andpara-doxes arise when concrete individuals,that is groups,areconsidered.10

Monochromatic waves were well defined in so far as oneprecise valueof the pulsation and of the wave number were associated with them.On the other hand they are denied any localization in time and space.The abstract character of such simple individuals directly results fromthespectral inequalities. Ithasno counterpartin classical mechanics ofcorpuscules wherea centreofgravity has atany timea well-determinedlocalization and momentum. For field magnitudes only groups maybe localized,butowing to the spectral inequalities, localization is cou-pled with complementary magnitudes. A system relevant to classical

54 JULESVUILLEMIN

mechanics does notnecessarily vibrate with a definite wavelength11 orpulsation. Therefore,in so far as continuity prevails, physical realityonly belongsto systems whichare not susceptible ofcomplete determi-nation,though their internal evolution follows a strictly causallaw.

ACKNOWLEDGMENT

Ithank YvanCuche andVincent Voirol for their criticisms.

Collegede France,11, PlaceMarcelin Berthelot,F-75231 Paris Cedexos,France

NOTES

1 VersuchdenBegriffdernegativenGrbfien indieWeltweisheiteinzufuhren,byJohannJacobKanter, Konigsberg, 1763.2 See, for example,Margenau,H. and Murphy, G. M.: 1956, The Mathematics ofPhysicsandChemistry, 2ndcd.,Princeton: Van Nostrand,Ch. 7-8.3 Cohen-Tannoudji, C, Div,8., andLaloe,F: 1973, Mecanique quantique, Paris:Hermann,I,pp.24-27; 11, AppendiceI,pp. 1446-1454.4 Inradio-electricity the inequality relationis written:

r " Ay ~ 1

where r is the time intervalseparating two successiveannulments of the signal, andAi/ is the extensionof the signal's spectrum (Blaquiere,A.: 1960, Calculmatriciel,Paris: Hachette,11, pp. 123, 124). It is rare, in classicalphysics, thata magnitudeisrepresentedby an operator. See, however, for the impedanceof a circuit: Blaquiere(1960, p. 104).5 Feynman, R., Leighton,R.H., and Sands, M.H.: 1963, The Feynman Lectures onPhysics,Reading,Mass.: Addison-Wesley, I,pp. 48^14.6 Blaquiere(1960,11, pp. 102-124). When asignal is linearly transmitted, let us callgain theratiobetweenthe amplitudesof theinput andoutputofFourier's terms havingthe same impulse. For imperfect filters the gainis a function of the impulse. For a(theoretical) filterthatshould letonly threeimpulsespass, if theoperatoris thesecondderivativewith respect to time, itapplies tothe input function

fi(t) =ao+y^ancos nu>t+N^bn sinnutn=l n=l

55CLASSICALFIELDMAGNITUDES

togive theoutput function(d2fi(t)/dt2 = fo(t): correspondingto the general termofrank n:

wehave

(2 x 3) +1terms

7Blaquiere (1960,pp.135-138); Feynman,Leighton,andSands (1963, 11, pp.22-21.8 Suppes, P.: 1970, A ProbabilisticTheory ofCausality, Amsterdam: North-Holland,pp.86-89.9 Vuillemin, J.: unpublished,Die MbglichkeitderErfahrung imLichtder zeitgenos-sischenPhysik,Trier (lecture).10 Vuillemin, J.: 1990, 'Physique pantheiste et d&erministe: Spinoza et Huygens',StudiaSpinozana,6, 231-250.11 Feynman, Leighton,andSands (1963,1,49^15).

COMMENTSBY PATRICK SUPPES

Formany yearsIhave benefitted fromconversations with Jules Vuille-min about the history of philosophy and the history of science. Thepoints about which he has instructed and corrected me are many andlarge in number. He is equally at home in discussing the physics ofthe Stoics or Aristotle, on the one hand, and on the other hand, thegreatdevelopmentsofmathematical physics from the seventeenth to thenineteenth centuries. The present article lays out in someconsiderabledetailthe wayin whichthebasic conceptualandmathematical apparatusof quantum mechanics was first developed in the classical theory ofoptics and electromagnetic fields, these developments in themselvesdependingupon the theory of wavesdevelopedbyHuygens andothers.Without knowledgeof this earlier history of classical physics it is tooeasy to think thatmuchofclassicalquantummechanics ismore originalin formulation than it is.

tn =ancosnut+bn sinnut

d 2 2 2 2? 2 2— -tn =— nv an cosnut —n v bn smnut =—n v to-at1

With n= 3

fa(t) =oxao — v (aicosut —b\ sinut —4u (a2cos 2ut+62 sin2ut) — 9u2(aicos2>ut +63 sin3ut).

56 JULES VUILLEMIN

However, the really important point brought out by Jules is that inmajor respects it isnotclassicalmechanics but classicalelectromagnetictheory, and more generally the theory of wave phenomena, that is thereal intellectual predecessorof quantum mechanics. This applies to thetheory of operators,as well as the fact thatif two operators in quantummechanics, such asposition and momentum, areFourier transforms ofeach other, then an inequality of the form of Heisenberg's uncertaintyprinciple must hold,but initially just in the domain of the classicaltheory of wave phenomena,especiallyelectromagnetic waves.

A point that Vuillemin does not emphasize, but Ithink he wouldagreewith,is thatquantummechanics resembles electromagnetic theorymore than classical mechanics in other respects. The concept of aparticle's unique trajectory isabandoned. Itis wellknown that it is notpossible to compute in classical quantum mechanics even the simpleautocorrelation of the positions of aparticle at two different times. AsIhave argued recently inmost explicit form, but also earlier as well,quantummechanics offers from acausal standpoint only a weak theoryof the mean probability distributions of particles, and nothing like afull probabilistic theory of sample paths (Suppes,1990). In classicalelectromagnetic theory, as Jules points out, there is no concept of adistinctparticle path, but only of fields that extend continuously overall of space, in the general case, and over a limited but continuousdomain in morerestricted cases. Itis of greatimportance, however, toinsist on the point that physicists do not really think about electrons,protons,and other particles just in terms of thesemean distributions orin terms of theapparatusgiven tothembyclassicalquantummechanics.It is completely natural and indeed necessary for the serious detaileddiscussion of experiments and of almost every kind of microscopicphysicalphenomenon to think of electrons,protons, andother particlesas moving inspace with trajectories. The concepts behind the exoticapparatusof high-energyphysics are testimony to this. The purpose oflinear accelerators is justthat, toaccelerateparticles to veryhigh speedsbut it wouldnot make any sense to talk of suchacceleration without theparticles having trajectories.

There are two remarks Iwant to make about Vuillemin's analy-sis of classical field magnitudes. The first is that the one thing thatquantummechanics did add to the classical theory is theprobabilisticinterpretation of wave phenomena. This means that what was classi-cally interpretedas wavedistributions,becameunder this interpretation

CLASSICALFIELDMAGNITUDES 57

probability distributions of particles. As far asIknow thisprobabilisticinterpretation does not occur anywhere in nineteenth-century physicsandissomething genuinelynew aspartofquantummechanics. What isironical is that it is the interpretation, not the mathematical formalism,that is the real contribution of quantum mechanics, even though theprobabilistic theory of quantum mechanics is a very weak one, in thesenseas alreadymentioned ofbeingonly a theoryofmean distributions.

The second remark is a historical one. Jules remarks that Kant'streatise The Metaphysical Foundations ofNatural Science has beenundulyneglectedby most scholars,exceptby afew personslike myself.But this is far too modest on his part. Iconsider myself an amateurscholar of this treatise compared to Vuillemin. He has fortunatelycaught several errors of mine before theyhaveappearedinprint; aboveallImention thathe himself published the most detailed work (1955)Iknow of on theMetaphysicalFoundations.

REFERENCES

Suppes,P.: 1990, 'ProbabilisticCausalityinQuantumMechanics', JournalofStatisticalPlanning andInference,25,293-302.

Vuillemin,J.: 1955 (2nd cd. 1987),PhysiqueetMetaphysiqueKantiennes,Paris: PressUniversitaires de France.

BRENTMUNDY

QUANTITY,REPRESENTATIONANDGEOMETRY

ABSTRACT. Suppeshas developedacharacteristicandconceptuallyunifiedapproachto the theoriesof physicalgeometry andphysicalquantity, basedon ideas andformaltechniques which he has also helped to develop within the representationaltheoryofmeasurement (RTM). These interrelatedideas, techniques and results are directlyrelevant to the philosophyof physics,butremainmostlyunknown withinthatcontext.HereIaimto explaintheseideasand theirrelevance ina clearandsystematic manner,andalso to extendthis general researchprograminseveral ways.

IncontrasttoSuppesIstresstheneedforaunified,realistandrevisionistapproachtothetheoryofphysicalquantity,as opposedtohis piecemeal,empiricistandconservativeapproach.Ialsostress theneedforintrinsicformalizationofwholequantitative theories,as opposed to his hybrid extrinsic set-theoreticapproach. As new contributions tothis ongoing program of researchIoutline anRTM-style representation theoryfordifferential geometry,leading toaxiomaticRiemanniangeometry. Thisalsoinvolvesanextensionof theRTM theoriesofuniquenessandmeaningfulness tosituationsinvolvingnonaturalsymmetry oruniquenessgroup.Ialsodescribesomenew typesofrelationistgeometrical theory,developedusing this sameRTM-based technicalapparatus.

1. THEORY OFPHYSICALQUANTITY

Suppes' work on geometryand space-time is closely linked tohis sys-tematic work on representational theory of measurement(RTM), inways explainedbelow. This work is summarized in the three-volumeFoundations ofMeasurement (Krantz etal,1971;Suppes et al,1989;Luce et al, 1990) co-authored with Krantz, Luce and Tversky, citedbelow as FM-1,etc.

In contemporary philosophy of physical science surprisingly littleattention is devoted to the generaltheory of physicalquantities,physi-cal measurement,and the methodology of quantitative physical theory.Historically these topics were much moreactively discussed: recall forexampletheEudoxian theoryofquantity,orGalileo's turn from the qual-itative method of Aristotle to thequantitative method of Archimedes,and theassociated philosophical theory ofprimary and secondaryqual-ities.

59

P.Humphreys (cd.), Patrick Suppes: Scientific Philosopher, Vol. 2, 59-102.© 1994KluwerAcademic Publishers. Printed in the Netherlands.

60 BRENT MUNDY

Theseolder debates weredrivenbysubstantivemethodologicalprob-lems innatural science. However, thequantitative methods of Galileo,HuygensandKepler were soon joined with the differential and integralcalculus ofNewtonandLeibniz andtheanalyticgeometryofFermat andDescartes toproduce whatIwill call the 18th centurymethodologicalconsensus,epitomized by the analytical mechanics of Euler, LaplaceandLagrange. Physical quantities werehenceforth to be representedbyreal-number variables,physical laws expressedby ordinary orpartialdifferential equations governing those variables,and physical science(as stated in the famous dictum of Kelvin) was to expressallphysicalfacts and laws using this quantitative conceptualapparatus. This con-ceptual apparatus has dominated subsequentphysical science,allayingthe methodological doubts which had motivated earlier philosophicalcriticism.

The modern situation, however, has changed again. On the onehand developments inmathematics and logichaveextended traditionalconceptionsof quantity inmany ways. The riseof abstract algebrahasshownthat 'quantity' isnot justone thing: thereare many abstractnum-ber systems and quantitative structures, just as there are many abstractgeometries. The question which of these possible quantitative formsfigures in the physical world is therefore ultimately an empirical one,just as the corresponding question for geometrical forms was finallyseen tobe. Moreover, the traditional real-number continuum itselfhasbeen found to conceal deep logical problems previously unsuspected(unsolvability of the continuum problem within ZF set theory); con-tradictions in the foundations of set theory itself)- Finally,of course,modern developments in general formal logic give new and far morepowerful tools for attacking suchproblemsof logical analysis.

On the other hand, modern theoretical physics has also movedsharply away from the18th centuryconsensus in quantitative method-ology. Both relativity and quantumtheory involve radical shifts in theunderlying conceptions of physical quantity and of quantitative laws(indefinite metric and variable curvature of Riemannian space-time;infinite-dimensional state spaces and Hermitian operators of quantumtheory). In this modern contextit seems clear that traditional questionsabout the nature, scope and content of quantitative methods in physi-cal science must arise again in new and deeper forms, and should beaddressed within the contextof a new general theory of the nature ofphysicalquantities and quantitative theories,taking account of modern

QUANTITYREPRESENTATION ANDGEOMETRY 61

developments in logic, mathematics and physics. This is an importanttask for philosophers of physicalscience.

2. REPRESENTATIONALTHEORY OF MEASUREMENT

Paradoxically, physical measurementhas recently been studied moreby psychologists than by philosophers of physics. The explanation(FM-1, p.xvii) is of course that the social sciences still face real prob-lems concerning the scopeand significance of quantitative methods,asthe physical sciences did before Galileo but not after Laplace, so thatthere the practical motivation for foundational analysis is still present.Although the main goal ofRTMis quantitative social science,the gen-eral framework has been developed in part through detailed analysisof physical measurement, yielding a body of work which essentiallybelongs to the philosophyof physics,thoughproducedand readmainlybypsychologists. (Forexample, thechapter 'Dimensional AnalysisandNumericalLaws' ofFM-1is themostcomprehensive existing treatmentof this topic). For a philosopher of physics it is abit disconcerting tofindpsychologistsdiscussing topics suchas velocity additionin specialrelativity, and there issome temptation to dismiss this work as amateur-ish dabbling in technical mattersbetter left to specialists. But in factthe RTM literature contains a more sophisticated treatment of physi-cal quantity and measurementthan anything in the current literature ofphilosophy of physics.

Representationtheory iscentral for theoryofquantitybecause itpro-vides the link between theartificialnumerical apparatusof the 18thcen-tury analyticalmethod(thenumerical 'variables' postulatedasprimitiveinallmodernquantitative theories) andthe underlyingintrinsic physicalreality. In the analytical method everything is already numerical: nophysicalphenomenoncan be discussedunless it isalready expressedinthe numerical language of classical mathematical analysis. Since theanalytical method assumes this translation to be given, i.e. assumesphysical facts to somehow already be representedby numerical vari-ables,itcannot tellus what about the world makes thatpossible,orhowthe world appears whennot so represented.

This is the first and most important theme inRTM: logical analysisof theprocess ofnumerical representation,andof the conditions underwhichit is possible oruseful. Todo this,RTMfirst gives a character-ization of the subject matterin terms of intrinsic relations among the

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objects, not dependent on or defined using any numerical representa-tion. (RTMcalls these empirical relations,butIreject the implicationof direct observability; rather they must simply be intrinsic to thephys-ical subject-matter.) Then a formal characterization is given of therepresentational conditions C which should be satisfied by a scale orrepresentingmap / from such a physicalor empirical relational struc-ture Stoacorrespondingnumerical structureM.Finally,asetofaxiomsA constraining the intrinsic structure of S are stated, from which onecanderive the representation theorem that for any S satisfying axiomsA there willexistafunction/ fromS toMsatisfyingconditions C. Thepractice of numerical representation of elements s of S by numericalscale values f(s) in Mis thenexplained and justifiedby theassertionthat the underlyingphysicalstructure S satisfies theintrinsic axioms A.Fromaphysicalviewpoint, therefore,RTMaims touncover andexpressthe intrinsic pre-numerical physicalfacts and laws which underlie thetraditional 18thcenturypracticeofnumerical representation, and whichcannotbeexpressedwithin thatframework becauseitpresupposesthem.

However,numerical representation is alsoarbitraryin various ways:thereare different possible scalesofmeasurementfor the same quantity.This wasrecognized within the analytical method, anddescribed usinga numerical group G of scale transformations: elements g of G gen-erate alternative scales as /'= go /. This is the older Klein-Stevensapproach torepresentation theory,inwhicheach scale or representationis postulated to have a numerical transformation group G associatedwith it, thecompositional actionof whichdefines the range of possiblealternative representations. This topic of uniquenessof representationis the second major theme in RTM. The group G is ultimately iden-tified as the structural automorphism group of M,and one proves theuniqueness theorem thatifstructure S satisfies axioms A then themaps/ satisfying conditions C from S toMare 'uniqueup to' compositionwith an element of G. This explains the role of thegroup G, which onthe Klein-Stevens approach was somewhat mysterious.

Finally, statements made usingnumerical scales (representationalpropositions) mayormay not be meaningful, i.e. have their truth inde-pendentofchoice of scaleorrepresentation. Asin the traditionalKlein-Stevens theory, the RTMconditions ofmeaningfulness is invariance oftruth-value under theactionofG ontherepresentationalpropositionsorsentences. However, in theRTMframework onecangiveamuchbetterexplanationand justification of this requirement, using the additional

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information about the group G and the intrinsic structure S given bythe RTMtheory of representation.This theoryofmeaningfulness is thethird major theme inRTM.1

Note that the same numerical transformation group G figures cen-trally in all three aspects of RTM, playing three different roles: assymmetry group of the representing space M; as uniqueness group ofthe representation, giving the range of possible alternative representa-tions ofSin M;andasmeaningfulnessgroup,expressingthe invariancecondition which defines the meaningful representationalpropositions.This pervasive role of asingle group Ghas led to a generalfeeling thattransformation groupsare central to representationalproblems,andhasmade it hard toapproach representationalproblems havingnoassociat-ed transformation groups.Iwill show in Section 7 that this feeling iswrong: the truly centralideasofrepresentationtheoryarequiteindepen-dentof transformation groups. When thosecentral ideas are formulatedin a sufficiently general way and the special conditions for group-basedrepresentationare recognized, the waybecomes clear for developmentofrepresentationtheoryin contextssuchas differential geometrywhereno symmetrygroups are present.

RTM laysout themethodological framework within whicha theoryof physical quantity may be developed, by showing how to mathe-matically ground extrinsic numerical representationalpropositions in anon-numerical structure of intrinsic physical facts. However, it fallsshortof actually presenting such a theory, in several ways. First thereare significant gaps in thatmethodological framework itself,connectedfor example with the justification of therepresentational conditions Cimposed ontherepresentingmaps /.RTMalwaysrequires / to bebothhomomorphic and faithful, i.e. to preserve the structural relations inboth directions,while the far moreextensive literatureofrepresentationtheory in algebrais based onpurely homomorphic representation. Onthe other handInoted in 1986b thepossibility ofrepresentationsbasedonfaithfulness alone, that beingintuitivelymore basic.

Second (Mundy, 1987a),physical representation theory should beapplied within a realist context,recognizing (unlike the narrow empiri-cistor operationalist tradition ofRTM) that fundamental physical quan-tities need not be in any sense empirical or observable. Moulines andSneed (1979) raise this objection, and in reply Suppes (1979,pp.207---208) concedes that we shouldnot"look for fundamental measurement"of thesequantities.Isay rather that weshouldandmust do this,but with

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the recognition (due toRTM itself) that 'fundamental measurement' isa mathematical relation between an intrinsic physical structure S anda numerical representation, not necessarily a directly implementableempiricalprocess.

Thus in seeking(for example) an RTM representation for time inmicrophysics, weask in realistphysical terms whether any structureofintrinsicphysicalrelations existsonanatomic level,sufficient to supportanRTM-style representation theoryand thus to ground intrinsically thestandardmicrophysicaluseof areal-valued time coordinate. These willcertainly notbe empirical relations,but theissue is whether theyexistatall,what theyare, and what axioms orpropertiesAof themground thisrepresentation. Unless this questioncan be answered,wesimply do notknow what it means to say of any theory using such a coordinate (e.g.standard non-relativistic quantum theory) that it is true of the actualworld.

Third, while presenting a unified representational methodologicalframework for a theory of quantity, RTM does not actually presenta single unified theory of physical quantities themselves (though theRTManalysis of the dimensional algebra of scalarphysical quantitiesisquite detailed). Essentially independent axiom systems arepresentedfor different kinds of quantities, with no real attempt at unification,for example,between scalar and vector quantities (discussed below),or between structurally similar but physically distinct quantities suchas the geometric and dynamic vectors. Such issues require deeperinvestigationof thephysical contentof thecorresponding fundamentalphysical theories thanis attempted within RTM.

3. INTRINSICFORMALIZATIONOFPHYSICALTHEORIES

RTMis essentiallya theory ofscaling, i.e. of numerical representationof individualquantities,not amethod of analysis of wholequantitativelawsand theories. Like theoperationalistbias ofRTM, this isdoubtlessa vestige of itspsychometric ancestry, wherein debate focused more onthe status of proposednumerical scales for particular variables thanonthe discussion of full-fledged quantitative theories using them (FM-3,p.263).

As aphilosopherofphysicsSuppesofcourse recognizes the needforlogicalanalysisof theories as well as scales,andunlike many has actu-ally undertaken this,using the method of set-theoretical formalization

65QUANTITYREPRESENTATION ANDGEOMETRY

(oftenalsocalledsemantic), whichhedevelopedin contrast to theoldermethod of syntactic formalization using a formal language. Semanticapproaches have since become fairly common. Ibelieve that the argu-ments purporting to show the generaluntenability or impracticality ofthe syntactic approach are misguided (Erkenntnis, 1987, p.173; 1988,p. 164; 1990, p. 345), and have outlined a syntactic formalization ofclassicalparticle mechanics (Mundy,1990). However,Iamhere con-cernednot with the contrastbetween semantic andsyntactic,but ratherbetweenextrinsic and intrinsic formalizations.

Alloftheexistingexamplesofsemantic formalization ofquantitativetheories seem to make essential use of extrinsic numerical representa-tional apparatus. This began with Suppes himself, and is a naturalconsequenceof therole playedby RTMinhisconceptionof theory for-malization (Moulines and Sneed,1979). Ineffect he takes the problemof logical analysis of quantitative structure to be solvedby RTM: oncethe relevantnumerical scales have been introduced by RTM methods,Suppes willproceed to formulatephysical laws as numerical represen-tational equations relating those numerical scale values, in essentiallythe standard18th century analytic manner. A modelof the theory willthen be aparticular numerical structure (e.g. a particular solution to adifferential equation), and the laws of the theory (e.g. the differentialequation itself) will impose numerical conditions on those numericalvalues,to be satisfiedin eachmodel of the theory.

However, thosenumerical valuescontaina mixtureofintrinsic infor-mationabout the physicalsystemitself,andextrinsicinformation whichdepends upon the numerical scales and coordinate systems. In otherwords,not all of the mathematical orset-theoretical structure present inthese 'models' of the theory actually represents something physicallyreal. At this point, therefore, quite in accord with the RTM tradi-tion,one mustdevelopa theory of meaningfulness to distinguish whichpropositions expressible using such a set-theoretic model are actually'meaningful', i.e. correspond to something intrinsic in the world, andwhich are not. This apparatus is closely related to the conditions ofdimensional and Galilean orLorentz invariance imposedin the analyt-ical tradition,andserves the samepurpose of rulingout any statementswhose truth dependsupon the choice of scale ofcoordinate system.

If the required theory of meaningfulness is properly developed,asin Suppes' own set-theoretic formalizations of physical theories, thenthe resulting formalization may be formally adequate. (Some other

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advocates of semantic approaches to theory-formalization, however,have overlooked this problem of meaningfulness.) Mypoint is ratherthat an analysis of this kind is philosophically unsatisfactory, since itdoes not succeed in presentingexplicitly the true intrinsic content ofthe theory as a set of conditions of some kind imposed directly uponthe world. Rather,itsays that the world possessesan intrinsic structuresufficient for theconstructionof variousextrinsic numerical scales,andthat one or more of these scales will then be found to satisfy certainnumerically expressed laws. However, if these are indeed laws ofnature, it must be possible to express them directly as propositionsabout that original intrinsic structure itself, with no detour throughthis extrinsic numerical apparatus. Such an intrinsic formulation willexpress explicitly someaspects of the contentof those laws whichareonly implicit in thenumerical formulation,and will deepenit in muchthe same ways that the RTM analysis of scaling deepens the olderKlein-Stevens one.

Intwo special casesRTMseems, somewhatbyaccident,tohavegiv-en intrinsic accounts of physicallaws or theories. One is the treatmentof dimensional algebra in FM-1, Ch. 10, which includes an intrin-sic formulation of the simple laws of proportionality (e.g. Hooke'slaw) which ground the dimensional algebra. However, the actual goalthere wasnot to find intrinsic formalizations of physical theories in thepresent sense,butrather simply to secure the standardrepresentationofcross-dimensional relationsbynumericalmultiplication of scalevalues.Indeed,inFM-3another wayis foundof achievingthat same goalwith-out intrinsic axiomatization of any individual laws,and this is regardedas an improvement (FM-3, pp. 85-86,315-316). The other context isaxiomatic geometry,discussed below.

Thefirstintrinsic formulationofasubstantivephysical theoryoutsideof geometry was thatof classical mechanics by Field (1980),extendedby Burgess (1984). However,Ihave argued (Mundy,1989b) that theseformulations donotcorrectlyexpress thephysicalcontentof that theory,because they dependessentially on assumptions about the structure ofphysicalspace which are notpart of classicalmechanics. Namely, theyassume the physical existence of space-time points, and of space-timeregionssatisfying strongexistence conditions (a set-theoretic 'compre-hensionaxiom'). Theintrinsic formulation of classicalparticlemechan-ics described inmy (1990) avoids these defects.


My fourth criticism (continuing from the end of Section 2) is thatRTMtakes anexcessivelyconservative attitudetowardsphysical theory.Onthe onehand thisappears intheartificial divisionbetween scalingandtheformulation ofnumerical laws. RTMrightly questions thetraditionalassumption ofdirectnumerical representability ofindividualquantities,constructing a detailed logical analysis of this procedure, while at thesame time accepting without discussion the traditional 18th centuryanalyticalapparatus for the numerical formulation of quantitative lawsand theories.

On the other hand this conservatism shows up in the criterion forsuccessof ascalinganalysis: namely, that itshouldyield 'theexpected'representation and uniqueness results, i.e. should lead simply to aformal justification of existing representational practice. In contrast,Istressed earlier that moderndevelopments in logic, mathematics andtheoretical physics cast substantial doubt on the correctness of manyaspects of traditional quantitative methodology. The ultimate purposeof formal and axiomatic analysis ofquantitativephysical theory,in thiscase as in all others, should therefore be not simply to explain andreaffirm existingpractices,but rather also to identify respects in whichtheymightusefullybe modified.

4. AXIOMATICGEOMETRY

AnimportantthemeofRTM,especiallyforphysical science,isitsassim-ilation of geometryto theory of quantity andmeasurement. Axiomaticgeometry wasan important technicalparadigm forRTMbecause itcon-tained the first explicit numerical representation theorems in the RTMsense (FM-2, pp.1-2). That is: Cartesian analytic geometry definesa numerical relational systemM having the structure of a qualitativeor non-numerical geometrical space S. The basic elements (points) ofMare n-tuples of numbers and the basic geometricalrelations amongthose points (congruence,colineality, etc.) are definedusing numericalproperties of those numbers (congruenceusing the Pythagoreanmetricformula, linealityusinglinear equations,etc.). In19th centuryaxiomat-ic geometry qualitative Euclid-style axioms A governinga qualitativegeometrical space S were shown to imply the existence of a systemof numerical coordinates satisfyinga condition C: that the coordinate-based Cartesian numerical definitions of geometrical concepts agreewith thecorrespondingprimitive qualitative concepts of the system S.

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This is a representation theorembecause itshows thatanymodel Sof axioms A canbe represented^the Cartesian numericalgeometricalspaceM: therepresenting map / is theconstructed coordinate assign-ment, whichcarries eachpointof the model toa 'point' in thenumericalspace, i.e. an n-tuple of real numbers, and satisfies the desired rep-resentational condition C. There is an associated uniqueness theoremshowing thatthealternative coordinate maps for suchamodel, i.e. thosesatisfying the samerepresentational conditions C,are just those obtain-able from the first one by composition with a numerical coordinatetransformation of aparticular group G, characteristic of thegeometryinquestion (e.g. for Euclidean geometry the orthogonal group). Finally,thereis a theory ofmeaningfulness(already formulated byKlein) iden-tifying the meaningful numerical statementsof the geometrical theoryas those invariantunder the transformations of this group G.

We see here one source of the conservatism of RTM mentionedabove,since inthis approach to foundations ofgeometrythecriterionofinterest or ofcorrectness for a geometricaxiom system is that it shouldyield the standard numerical representation.A basic test of the adequacy ofan axiomsystem for aparticularbranchof geometry isthat it should leadto the desirednumerical representation andthe desireduniquenesstheoremor automorphismgroupof therepresentation(FM-2, p. 81).

Such an approach obviously discourages lines of investigation (e.g.non-Archimedean geometries) leading to theories whose models lacknice numerical representationsin existingnumber systems. Thisspecialstatusaccorded to numerical coordinate representations in 19thcenturygeometryisperhapsanother manifestationof the18thcentury consensusin quantitative methodology mentioned above.

TheRTM treatmentof classical axiomatic geometrysummarizedinFM-2 does not add any fundamentally new ideas to this 19th centuryframework; rather,it extends its significanceby subsumingit under thebroader framework of representation theory. In particular, the RTManalysis of geometryprovides through its general theoryof representa-tion,uniqueness and meaningfulness a foundation for Klein's concep-tion of geometryas based on groups G of numerical transformationstakenas primitive. Interestingly, aclosely analogous Kleinian concep-tionofscale types asbasedon fixedgroupsof scale transformations wasgiven,apparentlyindependentlyofKlein,by thepsychologist Stevensin1946. RTMgives aquite satisfyinganalysis andunification of these twoanalogous theories of quantitative forms definedby numerical transfor-


mationgroups,together with theirassociated theoriesofmeaningfulnessas invarianceunder thesetransformations. (Seemy (1986b)for asimpleaccount; the full RTMtheory of scale-types is inFM-3,Ch.20.)

A significant defect in the geometric axiomatizations of FM-2, inmy view, is their failure to bring out fully the formal relations betweengeometrical quantities and other scalar and vector physicalquantities.Ibelieve that the drafts for FM-2 whichIsaw around 1980 includedattempts to formulate new axiom systems for the classical geometriesusing the RTM apparatus of scalar extensive measurement. However,the published axioms use only traditional geometrical relations suchas colineality andcongruence amongpoints, unrelated to theextensiveprimitives of order and addition used for scalar physical quantities inFM-1.

By far the most naturalunified approach, inmy view,is to interpretthe vectoradditionrelationofphysical vectorquantities asageneraliza-tionof theadditionrelation of scalar quantities,and themetric structureof physicalvectorquantities as a generalizationof the order relation ofscalar quantities. In the geometric context the relevant vectors are ofcourse the intrinsic displacement vectors relating pairs of points, andthe vector metric is the length-ordering of these segments. Indeed,essentially this kind of intrinsic vectorial approach to the axiomatiza-tion of geometry was mentioned aspromising in Suppes'(l972) (1973,pp. 387-388),though apparently not pursued. But this approach is notevenlisted among thealternatives atFM-2, pp.82-83.

On this approach onecan obtain a uniform intrinsic axiomatizationof all scalar and vectorphysical quantities overthe same twoextensiveprimitives, showing their fundamental unity of algebraic structure ascommutative groups equipped with an order relation, representableasvectorspacesover anormed field together with abilinear inner product(unpublished work of the author). This same theory comprises theEuclidean spacesofclassical geometricanddynamic vectors(forces andmomenta), the Minkowski spacesof special-relativistic geometric anddynamic vectors,and theHermitian spacesofquantum-mechanical statevectors over the complex field,and their respective non-Archimedeanextensions to Pythagorean or Euclidean fields. When aiming also atintrinsic representationofactual quantitativephysical theories,asinmy(1990),itismoreconvenient toreplace the orderprimitive by astrongerintrinsic metrical primitive corresponding to scalar multiplication bythe dimensionless length-ratio of two similar quantities. This allows

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polynomial equations to be expresseddirectly within the language oftheaxiomatic theory,as cannoteasily be doneinthequalitative languagebased on order and addition alone. (As anticipated by Suppes (1972),this isalsoaquantifier-free formulation inthe sense ofMoler andSuppes(1968).)

RTM always interprets geometry as simply another scaling theo-ry, the only difference being that the representing 'numbers' are nown-tuples (FM-2,p.81). As withscalarmeasurement,the workof formu-lating true quantitative laws or theories as numerical statementsusingscale values (e.g. thelaws of mechanics formulatedusing geometricalcoordinates) wouldthen notbeginuntil this initial 'geometricalscaling'stagewas completed.

However, there is an important sense in which this interpretationis incorrect: geometryby itself is already a true quantitative theory,while the theory of a single scalar quantity is not. This is intuitivelyrecognizedby physicists, but also has a formal basis in the fact thatmetrical geometry includes non-trivial quantitative laws: namely, thelaws of trigonometry relating the distances and angles in a triangle.(Allother geometrical laws in a classical geometry are consequencesof these.) In the scalar or 1-dimensional case,by contrast, these lawsreduce simply to the additivity ofdistances ('distance ab + distance be= distance ac),which does not expressa meaningfulphysical law butrather merely defines the numerical measureof distance,beingactuallya representationalcondition C imposed on the length scale /,not anintrinsic physical law. The trigonometric laws of a metrical geometryof dimension > 2,however, carry additional intrinsic contentafter thenumerical scales for measurementof length andangle havebeen fixed.

Mypoint, then,is thatunderSuppes'extrinsic set-theoretic approachto theformalization ofquantitative theories (Section3 above),onecouldgive a complete formalization of a metrical geometryby simply stat-ing the extensive measurement axioms for angle (FM-1, p. 76) andlength, and then asserting the trigonometric formulas of that geome-try as numerical laws constraining those scalar quantities. Indeed, thiswould producea very pretty anduniform treatmentof the various clas-sicalgeometries,showinghow they involvethe samebasic conceptsbutassert different lawsgoverning them. ButSuppes andRTMnonethelessemphasize the much more difficult traditional synthetic approach. Ithink this must be because the authors recognize intuitively that theseintrinsic geometric axiomatizations give a better account of the actual

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underlyingcontentof aclassicalmetrical geometrythan wouldbe givenby a simple numerical statementof its trigonometric formulas. This isan implicitacceptanceofmy earliercriticism ofSuppes' methodof for-malization ofquantitative theories: thatone shouldreplacenot only thescales but also thenumerical lawsbyintrinsic descriptionsof the under-lying facts whichground those extrinsic numerical representations.

5. MATERIALISM,SPATIALISM ANDRELATIONISM

There is another very important difference between the geometricalaxiom systems of FM-2 and the scalar axiom systems of FM-1, notdiscussed in FM-2but recognizedby Suppes elsewhere and stressedinFM-3,e.g. p.48. This is the difference between stronggeometry-styleaxiom systems yielding an isomorphism f from the representedspaceS to the corresponding numerical space M, and the weaker types ofaxiom system developed in FM-1 for scalar quantities, yielding onlyan embedding f (up to order-equivalence in S) of the space S into M,uniqueup to some group G of transformations ofM.

This disparitybetween ontorepresentationofgeometryandintorep-resentation of scalar quantities may be acceptable in the RTM frame-work, where each quantity is analyzed separately, but is surely notacceptable within a unified theory of physicalquantities. Since quanti-tative laws normally relate different quantities (as stressed in the RTMtreatmentof dimensionalalgebra),itis implausible andprobably incon-sistent to impose far strongerexistenceconditions on some thanothers.This isprecisely the typeofissue which transcends the piecemealRTMapproachand requires a trulyunified theory of allof the physicalquan-tities. (The intrinsic scalarmultiplicationprimitive ofmy (1990) theorymakesallquantitiessatisfy the sameexistenceconditions,over the samePythagoreanorEuclidean field.)

Since the actual world presumably does not containa continuum ofmaterial objects,an ontorepresentationof an actual physical structureS in a continuous M is only possible if the physical world containsimmaterial objects of some kind. Immaterial physical objects mightinclude: points or regions of physical space or space-time, propertiesor relations, possible objects, possible facts, and portions of physicalfields. Of course the physicalreality ofallof theseimmaterial objects iscontroversial.Iinsist,however, that whethersuch objectsexist isa realphysicalquestion,not avoidableby vague remarks about 'idealization.

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Inparticular, arepresentationaltheoryofquantity whose axiomsrequirethe existence of more entities in thephysical system S than its authoradmits as physically real is simply not doing its job: it is not a theoryof the actual world at all, but of some imaginary world. Calling thatworld an 'idealization' of the actual world cannotchange the fact thatthe stated theory is not trueof the actual world,so that results about ithave no power toexplainanyactual facts.

There are then two basic responses to the disparity between intoand onto representation: the immaterialist response of accepting anddefending theexistenceof some immaterial objects within one's theoryof quantity and geometry; and the materialist response of seeking aworkable theoryofquantity whoseaxiomsare satisfiablebyamaterialistphysical structure. Of course immaterialism arises not only for ontorepresentations,but whenever the axioms require theexistence of moreobjects than exist materially. Thus e.g. afinitist materialist must resortto immaterialismin order touse evenso weak a propertyas closureofS under addition, which forces its domain tobe infinite.Iwillconsidertheimmaterialist responses first.

The obvious immaterialist interpretation of the elements ofaquanti-tative structure Sis asquantitativeor geometricalpropertiesofmaterialobjects: geometrical points as location-properties (Horwich, 1978) andelements of 1-dimensional extensive structures as possible values ofscalar physical quantities, e.g. lengths. Such a second-order or Pla-tonist immaterialism is defended in my (1987b). The RTM literaturesometimes assumes such a second-order interpretation; for example,FM-3 (p. 281) places within therepresented structure S an element a\described as "the (qualitative) length of spring a-i\ thus assuming aphysical world containing immaterial 'lengths' in addition to materialobjects.

However, it must be stressed (Mundy, 1989b) that the immateri-al entities of a second-order theory of quantity need not function inphysical theory as monadic quantitativepropertiesof material objects;they may also function as quantitative relations of any given degree.Inparticular, geometricalquantities such as scalar distance and vectordisplacement are most naturally analyzedas second-order quantitativebinary relations: oneobject alonehasamass oracharge,but a distanceora vectordisplacement is aquantitative relation between twoobjects.This isespecially natural inthe contextof aunified extensive theoryof

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physical quantities as described above, with physical geometrybasedon the geometrical displacement vectors.

Insuch avectorial theory (e.g. my 1990)there arenopointsof spaceor space-time, since these correspond to monadic location-propertiesof objects. These are therefore relationist theories in precisely thetraditional sense: theories whichdeny the physical reality of points ofspace or space-time (spatialism), and instead assert that geometricalfacts are grounded solely in geometrical relations. Moreover, thereis no appeal topossible objects or possible configurations of actualobjects, suchas most relationists have invoked and some authors havethought they must invoke. The only immaterial entities required arethegeometrical relations themselves,and the second-order quantitativestructure (e.g. addition and order relations) over them. For any givenflat geometryone may construct such asecond-order vectorrelationisttheory which accounts for theobservablegeometrical facts in a mannerprovably equivalent to the monadic or spatialist theory. Moreover,the unified theory of physicalquantity obtainable usinga second-ordervector relationist theory provides a strong argument for its physicalsuperiority to the monadic alternatives: this is a new argument forsecond-orderrelationism derived from the theoryofphysicalquantities.

A second-order relationist theory need not use binary vector rela-tions; this is merely the simplest and most natural approach for a flatgeometry. Thequantitative relationist theory of my (1983), which has anatural second-order interpretation,was basedonathree-place extrinsicscalar relation (corresponding to a six-place intrinsic scalar relation).The quantitative formulations of classical metrical geometries usingtheir trigonometric laws asmentioned abovelead to second-order rela-tionisttheories basedonbinary scalar distancerelations and three-placescalar anglerelations among materialobjects. These theories tooallowthe full empirical contentof thecorresponding spatialist geometries tobe deduced from finitely many actual instances of these relations asinstantiated by actual objects. This trigonometric approach furnishessecond-order relationist theories ofhyperbolic andelliptic as wellas flatspaces.

It is instructive here to glance at the 'quantity theory of geometry'proposedby Teller. As a new alternative to spatialism and relationismhe suggests that

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weshouldtakespace-timeas aphysicalquantity differing fromquantitiessuch as massand temperatureonlyin detailsof structure,and we shouldtake space-timepoints tobethepropertiesthatconstituteparticular valuesof thespace-timequantity (1987,p. 425).

Teller's basic idea of assimilating geometryto quantity is sound,butpoorly developed for lack of the relevant technical apparatus. Havingnaively assumedthatphysicalquantitiesmustbemonadicproperties,heproduces a theory which (as many have observed) is simply a restate-ment of second-order location-property spatialism. In addition, bydisregarding those minor 'details of structure' he fails to notice thathisinitial analogy betweengeometry and quantitybecomes much strongerwhen cast in terms of the intrinsic quantitative structures definablewithin a geometric space -becoming strongest of all, as noted above,in terms of the additive group of displacement vectors. However, wealso then saw that theresulting theory is not a new alternativebetweenspatialism and relationism at all, but rather a new kind of relationism,namely second-order relationism,as inmy (1983 and 1989b).

Both spatialism and second-order relationism are forms of imma-terialism. They merely disagree over which immaterial entities exist:unoccupiedpoints (uninstantiated monadic location-properties, with asecond-orderstructure of standardgeometricalpoint-relations) or unin-stantiatedgeometricalrelations,withasecond-order structureofquanti-tative operations. Theonlyother preciselyarticulated formofimmateri-alist theoryseems tobe themodalimmaterialist relationism ofManders(1982), in which theinfinite setof points orrelations is replacedby theinfinite set ofpossiblefinitegeometricalconfigurations (finite structuresembeddable in thecorrespondinggeometricalspace). Manders (1982)seems to havebeen the first formally precise relationist geometrical the-ory articulated in print, followed closely by my (1983). Some authorssuch as Earman (1989) still question the very possibility of relationisttheories.

RTM remains neutral (or rather, inconsistent) onthe issue betweenmaterialism and immaterialism. Suppes himself, however seems tofavor a materialist response. This emerges most clearly in his 1972paper on space and time (1973a,pp. 391-395),leading to a quite dif-ferentkindof relationist theory.

Hestarts from Whitehead's now-familiar ideaofageometricaltheorywhose basic elements are not points but extended entities related byinclusion and overlap. Given strong enoughexistence conditions therole of points can certainly be taken overby convergent sequencesof

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suchentities ('regions'),but suchassumptionsareas dubious as those ofaspatialist theoryitself. Suppesinstead seeksa theory whoseaxiomsaresatisfiableby actual extendedmaterial objects ('bodies').For example,his axiom 8 implies that each bodyhas only a finite number ofminimalparts or 'atoms'. Heproceeds(p. 394),There are anumberof different waysto extend theaxioms...,andbyheavy-handedmethods, we can reach ordinaryEuclidean geometry fairly rapidly. We simply haveto postulate enoughbodies and atoms. We would of course not expect to get the fullEuclidean space,because of the finitedissectionproperty[i.e. axiomB],but we wouldwant tobeable to imbedin three-dimensionalspace, andto geta representationofthisimbeddinguniqueup to thestandard groupofrigidmotions.

Thereis no doubt that thisprogramcanbecarriedthrough. The techniques for theone-dimensionalcase of measurement exploitedinmanydifferentdirectionsinKrantzetal.,1971 [FM-1] providemore than adequatetools, butIdo not as yet see how topursue it in a simple and elegant fashion. At the same timeIam beginning to seea philosophicallyinteresting aspect of this programif it can be satisfactorily carriedthrough. Properly carriedout, it shouldprovide anew wayof lookingat the nature ofspace.

Suppes' subsequent discussion of the philosophical significance ofthis idea seems tome to be somewhat confused, drifting from finitistmaterialism toward modal immaterialism in the manner of his (1974).More important inmy view is the technical suggestionin the firstpara-graph; this defines whatIcall embedding relationism (seemy 1983,1986a) and is simply RTM representation theory applied to geometryfor the case of into rather than the traditional onto geometrical repre-sentations.Note here thekeyelements ofRTM:axioms,representationtheorem,and uniquenessup to the symmetry group G of the represent-ing spaceM (which in turnwilldetermine a theoryof meaningfulness).Suppeshere suggests that the RTM theory of representation providesthenecessary technical apparatus for developmentof amaterialist rela-tionist theory: one which will capture the empirical contentof a stan-dard geometricaltheoryusing axioms overastructureS containingonlymaterialobjectsandproperties orrelationsinstantiated bythem, withnoadditional structure involving any kind of immaterial entities,whetherthey bepoints, relations,possibilities, or what have you.Ibelieve that no such materialist embedding relationist theory has

been stated in print, by Suppes or anyone else. Space here permitsonly brief remarks on this line. Most important is an elementary butfundamental limitation due to the fact that by a simple counting argu-ment, the finite models of such a theory cannotreproduce exactly thefullrange of finitepossibilities describableby any standard geometrical

76 BRENTMUNDY

or quantitative theory. This is because (assuming a finite set of fini-taryprimitive relations, i.e. ones of finite degree), such a theory hasonly finitely many structurally distinct models of any given finite sizen,while any standardquantitative theory allows infinitelymany distinctquantitative configurations of n bodies (e.g. triangles in geometry).The point is that a materialist theory should allow finite models, sincethere might be only finitely many material objects; theproblem is thenthat for cardinality reasons finite models alone cannot replace all of thefinite modelsof any standard quantitative theory.

This problemaffects the materialist axiom system for scalar quanti-ties given inFM-1 (pp.81-85), in which closure under additionis notassumed, thus permitting finite models. By the resulting representationtheorem each finite model is embeddable in (R+,<,+)uniquely up toratio transformations, thus determining n — 1 independentreal-valuedratios among the n objects in the model. By the precedingcardinalityargument,however, thesemodels cannot representall of the size-ratiocombinations possible among nobjects in a continuous scalar theory.Inspectionreveals thatthe axiomscan onlybesatisfiedin a finitemodelifall its elements are exact multiples of the smallest one, so that theirratios to itare allpositive integers.

A.finitistmaterialist embeddingrelationist theoryover finitary prim-itives can thus never achieve exactexpressiveequivalence (preciselydefinable as in my 1986a, b) with any standard spatialist geometricaltheory. One line of response mentioned by Suppes (1979,p. 214) is tostress the approximation to the continuous theory as the finite modelsbecome larger.

Another response,developedinmy 1989afor thecaseofscalarquan-tities but also extendable to the geometrical case, yields finitist mate-rialist theories with exact expressiveequivalence to the correspondingcontinuous theories. This is done by introducing anew kindof quali-tativeprimitive relation: predicates ofvariable degree,which generatean infinite set of distinct atomic facts even over a finite model, thusavoiding the cardinality problem. The geometrical versions of thisapproachyieldafinitistmaterialist relationist theoryhavingfull expres-siveequivalence with the corresponding spatialist theories,but over anon-standard logic.

A thirdresponse, taking therevisionist attitude tothe theoryofphys-ical quantity whichIcontrasted in Section 3 with the conservatism ofRTM,is to takeseriously thedepartures from classical theoryofquanti-

QUANTITYREPRESENTATIONAND GEOMETRY 77

ty requiredby finitistmaterialism over finitary predicates. Forexample,thebreakdown of space-time geometryinmicrophysics couldconceiv-ably be simply a manifestation of the coarse-grained structure of thesmall finite models of such a finitist embeddingrelationist geometricaltheory.Iconclude by briefly mentioning another approach to embedding

relationism due toFriedman (1983). Friedman imagined, in contrastwith his own spatialist viewpoint, a relationist theory which takesembeddabilityof the physicalsystemSin the mathematical spaceMasits sole axiom.Iarguedbriefly in my (1986a) thatFriedman had over-looked the uniquenessproblem familiar from RTM: such a theory willnothave theexpressiveequivalencehe claimed for itunless the embed-dings of S inM are unique up to a symmetry ofM.Ialso noted thatRTMhad already developed a detailed methodology for dealing withsuch embeddingproblems, recognizing the need for structuralaxiomson the systemS and theneed for auniqueness theorem,and that Suppeshas already applied these ideas to embedding relationism, as quotedabove.

Inreply to thiscriticismCatton andSolomon (1988)defendedFried-man's idea,suggesting thathe wasindeedawareof theuniquenessprob-lem and had intended G-uniqueness of the embeddings as an implicitcondition on the primitives of the relationist theory (though the prim-itives chosen by Friedman for his own example do not satisfy thiscondition,asIhad already pointed out). Inmy 1991response (Mundy,1991)Ishowed,usinga variantof the counting argumentgivenabove,that in general this condition cannot be satisfied by any finite set offinitary primitives with embeddability as the soleaxiom;Ialsopointedout the general implausibility of such an approach to embedding rela-tionism. This episode, like the naive suggestion of Teller mentionedearlier, seem tome to show clearly how badly the philosophical litera-ture onrelationismis inneedof both thegeneral formal discipline andthe specific techniques and results of the RTM tradition (in agreementwith Suppes,1973).

6. AXIOMATICSPACE-TIME GEOMETRY (A)

Following Einstein,Minkowski showed how space and time togetherpossess geometrical structure (e.g. concepts of distance and straight

78 BRENT MUNDY

line); today space-time geometry(STG)studies such geometricalstruc-tures and theories (STGs) allowing spatiotemporal physical interpre-tation. While only pragmatically distinct from general mathematicalgeometry,thephysicalimportance of STGhas created aliterature moreinfluenced by physics and philosophy than by traditional mathemati-cal geometry. Thus, while Suppes' work on ST geometryfalls mainlywithin the sameRTM technical tradition as his work on generalmath-ematical geometry, its external significance is different because therelevantbackground literature is different.

Suppes' (1959a)paperon derivation of theLorentz transformationsdoes not fall within the RTM tradition,however, but rather within theearlier Klein-Stevens tradition in which a class of numerical represen-tations are assumed as given and the problem is to characterize thefamily of numerical coordinate transformations relating them. ThisKlein-Stevens framework meshes naturally with the standardphysicalapproach to space-time theory in terms of relativity, i.e. as a prob-lem of relating the coordinate descriptions of the same event givenby different observers in different states of motion. On this tradition-al approach the key to special relativity is to show these numericalcoordinate transformations to be Lorentzian rather than Galilean. Thestandard derivation assumes the transformations to be linear and toleave invariant the numerical values of the Minkowski metric forms2 = c 2t2 —x2—y2 — z2 (i.e. itassumes allobservers toagreeabout thespeedof light, c,as determinedusingEinstein'sopticaldefinition of thetime coordinate, t). Suppesshowed theLorentzian form to follow fromthe weaker assumption of invariance of s2 for positive cases (timelikeintervals) only, without assuming linearity. Zeeman (1964) ultimatelyshowedit tofollow from theeven weakerassumptionofinvariance onlyof thesignof s2,i.e. ofinterval-type orcausal ordering.

However, all such Klein-Stevens approaches take the existence ofnumerical representations (coordinate systems) as given, whereas thepointofRTMis todeduce this as arepresentation theorem fromintrinsicqualitative axiomsovertheempirical subject-matter, expressedin termsofqualitative primitives. RTMthusleads toaxiomatic orsyntheticSTG,analogous to classical axiomatic spatial geometry. This idea was firstpursuedbyRobb (1911and later), who gavequalitative axioms for theMinkowski STGof specialrelativitity using thebinarycausalprimitive,andprovedacoordinate representationtheorem in the standard fashion

79QUANTITY,REPRESENTATION ANDGEOMETRY

of axiomatic geometryandRTM.(My (1986d) contains a muchsimplerdevelopment of essentially Robb's material.)

The importance of Robb's work was often stressedby Suppes, andaxiomaticSTGwas incorporated intoRTMalongsideclassicalaxiomat-icgeometry. However,in contrast with the largemathematical literatureon that topic, the literature on synthetic STG wasrather sparse (FM-2,pp. 128-130),andconsiderable effort was devoted to the developmentofnew axiom systems for STGwhich would fitnaturally into theRTMcontext. (This includes my 1982 thesis, directed by Suppes.) Theunpublishedaxiomatic work of Dorling onMinkowski STG also datesfrom this period, as does the use in Field (1980) ofRTM methods toaxiomatizeclassicalSTG.Allof this workdiffers fromRobbin develop-ingSTGusingclassicalgeometricalprimitives (e.g. affine betweennessandmetrical congruencerelations) more thancausalones,thus empha-sizing the strong formal relations amongall of these geometries.

Axiomatic STG is particularly relevant to current philosophy ofphysicsbecause,unlike theories of other physical quantities,STGsarerecognizedas important physical theories in their own right. (Inotedtheunderlyingreason for thisearlier.) AxiomaticSTGthsbears directlyondebates concerningphilosophical andphysicalinterpretation ofSTGtheories.Iwillmention threeexamples,drawn frommy (1986c).First, the tra-

ditional physicalapproach to space-time in terms of 'relativity' among'observers' invites idealist interpretations of the theory as beingaboutour knowledge or observation rather than about any objective physi-cal subject-matter which exists independently of us (e.g. Eddington,1939). Wecanrebut such accountsby presentingan explicit axiomaticanalysis of the contentof the theoryas a setof statementsconstrainingsomeprimitives whichare entirely physicalin character andinvolvenoreference to the presence of 'observers'. Second,Reichenbach (1924,1928)andothers give causalanalysesofMinkowski STGandinfer thatsupraluminal causationis inconsistent with the theory, whereasaxioma-tizationin terms ofnon-causalkinematic primitives leads to the contraryconclusion. Third, Kuhn (1962) cites the transition from classical toMinkowski STG as an example of the supposed incommensurabilityof successive theories, while axiomatization shows the expressibilityof both theories in the same formal language, having the same physi-calinterpretation, and differing only by acceptanceor rejection of oneaxiom (constancyof the speedof light)expressedin that common STG

80 BRENT MUNDY

language. These examples again support Suppes' (1973b) opinion ofthe value for philosophy of spaceand time ofaxiomatic analysis in thetradition of the foundations of geometryorRTM.

It should be noted that axiomatic STG does require axioms, notmerely definitions. Here we may contrastSuppes' long-standing stresson Robb's axiomatic STG with the more recent flurry of discussionof Robb sparkedby Winnie in 1974 (see Winnie (1977) and severalother papers in the same collection). This discussion focused entire-ly on the definability of the Minkowski metric structure from Robb'scausal primitive: this follows from Zeeman's invariance proof, and ofcourse also follows from Robb's earlier explicit axiomatization usingthatprimitive. This wasphilosophicallyrelevant asanargumentagainsttheclaims ofReichenbach andGrunbaum that causal structure is objec-tiveor intrinsic while metric structure is conventional or extrinsic,andalso raised questions about the conformally flat STGs of general rel-ativity, which possess the same causal structure as Minkowski STGbut different metric structure. None of this discussion involved any ofthe details of Robb's axioms; it dependedsimply on the possibility ofsome suchaxiomatization,whichis essentially the content ofZeeman'sinvariance theorem. Ineffect Zeeman showed only the possibility of acausaltheory ofMinkowski STG,i.e. showed thecausalprimitive tobesufficient for this purpose. However, in order to know what such a the-ory actually says weneed to find some sufficient setofintrinsic axiomsover that primitive. This is the goal of axiomatic STGin theRTM tra-dition,following the generalpattern ofRTMaxioms andrepresentationtheorems.

The argumentfrom theend of Section 4 applies again for STG. ASuppes-style extrinsic numerical formulation of Minkowski STG canbe givensimply by conjoining the extensiveRTM axiomatizations forlength and timeintervals with thehyperbolicMinkowskian trigonomet-ric formulas as numerical laws governing those quantities. Suppes'long-standing emphasis on the importance of Robb's axiomatizationthus again implies a tacit recognition that such an intrinsic approach isdeeper,despite its greatercomplexity.

QUANTITYREPRESENTATIONAND GEOMETRY 81

7. REPRESENTATION THEORY OFDIFFERENTIALGEOMETRY

(a) TheProblemThe work in synthetic STGmentioned so far concerns onlyflat STGs,whoseuniformunderlyingvector-spacestructureplaysanessentialrolein the axiomatic developments. The nextnatural case, apparently notyetpursued, wouldbeSTGsof constantcurvature(used incosmology),presumably axiomatizable by analogy to the classical hyperbolic andelliptic geometries. However,much greaterphysical interest attaches toinhomogeneousdifferentialgeometries (DGs), suchas the RiemannianSTGsof general relativity (GR).

AxiomatizationofDGis a technicalproblem quite different in char-acter from those hitherto treated with the RTM tradition. On the onehand DGtheories arenotcategoricallike theclassicalgeometriestreatedinFM-2. Instead theycomprise many differentmodels of verydifferentgeometrical structure,and weseek only to axiomatize thebasic geomet-rical properties (e.g. Riemannian structure) common to allof them. Onthe other hand these different models are notallembeddable in asinglefixed Cartesian numerical geometryas with the non-categorical scalarembedding theories ofFM-1.

In standard RTM the intended representing numerical-space M isgiven in advance, and its known structure plays an important role inthe formulation of axioms. Every universalproposition true inMholdsin any S embeddable in it, so these necessary axioms must certainlybe included as axioms or theorems (FM-1,pp. 21-23). More deeply,the associated uniqueness result constraining the range of alternativerepresentations is linked to the symmetries of the fixed representingstructureM. If no such structure exists, therefore, it becomes unclearhow toproceed. Indeed,Teller (1987)assumesthe tasktobe impossible.("If the curvatureof space-time varies sufficiently irregularly, then norepresentation theorem can be brought to bear. Such cases lack thesymmetries needed to characterize the family of representations ..."p.435.)

Such a conclusion is naturally suggested by the RTM literature.Indeed, the extension of the RTM theories of representation, unique-ness and meaningfulness to situations involving few automorphisms isacknowledgedat the end of FM-3 (p. 333) as an important openprob-lem. ("Thereprobably are important connections between the problemof defining meaningfulness and that of formulating satisfactory theo-

82 BRENTMUNDY

rems about therepresentationand uniquenessof structures when thereare few or no automorphisms" FM-3, p. 333.) In this sectionIwilloutline (through application to DG) such a generalized form of repre-sentation theory whichis applicable inallcases regardlessofsymmetry,and which contains the RTM theory as a special case. However, inthis generalizedtheory theKlein-Stevens-RTMuniqueness group Gnolongerappears.Inotedin Section 2how in standard RTM the single transformation

group G plays three distinct roles, as: (a) a symmetry group of therepresentingspace M; (b) a uniquenessgroup of the representationalproblem, determining by functional composition the class of alterna-tive representations / meeting the representationalconditions C; and(c) a meaningfulnessgroup,determining meaningfulnessas invarianceunder its action on the representational propositions. The pervasiverole of thegroup G in standardRTMhas created abelief, illustratedbythequote from Teller, that the formal theory of representation,unique-ness and meaningfulness must in every case be dominated by a singletransformation group Gplaying all three of these roles.

However, there is no logicalreason why these three roles should belinked in this way, or indeed why roles (b) and (c) should be playedby numerical transformation groups at all. Roles (b) and (c) are ofcourselinked by thebasic idea thatmeaningfulness is invariance underchange of representation, and thus the form of the uniqueness theoryfor a given representational problem mustalways determine the formof the corresponding theory of meaningfulness. However, there is noapriori necessity that either of these should always feature some onenumerical transformation group G, whether given as the symmetrygroup G(M) or otherwise. This is simply an expectation based onthe theory of representation as hitherto developed in RTM, using afixed symmetrical M. (Apersonalnote: duringmy periodof informalcontact with SuppesatStanfordIwasparticularly struckbyhis tendencyto reason by association between specific examples, without verbalarticulation of corresponding general principles. Isuspect that thisunusual 'semantic' as opposed to 'syntactic' style of thinking is thesource of many of his most characteristic traits as a philosopher andtheoretician.)

This same tendency to identify the logically distinct representationalroles of a single group G has been even more pervasive and damag-ing in space-time theory. In addition to the three roles (a)-(c) above


(symmetry, frame transformations,meaningfulness), 'the group' Gof aspace-time theorymay alsoexpress: (d)arelativityprinciple,constrain-ing not only STGbutother physical theories as well (e.g. the familiarcondition of 'Lorentz invariance' imposedon dynamical theories). Itisvery often tacitly assumed that all four roles must be playedby someone group;forexample inphysics the derivation of theLorentzgroupofframe transformations is almost always taken to justify itsuseas asym-metry and relativity group as well. The best-known error of this kindis Einstein's confusion ofcovariance with generalized relativity, whichconflates roles (c)and (d) (seeFriedman,1983). My (1986d,Section 6)exposes these confusions by presenting a version of Minkowski STGwhich splits three of these four roles: it has the Lorentz group for itsframe transformation and meaningfulness groups but notits symmetrygroup,and allows denial of therelativity principle.Iwillnow sketch an account of the theory of coordinate representa-

tioninDG,leadingtoageneralizedtheoryofrepresentation,uniquenessandmeaningfulness whichdoes notuseanytransformation groupG,butwhich reduces to theRTM theory when appropriate special conditionsaresatisfied.

(b) Generalized TheoryofUniqueness

Coordinaterepresentation instandardDGessentially follows the Klein-Stevens format, in that the represented space is treated as a bare setS, and the geometric structure imposed on itby the theory is fixedentirelybymeansofnumerical representations/ofStaken asprimitive.(Actually Sis usually givenan antecedent intrinsic topology, while thedifferentiable and geometric structures are imposed using coordinaterepresentation,but it is more uniform to let the topology be inducedbythe coordinates as well.) As in the 18th century analytical method, thefamily ofnumericalmaps / issimplypostulated,andthetheoryproceedsentirely through specification of numerical connections among thoserepresentations. In particular there is a theory of uniqueness, whichcharacterizes the rangeof possible alternative representations. This isofcoursesimply theKlein-Stevens-RTMuniquenesscondition,namelythat /'is apossible alternative representationto/ justincase f =g°ffor some g in G, where G is the group defining the geometry or scaletype. G is always some group of transformations of M, usually itsautomorphism groupG(M).

84 BRENT MUNDY

However, DG has a different theory of uniqueness not fitting thispattern, though it will be seen later to be a natural generalization ofit. We find by examination of the DG coordinate apparatus that thereis no one fixed embeddingspace M: different representations / havedifferent coverings of Sby coordinate charts,and different values forthegeometricalcoefficients suchas the componentsof the metric tensoror affine connection;this implies that the actualnumerical structureMwhich serves to represent Sunder aparticular coordinate representationwill be different for different /. In a more detailed presentation thisstructureMcan bedefinedprecisely: it is itself aDG of the same typeas S, but with complex numerical entities for itspoints, like a Cartesiangeometry.

Since the different / map into different numerical DGs M, therecannotbe anyKlein-Stevensuniquenessgroup Grelating themall,andhence the uniquenessproblem must be attacked insomeother way. Infact, in DG that problem is normally solvedby a direct specificationof the conditions for two coordinate representations / and /'of S tobe equivalent, i.e. to impose the same geometric structure on S. Thisdefinition isdifferent for different types ofDGspace,but the basic ideais always the same.

We first define equivalence or compatibility for two individual coor-dinate charts cv,cv,within the intersection of their domains of defi-nition. Since the charts are 1-1 maps into Rn, the 'bridge function'kuv =cv o c~l willbe a1-1function fromRnintoRn,so that standardanalytical concepts are defined for it. The compatibility conditionrelat-ing different charts is then simply that kuv should preserve whateverproperties or structure we wish topull back onto S from Rnusing thecv: e.g. if Sistobe a topological manifold then the kuv need merelybe homeomorphisms, while for an m-differentiable manifold each kuvmust be ra-times partially differentiable in Rn. These compatibilityconditions ensure that the various geometric properties pulled back toS from Rnalong the different cv willallbe consistent,since the bridgefunctions kuv preserve them. Allof thecharts in anyone representation/ are required to be pairwise compatible wherever they overlap, andtwodifferent representations / and/'are defined tobe compatible ifallof their charts arepairwise compatible.

The only technicality is that thegeometrical(Riemannian or affine)structure pulled back from Rn along a given chart cv is not simplythe standard flat structure of Rn itself, but rather a modified structure


defined locally in Rnusing additional numerical functions associatedwith thechart cv.Thus the localmetric structure pulledback from Rn

to Sby cv is not that determined by the standard Euclidean 'metricform' ds2 = Y^i(dx1)2,which would simply induce aflat geometryon S,butrather that determined by theRiemannianmetric form ds2=Hii9ij&x%&xJ'» where then2 metriccoefficients gfj are functions takingdifferent values at different points in Rn,and are associated with anychart cv for aRiemannianS. The compatibility condition for two suchcharts is then that the bridge functions kuv should carry the lengthsdefinedby integration of thegfj quadratic differential form usingcv tothe lengthsdefinedbyintegration of the g\j formusingcv.The 'length-element' ds2 itself is then invariant, i.e. the same in all compatiblecharts.

My firstpoint, then,is thatby this definition the uniquenessprobleminDGis solveddirectly, without introduction of auniquenessgroup G,by a direct mathematical characterization of the relation which obtainsbetween two equivalent numerical representations / and /', i.e. oneswhich represent the same structure on S. The standard representationsof S are thus unique up to equivalence,i.e. any two standardrepresen-tations / and/'willbear thisrelation of equivalence. This in turn fixesthe form of theuniqueness theorem we should seek for an RTM-styleanalysis of DG - i.e. we should seek to show that any twomaps /meeting our representational conditions C are equivalent in this sense.Thus DGalready hasa uniquenesstheory whichdoes the same work asthe Klein-Stevens-RTM theory of G-uniqueness, whilemaking no ref-erence toanygroup G.Moreover, thisuniqueness theory is welldefinedeven though thealternative representations/ whichitcharacterizes mapSinto entirelydistinctnumerical structuresMandM.

Itmight seem that this 'direct' solution to the uniqueness problemin DG is merely an accident, with no real connection to the Klein-Stevens G-uniquenesstheory. Butin fact there is anatural andintimateconnection, due to the compositional action of the function g on fwhich generates thenew representation in the Klein-Stevens equation/;= Inthe generalDGcase wecannotcompositionally generateall of the alternative representations /'from any one / by theactionofagroup G, since this will not take us out of the numerical representingstructure Mused by /. However, the same basic equationf = k o fmay well hold, where k is a mathematical function of some broaderclass,not taken from one fixedtransformation group G.

86 BRENTMUNDY

There is an obvious guess, confirmed by closer analysis. Imen-tionedabove that the representingspacesMandM

'are themselvesDG

spacesof the appropriate type,e.g. Riemannian spaces, withnumericalelements. Thus the standard concepts of isomorphism for aDG spaceare defined for them, e.g. the conceptof aRiemannian isometry. Usingthese we findin generalthat thedefinition of equivalence of tworepre-sentations / and /' canalso be putin terms of connectibility by aDGisomorphism k. That is, two representations / inMand /' in M'arecompatible iff there is aDG isomorphismk of the corresponding typefromM toM' such that /'= k o /.

This enables us torecast theDGuniqueness theoryin acomposition-al form which makes explicit its status as adirect generalizationof theKlein-Stevens G-uniqueness theory. Instead of a group G actingon asinglenumerical CartesianspaceM,wenowhaveawhole categoryXinthemathematical sense, i.e. theset ofallof the differentnumericalDGsMof the given geometric type (e.g. all numerical twice-differentiableRiemannianmanifolds), together with the structure-preservingmaps ormorphisms among them (among which the isomorphisms are a specialclass). Our generalK-uniqueness condition then says that the repre-sentations f of S are unique up to composition with an isomorphism kof the categoryX, i.e. that for any equivalentor admissible /' thereissuchak in X with /'=k o /,andconversely any suchk generatesanequivalent representation /'. If for somereason werestrict attention torepresentationsall in a single elementMofX, thisK-uniqueness con-dition will reduce to the Klein-Stevens G-uniqueness condition, withG being the automorphism or symmetry group of M.But the presenttheoryis far more general,since the function categoryXismuch largerthan anysingle transformation group G(M).

Note that in the DG case, as in most geometry, we have been con-cerned only with onto representation. The conceptof if-uniqueness isinprinciple applicable alsofor into representation, where the conditionwould be that for any two maps /, /' into structures M,M' of thecategory X,both meeting conditions C, there is a morphism k in X(of some specified type) fromM toM' such thatf — k o /. WhenMis fixed this condition again reduces to uniquenessup to some type ofmorphism of M.Note however that as X becomes larger this unique-nessconditionbecomes harder to meet,and will generally require verystrong structural axiomson S andon theelements ofK.


(c) Generalized TheoryofMeaningfulness

First recall from Section 2 that a theory ofmeaningfulness within theRTMformat is essentially a conditionon representationalpropositionsaiming to distinguish a subclass of them whichexpress intrinsic infor-mation about the represented structure from a larger class which neednot. Followingmy (1986b) we may write representationalpropositionsas 'p[/]'» where p is an actual sentence in some numerical represen-tational language L. L contains terms of the form '/(s)' standingfor scale values determinedby a representation, and 'p[/]' means theproposition expressedbyp when the values of those scale-dependenttermsofLare determinedusing theparticular representingmap/.Thusp[f] andp[f] in general are different propositions expressible by thesame sentencepofL.

Forp in L we have (as in (**),p. 420 of my (1986b)) the intuitivecharacterization of meaningfulnessas independenceofrepresentation:

(Ml) p[f] <-► p[f% for all / and /'satisfying G.

However, (Ml) is a semantic condition referring to the meaning ofpunder the two interpretations / and /',andhencelacks clear criteria ofapplication. The basic goal of a theory of meaningfulness is to obtainaprecise syntactic criterion of meaningfulnessdefined directly for thesentencesp ofL.

The Klein-Stevens theoryof meaningfulnessdoes this in two steps.First weuse G-uniqueness to transform (Ml) into:

However,this too isstill asemantic statement,concerning twodifferentinterpretations of the same sentencep. The secondstepdepends onthefurther syntactic covariance assumption,

The right side here involves a new statementg(p) ofL, in which wereplaceevery termof the form '/(s)' inpbya termoftheform ig(f{s))\Here weassume thatLcontains a term(here writtensimply as '#') des-ignating the element g of G, and in the sentenceg(p) of L we prefix

(M2) p[f] *-+p[g o /], for allginG and/ satisfying C.

(SC) p[go/] <-> g(p)[f], for allgin G and/ satisfying G.

88 BRENT MUNDY

that term to the functional expressions of L which express the valuesf(s)of /inp.

This new sentenceg(p) is a syntactic transform of the original sen-tencep. Theintuitive justification for (SC),which canbemaderigorousfor a certain class of sentencesp, is simply that each value f(s) of /inp[f] will be acted on once more by g in g(p)[f], yielding the valueg{f(s)) at the same place in g(p)[f] where p[f] had the value f{s).This is the same substitutionas isperformed semantically when we usegof instead of / tointerpret theoriginalsentencep. Thusp[g of] andg(p)[f] willhave the same valuesg(f(s)) at the sameplaces,andhence'co-vary in truth-value. (SC) and (M2) yield the desired syntacticcriterion ofmeaningfulness:

(M3) p[f] <-► g(p)[f], for allginG and / satisfying G.

Since both sides of(M3)are now interpretedby the same /,a sufficientcondition for (M3) is interdeducibility in Lof the two sentencespandg(p). This is 'G-invariance' of astatementp in theordinary mathemat-ical sense: thatby actual algebraic manipulationone canderive pfromg(p) and vice versa. Thiscriterion involvesno appealto semantics,butstill implies the meaningfulness condition (Ml). A similar syntacticinvariance approach to representational meaningfulness was tried bySuppesinhis early paper (1959b), but was later discarded in favor ofthe RTMsemantic approach.

This theory uses the Klein-Stevens uniqueness group G in bothsteps: the appeal to G-uniquenessin the first step, and the appeal to asyntactic action p —

► g(p) of the group G on the sentencespof L forthe second step. However,both of these stepscan becarriedover to ourextended framework. First we invoke the K-uniqueness condition ofthe preceding section to rewrite the semantic meaningfulness criterion(Ml) as

Corresponding to (SC), we then must also seek some syntactic actionp —* k(p) of each k in X on the sentences p of L which mimicssyntactically theactionofasemantic changeofrepresentation/ =kofonthe propositionsp[f].

This looks hard (and it is), since the action of k on / in the DGcategories X is much more complex mathematically than the simpleaction of a numerical transformation group G on individual numbers.

(M2') p[f] <-> p[k o /], for allkinX and/ satisfying G.

89QUANTITYREPRESENTATIONAND GEOMETRY

But justsuch asyntacticactionk(p)hasalreadybeen workedout withinDG,beingknown as tensoranalysis. Consider any pairof compatibleDGrepresentations/ and /',andany if-isomorphismk with /'=kof.Tensor analysis defines a syntactic actionp — > k(p) over L satisfyingthe syntactic contravariance condition:

(SC) p[f] <-» k(p)[k o /], for allkinXand / satisfying G.

The condition (SC) is opposite from (SC),because the tensor analysisrules for 'transforming components' in sentences pofL are designedto preserve the semantic content of an interpreted sentencep[f] undera coordinate transformation / — * ko/.That is, thenew sentencek(p)will say thesame thingunder thenew interpretationkof thatpdidunderthe interpretation /, so the syntactic actionp

—► k(p) ofk counteracts

('contra-varies from') its semantic action / — > k o /. This is expressedin (SC).

However, either covariance or contravariance does equally well forour purposes,since in either case (M2') can be applied top or k(p) toyield:

(M3') p[f] *-*■k(p)[f], for allkinXand / satisfying G.

As before, a sufficient condition for (M3') is the interdeducibility ofthe numerical statementsp and k(p) in the language L. Since (M3')has been shown semanticallyequivalent to (Ml), this gives the desiredsyntactic criterion ofmeaningfulness, as invariance under the tensorialactions k(p) for all k in K. In tensor analysis it is shown,by directalgebraic manipulations within thenumerical languageL, thatacertainfamily of numerical sentencesp known as tensorequations have thisinvariance property,and are therefore meaningful.

Wehavethus shownthegeneralpattern forconstructionofa theoryofmeaningfulnessbasedonthegeneralX-uniquenessconditionof thepre-ceding sectionasopposedto theKlein-Stevens G-uniquenesscondition,andhave seenhow toderive fromit asyntactic k(p)-invariancecriterionfor meaningfulness analogous to the Klein-Stevens G-invariance con-dition,byageneralprocedure whichreduces to that in the G-uniquenesscase. The procedureis to construct for each k in X asyntactic transfor-mation k(p) actingon the sentencesof the language L, which satisfiesasyntactic co- orcontra-variancecondition. Either of these implies thedesired syntactic invariance criterion (M3').

90 BRENTMUNDY

There is no general recipe for constructing the syntactic operatork(p): this will dependonthe forms of the sentencespofLand the func-tions k ofK.Iremarked that the syntactic form of the tensorial actionk(p) is much more complicated than the simple substitutional actiong(p) of the Klein-Stevens groups G. This is because the sentencespof DG languages always involve derivatives of the coordinate values,so that on a coordinate change / — » k o / new terms are generatedinvolving the derivatives of k. The Klein-Stevens theory,in contrast,was essentially linear: first, the sentencesp rarely involved calculus,and second, the numerical transformations g are themselveslinear forthe most common G, so that even when substituted into p under aderivative or integral signno additional terms involvingderivatives orintegralsofg wouldbe generated.Thus eventhe G-uniquenesscasecanbecome more complex: if Gis non-linear andL involves calculus thenthesyntactic action g(p) willinvolve derivativesof g,like the tensorialk{p).

8. AXIOMATIC SPACE-TIME GEOMETRY (B)

Using the generalized concepts of if-uniqueness and k(p) invariancewe have seen that two of the three central elements of RTM,unique-ness and meaningfulness,are already present in standard DG.The firstelement, qualitative axioms yielding a representation theorem, is cer-tainly not. DG operates entirely at theKlein-Stevens level: coordinaterepresentation of S is simply postulated, and the theory is based oninvariance properties (uniqueness and meaningfulness) relating theserepresentations to oneanother.

This is also why we have had to postponeexplicit statementof therepresentationalconditions C on the maps / for DG. Suchconditionscanonly beformulated withinapre-representationalframework havingthe capacity tomake statements about the system S which are entire-ly independent of numerical representation,and not merely invariantunder changes of numerical representation. (AsInoted in my (1992),this important distinction between statementsabout S which are trulyintrinsic and those which are merely invariant is generally overlookedby physicists.) It is precisely this step which defines the line betweenthe earlier Klein-Stevens approach and the RTM one, as explainedinmy (1986b). We now wish to take this step in the DGcase.


This will require: (a) identification of a suitable set of intrinsicprimitives P in the DG spaces S; (b) articulation of representationalconditions C on the maps / in terms of the primitives P; (c) formu-lation of axioms A on the spaces S using the primitives P; (d) proofof a representation theorem that for any S satisfying A there is a rep-resentation / satisfying G; and (c) proof of a uniqueness theorem thatthe class of representations / satisfying G are K-unique, where X isthe correspondingDG category. This of course includes as a specialcase the problem of axiomatization of the Riemannian STGs of GR,postponedabove.

This axiomatization problem was never articulated within the RTMliterature, probably because of the departure from the group-basedparadigm of representation which it requires.Iknow of only one pub-lished line of attack on it,originated within GR to provide a more con-crete interpretationof the physicalcontentofSTG.Thecentralpapersinthis development areEhlers,Pirani and Schild (1972) and Woodhouse(1973). The motivations underlyingthis work areprecisely thoseofmySection 3 above: the authors recognize that extrinsic coordinate repre-sentation conceals the intrinsic physical content of space-time theory,and they believe that synthetic axiomatization will make that contentclearer (Ehlers etal,1972, pp.64-65; Woodhouse,1973,p.495).

However, theseaxiomatizations donot meet the technical criteria ofRTMandaxiomatic geometry. Both axiom systems include statementswhich cannot be meaningfully expressed in terms of finitary primi-tive relations alone, but rather require the antecedent specification ofsome differentiable structure. For example,AxiomDl of Ehlers etal.begins: "Every particle is a smooth,one-dimensional manifold." Theterm 'smooth' implies a differentiable structure, and is so used in thesubsequent derivations, whereas with finitary relations (e.g. order orbetweenness)onecan only characterize the topological manifold struc-tureofa \-dspace,but notanyoneuniquedifferentiablestructure. Thusthere are no intrinsic finitary primitives in terms of which this axiomcanbe meaningful. Whatever the authors mean by their primitive term'particle', ifa particle is topossess (as theproofs require) &particular\-d differentiable structure over its constituentpoints, then itcannotbedefined structurally using any finite set of finitary relations. A simi-larprobleminvolving an illegitimate (because non-unique) shift fromtopological to differentiable structure affects Woodhouse's Axiom5.

92 BRENTMUNDY

These axiom systems only yield the desired results if they are readas primitively postulatingcertainelements of differentiable structure onindividual particles. However, a differentiable structure is itself onlydefinable usingnumerical representation,sothese formulations ineffectpresuppose part of the standard apparatus of numerical representationwhichatrue geometricaxiomatization shouldderiveinitsrepresentationtheorem. Therefore theyreallyonlysucceedinreducing onepartof thatapparatus to another part, not in reducing the whole to something trulyintrinsic.

However, it is in fact possible topresent intrinsic axioms in at leasttheRiemanniancase. Thepossibilityofdoing soisshown byaZeeman-type invariance proof given in the Appendix of my (1992). The qual-itative primitives involved are natural Riemannian generalizations oftheclassicalprimitives ofgeodesic betweenness B(p,q,r)andmetricalcongruence C(p,q,r,s). These are true finitary primitives, relationsof degree three and four, respectively, over the points of the spaceS, and the theorem says that any two Riemannian spaces which areisomorphic as structures (S,B,C) are related by a standard smoothRiemannian isometry. In terms of if-uniqueness, this means that the(5,B,C) structure of a Riemannian space Sdetermines its coordinaterepresentation as a numericalRiemannian geometryM,uniquely up toRiemannian isometry.

Wecannow alsofinally state explicitly therepresentationalconditionG on the maps /, for the Riemannian case, which is of course simplythat the qualitative relations B and G in S be carried under / to thecorrespondingrelationsnumericallydefinable intherepresentingspacesMusing thegivencoefficient functions: that triples s, t, v in Sbearingtherelation BliealonggeodesiesinMasdefinedusing the gpq functionofM,and so on.

This invariance result is analogous to Zeeman's proof thatthe quali-tative causalstructureofMinkowski STGdetermines itscoordinate rep-resentation uniquelyup toaLorentz transformation,exceptthat Zeemanconsidered onlyrepresentations in one numericalMinkowski spaceM,and hence stated only the conclusion of Lorentz G-uniqueness ratherthan generalif-uniqueness for a categoryK. Justas Zeeman's resultimplies the sufficiency of the causalprimitive for an axiomatization ofMinkowski STG, so this result implies the sufficiency of the twoprim-itives B and C for a synthetic axiomatization ofRiemannian geometry(of any dimension and signature, hence including the STGs of GR).I


also have some explicit axiom systemsusing these two primitives, butthey are toocomplex to statehere.Ishouldstress that theproblem of axiomatizationofDGis different

for each different type or level ofDGstructure: thereisno way to knowin advance what ifany setof finitary primitives willprovide asufficientbasis for theaxiomatization ofa givenlevel ofDGstructure. Aninvari-anceproof like thatof my (1992) shows thata sufficient basis for affinemanifolds is provided by the geodesic betweenness relation B(p,q,r)together with an affine congruence relation C(p,q,r,s), which givescongruence relations along each geodesic rather than across differentgeodesieslike the Riemanniancongruence relation. For this case tooIalsohave explicitaxioms.

At the other extreme, it is easy toprove that no basis of finitaryrelations suffices for the axiomatization of any level of differentiablestructure alone, without additional geometricalstructure. This followsfrom the richness of the set of transformations relating these spaces,which preventsany non-trivial finitary relations from beingpreservedby all such transformations,as the basis relations for a geometry mustbe.

This last result shows how unphysicalis the procedure,common inGR,offirstpostulatinganunderlyingdifferentiable manifold asifit weresomethingindependent andself-subsistent,over which someadditionalgeometrical structure may then be imposed as an afterthought. Ifonebelieves thataphysical theory shouldbe expressibleusing finitary rela-tions, this result implies that the differentiable structure of space-timemust be determined by its higher levels of geometrical structure, andcould not exist physically without that. (This conclusion hasinterest-ing connections toMach's principle.) Conversely, the axiomatizatilityresults quotedabove show that the metric or affine structure definablein terms of theprimitives B and Gis sufficient to yield thatunderlyingdifferentiable structure as well, just as in the classicalgeometries. (Ofcourse in classical geometry no one ever would have thought it neces-sary to postulate an underlyingmanifold before proceedingtodescribeaffine or metric geometry. This odd and unphysical way of thinkingentered STGfrom moderndifferential geometry.)

Finally, to illustrate once more the philosophical and physical sig-nificance of intrinsic axiomatization in the RTM tradition,Iwillbrieflymention thataxiomatic STGinthis senseprovidesaclear andsimple res-olutionofaparadox whichoriginally troubled Einsteinand wasrecently

94 BRENTMUNDY

revivedbyEarmanandNorton (1987),called the 'hole argument. Thisproblem is in my view entirely an artifact of the standard extrinsiccoordinate representationemployedby physicists and thephilosopherswho follow them,and disappears completely when the theory of STGis properly expressed,in intrinsic axiomatic terms (Mundy,1992).

9. CONCLUSION

This paper ought to contain one more section, combining the variousRTM-based relationist ideas of Section5 with the intrinsic approach tocurvedspatialist STGofSection8toexploretheprospects for relationisttheories ofcurvedSTG.On this important topicIcanhere only refer tothebrief hints of my (1989b,pp.595-596) for the case of second-orderscalar relationism,and theobservationof my (1983,p.226) that such atheory will have to incorporate elements of GR itself in order to haveany non-trivial content.

By way of generalconclusionIwish to stress that all of thepreced-ingmaterial is rooted essentially in one idea and one technique,bothoriginating mainly with Suppes, though in my view not yetpushed farenough.The ideais thatof informal theoryofphysicalquantity(mislead-ingly called a theory of 'measurement'), including physical geometryasa special case. The techniqueis the theory of representation: intrin-sic axioms, representation theorems, uniquenessand meaningfulness.Development and application of that technique inpursuit of that goalleads to intrinsic axiomatization ofquantitative theories,unified theoryof scalarand vectorquantities,relationist geometricaltheories,axiomat-ic differential geometry,andsoon.

DepartmentofPhilosophy,Syracuse University,Syracuse,NY13244, U.S.A.

NOTES

1TheprecedingidealizedcharacterizationofRTMandits accomplishments isdevelopedanddefendedinmy(1986b), whereIalsoundertake to fillan importantgapinthe RTMliteratureby showing in a substantive and non-circular way that a representationalpropositionp[f] ismeaningful ifit isinvariantunder theactionof theuniqueness group

QUANTITYREPRESENTATION AND GEOMETRY 95

G.Istandby thisclaim, whichis not addressed in thediscussionof meaningfulness inFM-3.

The argumentdependsessentiallyon treatingmeaningfulness as apropertyof indi-vidual representationalpropositionsp[f],as wasdonelongagoinSuppes (1959b). Incontrast,RTM takes meaningfulness as a property of the relations occurring as pred-icates in such propositions (FM-3, Ch. 22). My approach is more generalbecausea relationis meaningfuliff each instance of it is, whilenot every meaningfulpropo-sition is an instance of a meaningful relation, since a predicate may be meaningfulfor some instances butnot others. This difference in approachmay havehamperedcommunicationwhenIpresentedthematerialofmy (1986b) at an RTMconference onmeaningfulnessin1984; certainlyIdidnot thensee its full significance,ortheinherentlimitationsof the RTM approach. We willsee later that this differenceof approach iscrucial for theextensionof the theory of meaningfulness todifferentialgeometry andothercases wherenouniquenessgroupexists. These twosuccessesof thepropositionalapproachtomeaningfulness illustrateits advantagesover theRTM relationalapproach.

REFERENCES

Burgess,J.: 1984, 'SyntheticMechanics',JournalofPhilosophicalLogic,13,379-395.Catton,P. andSolomon,G.: 1988, 'Discussion:Uniqueness ofEmbeddings andSpace-

TimeRelationism',Philosophy ofScience,55, 280-291.Earman,J. andNorton,J.: 1987, 'What PriceSpace-TimeSubstantivalism? The Hole

Story',BritishJournalofPhilosophy ofScience,38, 515-525.Earman,J.: 1989, WorldEnough andSpace-Time,Cambridge,MA: MITPress.Eddington, A.S.: 1939, The PhilosophyofPhysicalScience, Cambridge.Ehlers,J.,Pirani,F. A.E.,and Schild,A.: 1972, 'The Geometryof FreeFallandLight

Propagation',in: L. O'Raifeartaigh(Ed.), GeneralRelativity: Papers in HonourofJ.L. Synge, Oxford: ClarendonPress, pp.63-84.

Field,H.: 1980,Science withoutNumbers,Princeton: University Press.Friedman,M.: 1983, FoundationsofSpace-Time Theories,Princeton: PrincetonUni-

versityPress.Horwich, P.: 1978, 'On the Existence of Time, Space and Space-Time',Nous, 12,

397-413.Krantz,D., Luce,R., Suppes,P., andTversky, A.: 1971,FoundationsofMeasurement,

VolumeI:AdditiveandPolynomialRepresentation,New York: AcademicPress.Kuhn,T: 1962, TheStructureofScientificRevolutions, Chicago: UniversityofChicago

Press. Secondenlargededition,1970.Luce, R. D.,Krantz,D. H., Suppes,P., andTversky, A.: 1990 FoundationsofMea-

surement, VolumeIII:Representation,Axiomatization,andInvariance,London andNew York:AcademicPress.

Manders, X.: 1982, 'On the Space-Time Ontology of Physical Theories',PhilosophyofScience,49, 575-590.

Moler,N.andSuppes,P.: 1968, 'Quantifier-FreeAxioms forConstructivePlaneGeom-etry', CompositioMathematica,20, 143-152.

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Moulines, C.-U.andSneed, J.: 1979, 'Suppes' Philosophy of Physics', in: R.Bogdan(Ed.), PatrickSuppes, Dordrecht: D.Reidel.pp.59-91.

Mundy, B.: 1983, 'RelationalTheories of Euclidean Space andMinkowski Space-Time',Philosophy ofScience,50,205-226.

Mundy, B.: 1986a, 'Embedding and Uniqueness in Relational Theories of Space',Synthese, 67, 383-390.

Mundy, B.: 1986b, 'On the GeneralTheory ofMeaningfulRepresentation',Synthese,67,391-437.

Mundy, B.: 1986c, 'The Physical Content ofMinkowski Geometry', British Journalfor thePhilosophy ofScience,37, 25-54.

Mundy, B.: 1986d, 'Optical Axiomatizationof Minkowski Space-Time Geometry',Philosophy ofScience,53, 1-30.

Mundy,B.: 1987a, 'FaithfulRepresentation,PhysicalExtensive Measurement Theoryand ArchimedeanAxioms',Synthese, 70, 373^-00.

Mundy, B.: 1987b, 'TheMetaphysics of Quantity',PhilosophicalStudies,51, 29-54.Mundy, B.: 1989a, 'Elementary CategorialLogic,Predicatesof VariableDegree,and

Theory of Quantity', JournalofPhilosophicalLogic,18, 115-140.Mundy,B.: 1989b, 'OnQuantitativeRelationistTheories',PhilosophyofScience, 56,

582-600.Mundy, B.: 1990, 'MathematicalPhysics and Elementary Logic', in: A. Fine, M.

Forbes,andL. Wessels (Eds.),PSA 1990, Vol.1, EastLansing,MI:PhilosophyofScience Association,pp. 289-301.

Mundy, B.: 1991, 'Discussion: Embeddingand Uniqueness inRelationist Theories',Philosophy ofScience,58, 102-124.

Mundy, B.: 1992, 'Space-Time and Isomorphism', in: D. Hull,M. Forbes, andK.Okruhlik (Eds.), PSA 1992, Vol.1,EastLansing,MI:PhilosophyofScience Asso-ciation,pp. 515-527.

Reichenbach,H.: 1924, Axiomatizationof theTheory ofRelativity(in German),EnglishtranslationUniversity of CaliforniaPress,Berkeley andLos Angeles, 1969.

Reichenbach, H.: 1928, The Philosophy of Space and Time (in German), EnglishtranslationDover,New York,1957.

Robb,A. A.:1911,OpticalGeometryofMotion: ANew ViewoftheTheoryofRelativity,London: Heffer.

Robb, A. A.: 1936, GeometryofTime andSpace,Cambridge:Cambridge UniversityPress.

Stevens,S. S.: 1946, 'On the Theoryof Scales ofMeasurement', Science, 103, 677--680. Reprinted in: A.Danto andS. Morgenbesser(Eds.),Philosophy ofSciences,New York: New AmericanLibrary, 1960, pp. 141-149.

Suppes, P.: 1959a, 'Axioms for RelativisticKinematics with or withoutParity', in:L. Henkin, P. Suppes, and A. Tarski (Eds.), The AxiomaticMethod with SpecialReference to Geometry and Physics, Amsterdam: North-Holland, pp. 291-307.ReprintedinSuppes (1969,pp. 194-211).

Suppes,P.: 1959b, 'Measurement,EmpiricalMeaningfulness, andThree-Valued Log-ic', in: C. W. Churchmanand P. Ratoosh (Eds.), Measurement: Definitions andTheories,New York: Wiley, pp. 129-143. Reprinted in Suppes, P., Studies in theMethodology andFoundationsofScience,Dordrecht: D.Reidel, 1969,pp. 65-80.

QUANTITY,REPRESENTATIONAND GEOMETRY 97

Suppes,P.: 1972, 'SomeOpenProblemsinthePhilosophyof SpaceandTime',Synthese,24, 298-316. ReprintedinSuppes (1973a),pp. 383-401. Suppes,P. (Ed.): 1973a,Space, Time andGeometry,Dordrecht: D.Reidel.

Suppes,P.: 1973b, 'Introduction',in Suppes (1973a),pp.ix-xi.Suppes,P.: 1974, 'TheEssentialbutImplicitRoleofModalConceptsinScience',in:K.

F.Schaffner andR.S.Cohen(Eds.),PSA1972, Dordrecht: D.Reidel,pp. 305-314.Suppes, P., Krantz, D. H.,Luce, R.D., andTversky, A.: 1989, Foundationsof Mea-

surement, VolumeII:Geometrical, Thresholdand ProbabilisticRepresentations,LondonandNew York: AcademicPress.

Suppes,P.: 1979, 'Replies',in:R.Bogdan(Ed.),PatrickSuppes,Dordrecht: D.Reidel,pp.207-232.

Teller,P.: 1987, 'Space-Timeas aPhysicalQuantity', in:R. Kargonand P. Achinstein(Eds.), TheoreticalPhysics in the 100 Years Since Kelvin's BaltimoreLectures,Boston: MITPress.

Winnie, J. A.: 1977, 'TheCausalTheory of Space-Time',in: J. Earman, C.Gylmour,and Stachel,J. (Eds.),FoundationsofSpace-Time Theories,MinnesotaStudiesinthe Philosophyof Science, Vol. VII,Minneapolis: University ofMinnesotaPress,pp. 134-205.

Woodhouse, N.: 1973, 'On the Differentiable and Causal Structure of Spacetime',JournalofMathematicalPhysics,14,495.

Zeeman, E. C: 1964, 'CausalityImplies the LorentzGroup',JournalofMathematicalPhysics,5,490^193.


Reading Brent Mundy's longpaper on the representational theory ofmeasurementrecalls, for me, in a vividway our manylively andargu-mentative discussions about the theory ofmeasurementand many othermatters in thephilosophy of science duringhis yearsas a graduate stu-dent at Stanford. For those who do not know Brent, the style of hisarticle conveys well his style of thinking. He writes well and clearlyabout a variety of technical matters. He has in fact a considerable giftfor expositionof technical concepts. Interspersed with this expositionare quirky andhighly idiosyncratic views. Nowhere else in the litera-ture onthe foundations of measurementwill you find anything like hisSection 5, entitled 'Materialism,Spatialism and Relationism'. On theother hand,byanalysing critically and with care the writings ofa num-berofphilosophers about space-timeandrelated topics,he has used theconcepts perfected in the last few decades in the theoryof measurementto criticize inadequacies in a variety of philosophical work. In spiteof this yeoman service,Ifind his views aboutmaterialism and realism

98 BRENT MUNDY

toodogmatic and sharply drawn for my taste. Inhischaracteristic wayIam sure he would reply that this is just an expressiononmy part ofmy piecemealand pluralistic approach tophilosophy, which toooftenavoidsthebigissues.Ihaverespondedto thisbroadsort ofviewpoint invarious other commentsin these three volumes. What Mundydoes notdiscussasanalternative to thekindof foundationalist view headvocatesis the instrumentalist andproblem-solving approach thathas its lineagein Peirce and Dewey as an alternative fundamental way of thinkingaboutphilosophical matters.Inow turn to some detailed remarks onMundy's paper.Ilist them

in their approximate order of occurrence inhisexposition,notin termsof their relative importance.

Eighteenth-Century Consensus. Ivery much agree with and likeMundy's point about the eighteenth-century consensus on quantitativemethodology. From aphilosophical standpoint, one thing that isimpor-tant to noteabout this consensusis that it tookplace before the founda-tionsof analysis hadbeenput inproperorder,orevenbefore theconceptof function had been satisfactorily clarified. What was apparent wasthat the methods worked and provided new anddeepinsights into theapplication ofmathematics to natural phenomena.

AncestryofRTM.At the beginningof Section 3 Mundy remarks thatRTMis essentially a theory of scaling with psychometric ancestrypri-marily, but the fundamental papers onthe representational theory werewritten completely outside the psychometric tradition and even beforeit was developed.Iam thinkingespecially of the fundamental work ofHelmholtz (1887) in thenineteenthcentury,andaboveall the importantpaper of Holder (1901). Moreover,without any question this traditionas reflected inHelmholtz andHolder was very much influenced by theslightly earlier geometricaltheory ofrepresentationas formulated clear-ly and philosophically in a very satisfactory wayby Pasch (1882) andlater given wide currencybyHilbert.

RepresentationTheoremsinGeometry. InSection 4Mundy discuss-esin anexplicit andstandard fashionrepresentation theorems for givengeometriesin terms ofCartesian orother numerical geometrical spaces.It is important to recognize,however, that early in the geometrical tra-


dition more general representation theorems were proved over fieldsthat were not numerical fields and in fact satisfied no Archimedean orcontinuity condition. Good modern examples are representations forweak affine spacesover ternaryfields orquasi-fields, whichdo notevenhave standard commutative and associative operations of addition andmultiplication. Early examplescan be found already in the nineteenthcentury as well. This is,in fact, avery richsubjectgeometrically withavast modern literature of representation theorems that are not properlynumerical in character.

Intrinsic Formalization of Physical Theories. Iagree with many ofMundy'sremarks about the desirability ofgivingan intrinsic formaliza-tion of physical theories, justas we do for geometry,but his objectionsto such extrinsic set-theoretical formulations are too strong, or at leastare not sufficiently developedin the present paper. Certainly for manyphysicalphenomenait would be very awkward to give suchan intrinsicformalization and from the standpoint of a very extensivefoundationaltradition it would seem rather bizarre in the case, for example, of sta-tistical mechanics, just toname one significant instance. Thesamecanbe said for other parts of physics in which very extensiveuse is madeofprobability theory. It is extremely awkward -and, in fact, not doneanywhereinpractice-to formulate complicatedphysicalprocesses thatare also stochastic processes in a purely intrinsic qualitative way froma foundational standpoint.

Finitism in Physics. AgainIagree with many of Mundy's remarksaboutfinitism whichIfindmore interesting than the discussion ofmate-rialism,butIdo wanttoemphasize my own viewpointonthesematters.IncreasinglyIlook upon the choice between discrete and continuousmodels as one driven by computational, and only computational con-siderations. There is no real way of deciding ifphysical quantities areactually continuous or actually discrete. Once a sufficiently fine dis-cretemesh isconsidered, discrimination is notpossible. Theone thingthat could turn out tobe the case is proof that there is indeed a lowerlimit anddiscreteness need not gobeyondthis lower limit toprovide acompletely satisfactory account. Letus suppose that this lower limit isroughlyon theorder of Planck's constantand weshall postulate that intermsof lengths it is of theorder of 10-40 meters. Even though we are

100 BRENT MUNDY

convinced of thisby experiment andbelieve it in our hearts, we wouldstillcontinue throughout large domainsof physics to use inunimpededfashioncomputationbaseduponassumptionsofcontinuity, notbecausewe believe the world was constructed this way,but because this wasthe only sensible way to do any elaborate quantitative analysis. Oneof the oldest and best examples of this is DeMoivre's proof that forlarge n thebinomial distribution is approximated very well by the nor-mal or Gaussian distribution of the same variance. (DeMoivre's proof(1733/1738) was only forp=1/2.) This kind of approximation wouldstill beused without even a shrugof the shoulders in a great range ofphysical applications. These computational aspects of matters are notemphasized atallbyMundy,andinmy view are toomuchneglectedbyhim.

GeneralizedTheory ofMeaningfulness.Ilike Mundy's developmentof differential geometry and the way he uses it to challenge standardconcepts of uniquenessand meaningfulness.Ido think the story aboutuniquenessand invariance in differential geometryis inpracticeagooddeal more complicated than his brief overview here suggests. Thestudy for exampleofsomethinglike infinitesimal affine transformationsin differential geometryhas been a major topic and is driven by thesameconsiderations of invariance that havedriven the standard theory.Even thoughIthink the developmentsin Chapter 20of Foundations ofMeasurement, Vol. 11l represent an excellent analysis of the standardtheory,asLuce and Narens alsopoint out in their article in this volume,there remain a number of problems of meaningfulness that are not yetsatisfactorily resolved. Mundyhas put his fingeronone setofproblems,namely, when there is nota naturalgroup expressinguniqueness of therepresentation.Iwantto point to somethingrather different, drivenby more physi-

cal and less mathematical sorts of problems. In the standard theoryofextensive measurement without anyplace for error, itis meaningful toask the question whether the ratio of the mass of the sun and earth isrational or irrational or whether, if it is irrational, it is transcendentalor not. Such questions seemby ordinary physical standards of experi-mentation and the like,nonsensical. The reason for that view is clear.Therepresentationasgiven in detail is notin terms ofaunique numberrelative toagiven scale,butin terms of adistribution representingerrorin the measurement. This seems to me a philosophically better and


more important direction to go in generalizing the theory of meaning-fulness than that taken by Mundy, without my needing to quarrel withthe correctness of many of his points.

If werepresentphysicalquantitiesby random variables,as for exam-ple in Suppes and Zanotti (1992), or in Chapter 16 of Foundations ofMeasurement, Vol.11, then each physicalquantity is representednot byauniquenumber butbyaprobability distribution. We can ofcourseaskthe samenonsensical mathematical questions about the transcendentalcharacter of the parameters of the distribution representing a physicalobjectbut wereally do notknow theseparameters in thismathematicaldetail. The parameters themselves are estimated from finite samplesof data with error terms of their own, and we have no strong convic-tion about the transcendental character, for example, of any estimatedparameterof anerror distribution.Idonot mean to suggest thatmovingfrom the representation of physical quantities by numbers to randomvariables solves all the problems of meaningfulness, but it does takecare of some the problems that do not arise in the purely mathemat-ical or geometrical theory of meaningfulness but that are natural toraise from a physical standpoint. Notice that such matters as the ratioof two masses has a natural expressionin terms of error distribution,although often not made explicit. That is,once we think of the massesof two objects represented as random variables then we will take asfirst approximation the ratio of the means of these distributions as theratio of their masses,but we willalso attachan error distribution to thisestimation. Ofcourse, whatIhavecallederror willin the caseof manynaturalquantitiesbe areflection of their natural variation over time, noterrors in their measurement. So, for example,it is natural to representthe diurnal and secular change in temperature of a given location asa stochastic process and thus as a family of random variables, or forsimple representation,as asingle random variable,changing with time.

AxiomaticSpace-Time Geometry. AgainIfindmyself agreeing withmuch that Brenthas tosay about theharder problems of axiomatizingproperly space-time geometry. He brings out well the complexities ofproviding an intrinsic axiomatization for generalrelativity.Ithink thedirection of the various unpublishedresultsof his own thathe cites is apractical one to obtain specific results for particular cases. Ialso thinkthat his strictures aboutbeginning with a differentiable manifold alonebeingaphysicallyunsound way to beginare correct. Itis just notclear

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whether anything like a genuinesynthetic tradition will ever get a verysolid foothold in general relativity, or more generally, in differentialgeometry.

To end on a note that is dear tomy heart and in certain waysmoreradical than Mundy's suggestion, let me take up again the idea of dif-ferentiable structures. If we take seriously the concept that we cannothope to distinguish physicallybetween discrete spaces with a very finemesh and continuous spaces, then to take seriously details of a dif-ferentiable structure seems physically unsound,unless a case is to bemade purely on computational grounds. But if thecase is to be madepurely on computational grounds then we would want to end up withtheorems showing that wehave selected a differentiable structure as apossible approximate structure just for computational purposes. Thisrequires abroadeningof the conceptofrepresentationandaskepticismabout taking seriously much of the mathematical apparatus developedfor manifolds. This seemslike aheterodox view that goes too far, andnot too longagoIwould have said so myself, but the modern moveto digital computation of solutions of all kinds of physical problemsformulated in terms of partial differential equations, etc., suggests thatthe mathematics of the future will be more serious in not acceptingas an unbridgable chasm the difference between the discrete and thecontinuous,or even more, the differentiable.

REFERENCES

DeMoivre,A.: 1733, Approximatio terminorum binomii(a +b)n inseriem expansi,pamphlet,London.Translatedandcommentedoninhis TheDoctrine ofChances,2ndcd.,London,1738.

Helmholtz, H. yon: 1887, 'Zahlen und Messen erkenntnis-theoretischbetrachtet',Philosophische Aufsdtze Eduard Zeller gewidmet, Leipzig, 1887. Reprinted inGesammelte Abhandl.,Vol.3, 1895, pp.356-391.

Holder,O: 1901, 'Die AxiomederQuantitatunddieLehre vomMass',Ber. Verh.Kgl.Sachsis. Ges. Wiss., Leipzig,Math.-Phys. Classe, 53, 1-64.

Pasch, M.: 1882, Vorlesungen iiberNeuere Geometric,Leipzig: Verlag yon JuliusSpringer.

Suppes, P. and Zanotti,M.: 1992, 'Qualitative Axiomsfor Random VariableRepre-sentationsofExtensiveQuantities',in: C. W. SavageandP. Ehrlich(Eds.), Philo-sophicalandFoundationalIssues inMeasurementTheory,Hillsdale,NJ: LawrenceErlbaum,pp. 39-52.

PAULHUMPHREYS

NUMERICAL EXPERIMENTATION

ABSTRACT.Iargueherethatthecomputationalmethodsofnumericalexperimentationconstitutea distinctively newkind of scientific method, intermediateinkindbetweenempiricalexperimentationandanalytictheory. Aparallelisalsodrawnbetweenextend-ing our senses with scientific instrumentsand extendingour mathematicalpowersbyusingcomputationalinstruments. A specificapplicationofthese methodstoIsingmod-els is describedindetail.

Throughout his career, Patrick Suppes has emphasized the centralityof probabilistic methods in modern science. He has also insisted onmaintaining a commitment to a rigorous scientific empiricism withinthe philosophyof science. Equally important is his repeatedinsistencethat we keep in mind the fact that real science is complex, messy,and highly sophisticated. We should not, as he often put it, indulgein 'philosophical fantasies' about the nature of science but addressourselves to the contentof actual science. Ithink thatPat never reallycured me of my tendency towards philosophical fantasies, but Ihopethat thispaper at leastcombines these threecomponentsofhis approachin a novel way.

I

Overthecourse ofthepastcoupleofdecades,computationalsciencehasbecome anincreasinglyimportantmethodinmany areasof thephysical,biological, psychological, economic and other sciences. Because ofthe wide variety of methods subsumed under the general frameworkof computational science, it is not easy to characterize the methodssuccinctly, but for the purpose of this paperIshall use the followingworking definition:Computationalscienceis thedevelopment,explorationandapplicationof mathematicalmodelsof non-mathematicalsystems using concretecomputationaldevices.

To flesh out this working definition,Iinclude in the class of con-crete computational devices both digital and analogue computers (i.e.

103

P.Humphreys (cd.), PatrickSuppes: Scientific Philosopher, Vol. 2, 103-121.© 1994 KluwerAcademic Publishers. Printedin the Netherlands.

104 PAULHUMPHREYS

analogue devices that are used for computationalpurposes andnot justas physical models). Humans are one special kind of computationaldevice,but the kind of novel philosophical issues thatIwant to dis-cuss here occur only when we move beyond the category of humancomputers. Moreover, it is important that the computation is actuallycarried out on a concrete device, for mere abstract representationsofcomputations do notcount as falling within the realmof computationalscience. To be counted as part of computational science proper, thecomputations must have a dynamic physical implementation. (I do,however,include computations run on virtualmachines in the class ofconcretecomputations.)

This workingdefinition is simple,but it already subsumes suchpro-cedures as applied finite difference methods, Monte Carlo methods,molecular dynamics, Brownian dynamics, semi-empirical methods ofcomputational chemistry,and a wholehostof other, less familiar meth-ods. For those of us who are interested in how theories are appliedtonature, the most important immediate effect of using thesemethodsof computational science is that a vastly increased number of modelscanbe brought into significant contact with real systems,primarily bycircumventing the serious limitations that our present restricted set ofanalytical mathematical technique imposes on us. These constraintsshould not be underestimated, for once we move past the realm ofhighly idealized systems, the number of mathematical theories that areapplicable using only analytic techniques devoid of approximations isvery small. Ishall not dwell on this aspecthere,but refer the reader toHumphreys(1991) for examples of thisanalytic unsolvability.

Many of the other features of computational science have no realinterest for philosophy, important though they are to the working sci-entist. But some aspects of this methodology give rise to questionsthat have clear philosophical content. Amongst these Ishall focuson two specific issues. First, it is often claimed that computationalscience provides us with a new kind of scientific method, one that iscomplementary totheexistingmethodsof theoryand (empirical)exper-imentation. Such claims are often rather vague about just what is newin these methods,ranging through a variety of claims such as: (1) thatcomputational science allows a reduction of the degreeof idealizationneeded in the models as well as a check on those remaining idealiza-tions that are made; (2) that they allow far more flexibility and scopefor changing boundaryand initial conditions thando real experiments;

105NUMERICAL EXPERIMENTATION

(3) that they are precisely replicable;(4) that many parameters can bevaried that could notbe altered by real experiments,perhaps becausesuch achange would violate a law of nature,perhaps because as in thecase of astrophysical systems the very idea of large scale experimentsis absurd;and soon.Ishallargue here that at least one specific aspect of computational

science does indeed introduce a genuinelynovel kind of method intoscience. That method is numerical experimentation, andIshall arguefor thisclaimwith reference to aparticularclass ofmathematical models-thelatticemodels ofstatisticalphysics;usingaparticular typeofcom-putational method- the Monte Carlo method; and applying a specificsolution procedure to thatmethod -theMetropolisalgorithm. Becauseof the wide applicability ofMonteCarlo methods within computationalscience, this example constitutes a lesser degree of special pleadingthanmight appearat first sight. Fritz Rohrlich (1991),followingNaylor(1966) and others, has similarly arguedfor the importance of comput-er simulations as akind of 'theoretical model experimentation. Theexamplesused here reinforce his perspective while exhibiting a some-what different aspectof computational experimentation, and they havethe additional advantage that they tie together two topics of perennialphilosophical concern,probability and empiricism, in a rather unusualway.

The secondfeature ofphilosophical interest involves aparallel that issometimes drawn betweenthe useofcomputational scienceand the useof scientific instruments. Whereas the latter enableus to transcend thelimitations ofourunaided sensorycapacities,theformer, so itis claimed,enableustoextend ourlimitedmathematical abilities,especiallyincaseswhere computational processesplay an important role. The degree towhich this parallel holds, if at all, is important at least because theinstrument sideof theparallel has hadprofound and well-known effectson thedevelopment of contemporaryscientific empiricism. We are allfamiliar with the arguments that were developedin the middle thirdof this century which produced serious difficulties for certain kindsof foundationally inspired empiricisms. Many things too small or toofar away tobe seen with the naked eye and things emitting radiationoutside the realm of the visible are now routinely considered to beobservable. For example, observing a cold virus under an electronmicroscope seems to many of us to be not only a perfectly legitimatepartof scientific practice,but one thatought to beacceptable to aliberal

106 PAULHUMPHREYS

empiricist. There is no need to rehearse here the consequences thatextendingthe conceptof 'observable' to include what is detectable byscientific instruments has had on empiricism. It is enough to remindoneself that they have beenprofound. So we must ask: if the parallelbetween computational science and instrumentation is soundly based,what consequences are there for how we should construe the use ofmathematical models in science? Ishall restrict myself here to theissue of how the parallel affects numerical experimentation, and inparticular, whatkinds of epistemological criteria are appropriate to thenew methodology. But enoughof the preliminaries. Letus turn to thedetails of the model we shallexamine.

II

Ferromagnetism involves spontaneousmagnetization ofmaterials suchas iron andnickel when the temperature is lowered below the Curie orcritical temperature Tc. The exchangeenergy involved in ferromag-netism is a specifically quantummechanical phenomenonmanifestingitselfat the macroscopic level,and oneof the centralapplications of thelattice models is to study this phenomenon.2 These models consist ofanra-dimensional lattice (m < 3), with thenodesoccupiedby particleshaving spin values Si. The Hamiltonian for the Ising model, whichdescribes a ferromagnet with stronguniaxialanisotropy is givenby:

Here Jis the exchangeenergyconstant,restrictedinitially tointeractionbetween nearestneighbours. If J > 0, then the pairs (up,up), (down,down) are favoured for neighbourpairs and this gives rise to the spinalignments thatproduce ferromagnetism. IfJ< 0, then the pairs (up,down),(down, up)are favoured,resulting in antiferromagnetism. Histhe external magnetic field, ft is themagnetic momentofa spin and thesecond termon theright is the Zeeman energy. When generalized to afully isotropic ferromagnetin the Heisenbergmodel we obtain

where

Rising=—J z2 &iSj ~ z2"%"ij i

"^Heisenberg —— J2_^(§i 'Sj) -pHz S?ij i

NUMERICALEXPERIMENTATION 107

Here,as opposedto the Isingmodel in which the spins have values+1or -1along thepreferredaxis,in theHeisenbergmodel the spins cantake on continuously many orientation values. Among the simplifyingassumptions used in these models are: (a) the kinetic energy of theparticles associated with the lattice site is neglected; (b) only nearestneighbour interactions are considered (although this can be relaxed toinclude nth neighbour interactions); (c) spins have only two discretevalues;(d) J,Hare considered tobe uniform (thiscan alsobe modifiedto allow randomexchangeconstants J{janddifferent magnetic fields ateach point in the lattice).

Contact with empirical data comes through the calculation of aver-age values for macroscopically accessible quantities, such as the meanmagnetization. In order to calculate these averages, the model is sup-plementedbyachoice of thermodynamicalensemble withits associatedprobability distribution over states of the system. A common choicehere is the canonical ensemble, i.e. an ensemble of closed systems inthermal equilibrium with a heat reservoir. Then for a (macroscopic)observable A and a system with continuously many degrees of free-dom, the thermalaverage is givenby

In the isotropic Heisenbergmodel, we have continuous degrees offreedom leading to aproper integral,but inmodels where there are onlyfinitely many degreesof freedom, such as theIsingmodel, the integralin (1) will,of course,be replacedby a summation.

The integration or summation is over the configuration space X,which in the lattice model has as points Af-dimensional vectors X =(5i,...,Sn) specifying the spin state for each particle in the lattice.This introduces in an explicit way one probabilistic aspect of the mod-el in that the normalized Boltzmann distribution exp[— H(x.)/kBT]/Zdescribes the probability of theconfigurationxin thermal equilibrium.

TherepresentationgiveninEquation(1) is familiar,tidy,andexplic-it. Yeteven given the severe simplifications involved in (a)-(d) above,

(Sf)2+ (Sf)2+ (Sf)2 = l.

(1) (A(x))T = Z~x [exp[-H(x)/kBT]A(x) dxJx

where

Z= / exp[-#(x)/fcfiT] dx.Jx

108 PAULHUMPHREYS

Equation (1) is impossible to apply in any realistic context. The inte-grals involved in the Heisenbergmodel are unsolvable in any explicitway,and although there exist analytic solutions to the Isingmodel forone- and two-dimensional lattices, there is no such analytic treatmentfor the three-dimensional lattice. Moreover, even in the discrete case,for very simple models oflattices with 102 nodes in eachspatial dimen-sion,there are 210 possible states to sum overin the three-dimensionalcase. Analytically intractable integrals produce one standardcontextinwhichMonteCarlo methods areused,andin employing these methodsweswitch froma treatmentin which adeterminate solutionis producedbyanalyticmethods to onein whicha statisticalestimate of thatsolutionis provided. To be clearabouthow probabilities operatein this context,itis essential toseparatethe fact thatwe beganwith aprobabilisticmod-el involving aprobability distribution for the canonical ensemble overconfigurationspace [x] from the probabilistic techniques that areused toproduceasolution to themultiple dimensional integral (1). We are hereconcerned with the second of these probabilistic aspects. Traditionalintegration methods,including thoseusing approximation methods, donot treat theseaverages as stochastic quantities tobe estimatedbyprob-abilistic methods. Incontrast,Monte Carlo methods do just that,andinfactMonte Carlomethods can be applied equally well to solvemodelsthat are based onprocesses that are themselves deterministic. A briefoutline of how these methods are applied is neededhere.

11l

The bridgebetween traditional methodsof integrationandMonteCarlomethods is providedby themean value theorem, givenhere for theonedimensionalcase: Ifv,v are continuous functions on [a, b] and v doesnotchangesign in the interval [a,b], then thereis an £ in [a,b] such that

6 b

/ u(x)v(x) dx =u(£) / v(x) dx.a a

Thus,when v(x) =1everywhere,we haveb

(2) / u(x)dx = u(Z){b-a).a

109NUMERICAL EXPERIMENTATION

Here w(£) is themean value of v on [a, b].Hence the point of Monte Carlo methods is to statistically estimate

the mean valueu(£). then to calculateb

u(x) dxa

using (2). Now suppose we consider the quantity u(X) tobe arandomvariable,and we sample valuesU{ = u(Xi)at random. Then

Finally, by appeal to the strong law of large numbers,one can justifyprobabilistic convergenceof the statisticalestimatoru(x) to the analyticmean valueu(£).

Now this argument clearly has nothing to do with computationaldevices; it is a result that derives solely from familiar facts about thecalculus andprobability theory. But there are two facts that make thisotherwise routine piece of numerical mathematics potentially interest-ing. Thefirst is thatbecause of the large dimension of the configurationspaceinvolved inthe latticemodels (andin mostother models withanypretence to realism) thecomputation involvedin(3) has tobe carriedoutin practice by resorting to computational devices using the techniquethat is our principal focus of attention here, numerical experimenta-tion. This technique can be illustrated by means of the Metropolisalgorithm. Within the generalprocedures for Monte Carlo estimators,drawing sample points from configuration space for multidimensionalintegrals in a non-random way often results in large portions of theconfiguration space beingsampled negligibly often. Using a uniformprobability distribution on the points of configuration space is usuallyinefficient,becausemanypoints of the spaceare highly improbable andcorrespondto negligibly small contributions to theestimator. Giventhelargedimensionalspace over which theestimation takes place,this willproducearate of convergencethat is far tooslow for the computation tobe carried out in practice. If, instead, thepoints in configuration spaceare selected by a distribution that mimics the profile of the functionthat is already beingused to weight the quantity whose mean we areinterested in,then this makes much more efficient useof the samplingprocedure.

N(3) u(x) =

i=\

110 PAULHUMPHREYS

Tosee this,suppose wehavechosen anensemble (here the canonicalensemble) thathas a probability distribution f(H(x)) over the energystates. Then,in simple sampling, we would approximate theensembleaverage for an observable A by

where x;is some element from phase space (e.g. the Af-dimensionalspin space [x; = (5i,...,Sat)]. But to avoid wasting time on ele-ments ofphase space with low probability, wechoosetheelements withprobability P(xi) to get

M M(A)

jA{xl)p-\xl)f(H{xl))lYjP-\xi)f{H{xi)).

(Note: this comes from choosing points according to the measureP(xi) dx instead of theuniformmeasure dx.)

Now in the case of Isingmodels, suppose we choose P(xi) tobeproportional to theequilibrium distribution f(H(x)), which here is theBoltzmann distribution. So, for example, if in the Ising model wechoosea sampling distribution

then the weightedaverage

reduces to

But inmany cases the exactvalues of theequilibriumdistribution

cxp[-H(x)/kBT]

are not known for a specific Hamiltonian, and so we would seem tohavemade noprogress.

M M(A) «X>(xO/(#(xO)/£/(tf(x,))

i=i ;=i

t=l t=l

P(x)= Z~x exp[-H(x)/kBT]

(A) Z&AMP-^expl-HM/kBT]

M=I=l


To circumvent the difficulty, the Metropolis algorithm constructs arandom walk in configurationspacehavingalimit distribution identicalto that of the sampling function P. If this were merely a piece ofabstractprobability theory aboutMarkov processesit would,of course,be nothing exceptional. The key fact about this method is that thelimit distribution is not computedby calculating apriori the valuesof the limit distribution. Rather, by running concrete random walkswithin numerical models of configuration space, the limit distributionsogeneratedis proportional to theequilibrium distribution.

The Metropolisalgorithm iseasily described. First,pick an arbitrarypointin configurationspace,Xq. Then,ifweare atpointX{,to generateamove in the random walk,chooseanew point in configuration space,Xj. Compute the transition probability w(Xi —► Xj) = r,for whichonecommon choice is theMetropolis function:

Next, generate a uniformly distributed random number md, and ifr> md let Xn+\ =V, otherwise let Xn+\

= Xn. Iterate the pro-cess for sufficiently many steps until equilibrium occurs. Then repeattherandom walk choosing adifferent initial starting point. Iterateuntila frequencydistributionof equilibrium stateshasbeen generatedby theequilibrium end states of the repeated random walk. (For a proof thatPeq(Xi) ~ w(Xi),seeMetropolis etal. (1953)).

Coupled with the transition probability is aphysically interpretabledynamics inconfigurationspace that generatesnew states fromold. Twocommon choices for the Ising model are the single spin flip dynam-ics, within which the spin at the selected point is reversed, and thespin exchange dynamics, within which the spin at the selected pointis exchanged with a neighbour. Putting this in the concretecase ofspin flip dynamics for the Isingmodel, an initial configuration for thelattice is chosen, and then a random lattice site is picked and the spinthere is flipped. If this results in a decrease in total energy, then theMetropolis transition probability (which in that case willbe unity) willchoose the new configurationand thenext stepin the random walk willbe generated. If,in contrast, the spin flip results in an increasein totalenergy, then the Metropolis transition probability is compared to theuniformly distributed random number, and if the transition probabilityis greater,the new configuration is accepted. Ifnot, the process retainsthe old configuration. What the Metropolis algorithm does, then, is

w(Xi-

Xj) =mm{l,exp(-[H(Xi)-H(Xj)]/kBT}}.

112 PAULHUMPHREYS

to get from an initial configuration to one in which (a) the energy isglobally, rather than locally, minimized and (b) the probabilities aredistributedaccordingto theequilibrium distribution.

IV

There arecertain distinctive features of numerical experimentation andthe Metropolis algorithm thatneedemphasis. First,the equilibrium dis-tribution that the algorithmhas as a limit is not in generalanalyticallycomputable in the sense that its values canbe calculated explicitly for aclosed form representationof the probability. Theonly way tocomputetheseprobabilities isby meansof the numerical random walk generatedby the Metropolis algorithm. Although thisrandom walk isamodelofaphysicalprocess, where thegenerationof the random numbers couldinprinciplebe carried outbyphysically indeterministic means, thusresult-ing in a traditional kind of empirical experimentationusing stochasticprocesses, themethod describedhere is carried outpurely numerically.It is this simulation bynumerical methods of physicalprocess that haslead to the term 'numerical experimentation. There is no empiricalcontent to these simulations in the sense that none of the inputs comefrom measurements or observations on real systems, and the latticemodels are just that, mathematical models. It is the fact that they haveadynamical contentdue tobeingimplementedon a realcomputationaldevice that sets these models apart fromordinary mathematicalmodels,because the solution process for generating the limit distribution is notoneof abstractinference,but is available only byvirtue of allowing therandom walks themselves to generatethe distribution. It thus occupiesan intermediate position betweenphysical experimentation andnumer-ical mathematics,being identical withneither.

Aquick comparison with the well-knownBuffon's needleprocessisilluminating. Although this process is perhapsbest known to philoso-phersfor havinginducedBertrand toinventhisparadoxesof equiproba-bility,Buffon's originalprocess isoften considered to be theprogenitorof empirical Monte Carlo methods. In its simplest form, the processconsists of casting a needle of length L onto aset of parallellines unitdistance apart. (HereL < 1.) Then,by an elementary calculation,onecan show that the probability that the needle will intersect one of thelines is equal toILJ-k. As Laplace pointed out, frequency estimates


of this probability produced by throwing a real needle can be usedto empirically estimate the value of n, although as Gridgeman (1960)notes, this is a hopelessly inefficient method for estimating -k, since ifthe needle is thrown once a second day and night for three years, theresulting frequency will estimate -k with 95% confidence to only threedecimalplaces. Be this asit may, the Buffon-Laplacemethodisclearlyempirical in that actual frequencies from real experiments are used intheMonte Carloestimator, whereas the standard methods for abstract-ly approximating the value of 7r using truncated trigonometric seriesare obviously purely mathematical in form. In contrast, the dynamicimplementation of the Metropolis algorithm lies inbetween these twotraditional methods. It is in this senseat least thatIbelieve numericalexperimentation warrants the labelof a new kind of scientific method.

v

This point needs further discussion because it can easily be miscon-strued. What of the fact that this is,after all,an algorithm within whicheven the random numbers are generatedaccording to a deterministicformula? Does this not mean that inprinciple, this is no different fromany traditionalmathematical modeland thatourinability tocarry outtherequisite computations ourselves,and thus delegating them to an artifi-cialcomputational device,is nothing buta practical concern andhencedevoidofphilosophical interest? Here theparallel withinstrumentationis relevant. It is natural to respond to this fact by appealing to an 'inprinciple'argument,and touse thecontingencyofourlimited computa-tional powers toargue that the inability to carry outsuchcomputationsin practice is philosophically irrelevant, no matter how important itmight be for thepurposeof applied science. Buthere it is necessary tomake a sharp distinction between on the one handphilosophical issuesconcerned with mathematics conceived as an autonomous disciplinedivorced from its scientific applications and on the other hand philo-sophical issues concerned with that subset of mathematics needed for,and sometimes developed inexplicit recognition of, its applications inscientific models. One couldconceive of this subset as merely beingapartof puremathematics,and that the philosophically important issuesoccur when considering therelation between themathematics andreal-ity. But that way ofviewingcomputational sciencehas the potential for

114 PAULHUMPHREYS

transferringphilosophical positions that are appropriate for the analysisof pure mathematics to the areaof appliedmathematics, where theyarenot so obviously appropriate. For us, the key concept is that of 'com-putable. When philosophers use this term, they almost always havein mind the purely theoretical sense centred around Church's Thesis.But the dispositional aspect of the concept 'computable' can also beconstrued as involving what can be computed in practice on existingcomputational devices. Consider now the parallel with the concept of'observable' mentioned inSectionI.Many of thedisputes withinempiri-cism about the conceptof observability hingedon whetherit should betaken as fixed or as changeable. Those minimalist empiricists whotook 'observable' to be whatever is detectable by theunaided sensesofhumans rejected the idea that what was observabledependedupon thecurrent state of technological enhancement of our sensory apparatus.For theseminimalists,the epistemic acceptabilityof thingsbeyond theimmediately accessible was a matter of whether the terms referring tothose perceptually inaccessible entities were reducibleby definition totermsreferring to things that were immediately accessible to the senses.Incontrast for the non-minimalists,what was observable was (at leastin part) a matter of what contingently available instrumentation wasavailable.

The situation with the conceptof 'computable' is in some ways theinverse imageof this situationregarding what isobservable.Minimalistempiricists hadaverynarrow conceptionof what is observable,whereasthe realm of what was viewed as observable by the technological non-minimalists wasmuch wider. With therevisedconceptof 'computable'the wide domain of recursive functions is drastically reducedin size toobtain the conceptof computable that isappropriate for technologicallyaccessible computational science. But within the domain of this tech-nologically enhanced conception there remains a parallel, for what isconsidered computable goes far beyond whatis actually calculable bythe limited powersof ahuman.

VI

Itshould be noted that there is adouble element of contingencyin thisappeal to actual computability. It is obvious contingentupon what iscurrentlyavailable tousin terms ofartificial computationaldevices,but


it is also contingent upon the complexity of the world itself. For, ifthe number of degrees of freedom involved in physical systems wasalways very small, we should not need augmentation of our nativecomputational abilities to employ the Monte Carlo methods describedabove. (We might need such augmentation to deal with other kindsof analytically intractable problems,but weare dealinghere only withthe case at hand. For some other cases where analytical intractabilityforces computational science on us, see Humphreys (1991).) But theworld is not simple in that particular way and in order to apply ourtheories to systems with large number of degrees of freedom, suchaugmentation is forced uponus. This double element of contingencyis not a reason to reject theposition for whichIhave argued. In fact itis exactly thekind of consideration that shouldplaceconstraints on anempiricist's epistemology, because what is acceptable to an empiricisthas to be influenced by the contingencies of our epistemic situationrather than by appeal to superhuman epistemic agents free from theconstraints to which we are subject. For in fact, an exactly paralleldouble contingency holds in the case of observability. How far theconcept of 'observable' gets extended beyond the realm of what isaccessible to theunaided sensesdependsnotonly upon the current stateoftechnological developmentin instrumentation,butuponwhatis in theworld. If the world were composedentirely of relatively close mediumsizedobjects devoid of microscopic internal structure, and itpossessedonlypropertiesperceivablewith ourfive senses,then wecouldnotarguewith theminimalist conceptionof observability. Butofcoursethe worldis not that way, and so the minimalists do have to defend their limitedconception of theobservable.Inote in passing here that van Fraassen's recent (1980) attempt

to return to a minimalist conception of what is observable is subjectto objections along the above lines. For van Fraassen, the moons ofJupitercountasobservablebecause we,aspresentlyconstitutedhumans,could see them directly from close up. But this possibility dependson technological advances just as much as does the observability (inthenon-minimalist sense) of a chargedelementary particle, which vanFraassendenies is observable. But why allow the useof technology inonecaseandnot in theother? Evenifit weresaid that the technologicalenhancement provided by space craft is an enhancement of a non-sensory faculty wehave,thatof locomotion,andis thus different inkindfrom theenhancementprovidedbybubble chambers, we could respond

116 PAULHUMPHREYS

that our ability to observe something directly is a relation between usand the object, and it simply happens to be a linear spatial relation inthe moons of Jupiter case, and a relative spatial size relation in theelementary particle case, ifwe are observing theparticle, that is,ratherthan its charge. But these are not clearly relations of adifferent type.

So, there isindeeda significantparallel thatcanbe drawnbetween theissues involvedin instrumentational enhancement andthoseinvolved incomputational science. Themost immediate conclusion to draw is thatin dealingwith issuesconcerning theapplication ofmathematical mod-els to the world,asempiricists we shoulddrop theorientationofan idealagent who is completely free from practical computational constraintsof any kind, but not restrict ourselves to a minimalist position wherewhatis computable is always referred back to the computational com-petenceof human agents.If, asmany minimalist empiricists believe,itis impermissible to argue thathumans could have had microscopes foreyes or could evolve into such creatures, and it is impermissible to soargue because empiricism must be concerned with the epistemologicalcapacitiesof humans as they are presently constituted, then it ought tobe equally impossible to argue that inprinciplehumans could computeatrates 106 faster than they actually do. But that is justnot plausible.

The position for whichIhavearguedshouldbe keptseparate fromanapparently similar issue that has been much discussed in the literatureon connectionist architectures in artificial intelligence. One of the firstapplicationsof theMetropolis algorithm was to theprocessofsimulatedannealing, where a liquid that is cooled slowly ends up in an orderedstate corresponding to an energyminimum. (In contrast, if the liquid iscooledquickly, itendsupina state with ahigherenergy level.) Thesim-ulatedannealingmethodhas alsobeenemployedby connectionists,forexampleinHopfieldnets, to generaterelaxationschedules for multipleconstraint problems.3 Because it is often asserted thatoneof the majordifferences between connectionism andclassical (rule-based) artificialintelligence lies in the fact that the latter employs algorithms that areexplicitly computable,whereas the former employsnon-linear functionsthat are not, and in consequence systems have tobe allowed to settlethemselves into the appropriateoutput state,it might seemthat there isa similar lack of computational transparency in the Ising models. Butthis isnot so. Although there are interestingproblems posedby theuseofcontinuous analog computational devices, the MonteCarlo methods

NUMERICAL EXPERIMENTATION 117

described here are all computationally transparent, implementable ondigitalmachines andgive finite approximations to thelimit distribution.

ACKNOWLEDGMENT

Iam grateful toFritz Rohrlich for comments that helped improve thispaper. The preliminary work for thepaper was done under NSF grant# DIR 8911393, and an earlier version was read in June 1992 at theVeniceconference onProbability andEmpiricismin theWork ofPatrickSuppes.

DepartmentofPhilosophy,University of Virginia,Charlottesville,VA 22903, U.S.A.

NOTES

1 Those uninterested in the technical details of these models can move directly toSection IV, referring back as necessary. It is not possible to fully understand whynumerical experimentationis forced on us without a grasp of the physicalmodels,however.2 The expositionof the formalaspectsof latticemodelsgivenhererelies heavilyon thetreatmentin(Binder andHeerman, 1988).3 See e.g. (Rumelhart andMcClelland, 1986, Ch. 7).

REFERENCES

Binder,K. andHeerman, D. W.: 1988, Monte CarloSimulation inStatisticalPhysics,Berlin:Springer-Verlag.

Gridgeman,N.: 1960, 'GeometricProbabilityandtheNumbertt',ScriptaMathematica,

25, 183-195.Humphreys,P.: 1991, 'Computer Simulations',in: A.Fine,M.Forbes, andL. Wessels

(Eds.),PSA 1990, Vol. 2,EastLansing: Philosophy of Science Association.Metropolis,N., Rosenbluth, A., Rosenbluth, M., Teller, A., andKeller, E.: 1953,

'EquationofStateCalculationsforFastComputingMachines',/.ChemicalPhysics,6,1087ff.

Naylor,T.H.: 1966, Computer SimulationTechniques,New York:John Wiley.Rohrlich, F: 1991, 'Computer Simulationsin thePhysical Sciences', in: A.Fine,M.

Forbes, andL. Wessels (Eds.), PSA 1990, Vol. 2, East Lansing: Philosophy ofScience Association,pp.507-518.

118 PAUL HUMPHREYS

Rumelhart, D. and McClelland, J.: 1986, ParallelDistributedProcessing, Vol. 1,Cambridge,MA:MITPress.

VanFraassen,B.: 1980, TheScientific Image,Oxford:The ClarendonPress.


Paul Humphreys is undoubtedly right in emphasizing the increasingimportance of numerical experimentation, an importance sufficientlygreat as towarrant the label for a newkindof scientific method thathasbeen introduced. Italsoseemsapparent that it is going to be some timebefore we have a stable typology of the different ways of approach-ingnumerical experimentation. On the one hand,numerical methodsfor solving differential and integral equations alreadyhad arespectablebeginning inthe nineteenth century,but the serioususeofMonteCarlomethods hardly predates the significant use of such methods by yon

NeumannandUlaminthe 19405. The generalrecognitionof the impor-tanceof finite element methods for solving complexphysicalproblemsis even more recent. Stillmore recently, there is hope that therapeuticdrugsof the futurecanbe designeddirectly bylarge scalecomputationalanalysis ofmolecular structures. Instillanother direction, the statisticalanalysis of large bodies of data has been completely transformed bythe possibilities of large scale computations. In fact, it is fair to saythat no large scale statistical analysis of data takesplace today withoutextensiveuse of numerical computational facilities.Ialso like theparallel that Paul draws between the use of computa-

tional methods and the use of scientific instruments. Modern scienceis wholly dependent upon high technology for both instruments andcomputations. Current theoretical and experimental work would beunthinkable without them. It is worth noting,however, that in philo-sophically simple concepts of rationality much more attention is paidto observation than to computation which is,Ithink, an emphasis thatis mistakenand has been only partly addressed by ahandful of seriousarticles. Extensivecomputation, asanyone familiar with the Americantaxcode wellknows,canbeas important toindividual rationalbehavioras it is tomodern scientific research.

Thereare threeconceptualremarksIwant to make about numericalexperimentation. Theydo notgo against the grain of what Paul has tosay,but areintended toamplify ourperspectiveonfuture developments.


Computationand Complexity. The groundbreaking work of Turing,Churchandothers establisheda sharp dividingline between that whichis computable and that which is not. This is one of thegenuinelynov-elmathematical results of the twentieth century, but the distinction isnot the right one for almost all numerical experimentation. We mustdraw conceptual distinctions within that which is computable and thistheoryin itselfis closely related to Kolmogorov's theoryof complexity.A familiar example tophilosophers is Tarski's decision procedure forelementary algebra and geometry. This procedure is not feasible, inthe sense that it is exponentially explosive as the size of formulas tobe decided as true or false increases. Within computer science we nowhave the standardnotion of afeasible computation, that is,one that canbe made in polynomial time or space. But even this general notion offeasibility is probably not really satisfactory for down-in-the-trenchescomputation, when in many cases something close to linear,not justpolynomial, constraints is needed. There is no questionabout the con-ceptual importance of the distinction between polynomial as opposedto exponentialconstraints as lower bounds,but whether the constraintsbelow polynomial willplay acorrespondinglycritical conceptualroleinour thinking aboutnumerical experimentation is probablynotyetclear.

NumericalUnease. Itseems to me that the kind ofdetailedmodelpre-sentedby Humphreysin the secondpartofhispaperon ferromagnetismis typical of what we shall see in the future and it undoubtedly hasalready created and will continue to createa great sense of conceptualunease about the nature of our scientific theories of complex phenom-ena. We simply will not be able to computeeven themost elementarycases in an analytically closed form. Almost everything willconsist ofnumerical computations in one form or another. The highly tentativeand experimental character of much of the results will consequentlybring about rather sharp psychological changes in the attitude towardtheory. This can be seen very clearly in Sparrow's (1982) numericalanalysis of the now famous Lorenz equations in chaos theory. This isnotnumerical analysis in the traditional sense but experimentalmathe-matics,or touseHumphreys' phrase, numerical experimentation in themodern sense. The Lorenz equations define a three-dimensional sys-temof ordinary differential equations dependingon three real positiveparameters. What isimportant is that different valuesof theparameters

120 PAULHUMPHREYS

radically change the behavior of the solutions. The Lorenz equationsare especially of interest because for certain values of the parameterswe get what we now call chaotic behaviour. What is important aboutthis kind of example is that in spite of the apparent simplicity of theequations we are able only to understand and study arelatively smallrange of values of the parameters with any thoroughness. Even thoseexplorations are necessarily tentative in character.

IfSparrow's work and others' of asimilar sort,much more orientedtoward practical problems, are any guideline, it may well be that thegreattradition of nineteenth-centurymathematical physics-to focusona smallnumber of carefully-selectedproblems whose solutions can bewritten down in closed form -may be coming to an end. The future,at least within the present range of mathematical and computationaltechniques, may be much messier and inconclusive in the analysis ofnatural phenomena. If so,history will record the period running fromEuler to LordRayleighasoneof the greatperiodsofintellectual history,comparable to the developmentof astronomyin Hellenistic times. Weshall not return to that paradise of elegant mathematical analysis butcertainly we can lookback upon it with nostalgia.

Will Theories Change? But more than questionsof nostalgia are in-volved. Itis easy to trace the decayof geometricalmethodsusedstillinsuchbeautiful form by Newton,as the powerof the calculus took over.The samemaybe trueof thedifferential equationsand integralequationsthat have been the mainstay of mathematical methods in the physicalsciences since the eighteenth century. If the actual solutions are reallygoing to be computed in terms of discrete models or by probabilisticapproximations of a similar sort, then it would be a natural predictionthatas thesemethods takeover, increasinglythe old waysofformulatingtheories will disappear. New waysmore congenial to the modern styleof computing rendered necessary by the complexity of the problems,will change the shape of theory itself. AsIhave already remarkedindiscussion of the future of theories of measurement, the continuummay disappear asan object of great interest,and the same may be truefor the use of the continuum of real numbers in the theoryof physicalphenomena.

ButIdonot mean tosuggest that the useofcontinuous mathematicswill vanish. The interplay will surely be more subtle than that. Letme just give one simple example, the approximation of the binomial


distribution by the normal distribution for n of any size. The asymp-totic correctness of this approximation for p = 0.5 was proved at thebeginningof theeighteenthcenturyby DeMoivre andhasbeenextraor-dinarily useful ever since. Even though the normal density is itselfnot integrable, it is most convenient to compute the normal deviatesrather than to compute the binomial distribution for large n, which istechnically unfeasible. Insimilar ways the host of special functions inmathematical physics will have acontinued active use. What has real-ly changed is thekind of philosophical or ontological commitment tonaturereally beingnotonlycontinuous but almost everywheredifferen-tiable. Therole of the continuumof realnumbers andall theassociatedmathematics will continue tobe of great importance in various specialways, but only for computational purposes. This is a conjecture thatgoes against the grain for many people,but it is at least reasonable tomakein view of the massive changes in the way scientific theories areactually applied to empirical data.

One addendum to this story. Ofcourse theapplications ofmathemat-ically formulated theories to data have always been more complicatedthanphilosophers of science would like torecognize. This iseasily ver-ified from close examination of much scientific work in the nineteenthcentury. Letmegivejustoneexample. Thereallycomplicatedcharacterofcomputations, bothanalytic andnumerical,required to solveany realproblems in the classical theory of waveoptics was alreadyrecognizedandstudiedcarefully in the last century. LaPlace's massivecalculationsinhis great work CelestialMechanics providea muchearlier example.This list is easily extendedinmany different directions.

REFERENCE

Sparrow,C: 1982, TheLorenzEquations:Bifurcations, ChaosandStrangeAttractors,New York: Springer-Verlag.

PARTIV

THEORY STRUCTURE

RYSZARD WOJCICKI

THEORIES ANDTHEORETICALMODELS

ABSTRACT. The central question discussed in this paperis howempirical theoriesare related to the empirical phenomena for which they are supposed to account. Iargue thateach applicationof a theory to anempiricalevent or an empiricalstate ofaffairsrequires forminga theoretical(mathematical)modelfortheproblemsought tobesolved. Differentproblems mayrequireformingdifferentmodelsrelevant to the samephenomenon. The wayin whichtheoreticalmodelscanberelated tosemantic modelsis examined.

In the context of the above ideasIsurvey some of Suppes' views on theories,set-theoreticalmodels,and theroleof bothinempiricalscience.

1. RULES OFCORRESPONDENCE

A shortreminder of some tenets of logicalempiricism can beuseful forthe following considerations.

For the logicalempiricist tointerpret a theory Tin order to makeitapplicable toempirical phenomena was to define how touse the princi-plesof the theory in order to derive from them therightconclusions ontheobservable states of affairs. Since the descriptive terms ofa theorymaynotrefer to observableentities, the derivation mayrequire appeal-ing to rulesofcorrespondence,alsocalled coordinatingdefinitions,i.e.extratheoreticalprinciples whose task is torelate the statementsof thetheory to the hypotheseswhich can be decidedby observation.

Ifthenotionofobservability is interpretedinanorthodox way, famil-iar from the writings of representativesof logical empiricism, notablyCarnap, in the task of reducing theoretical knowledge to observationalknowledgeone encountersserious difficulties which,as theempiriciststhemselves werebound toadmit, cannotbe overcome.

Rather than discuss the ideas of logical empiricism in their origi-nal form,concerning therelationbetween theoretical andobservationalsentences,Ishall focus on the relation between theoretical claims andexperimentalhypotheses, the latter being any sentences which a com-petent experimenter can decide by means of some reliable empiricalprocedures. How sophisticated those procedures are and how much

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involvedare the theories theypresupposes,does not matter. Measuringadistance with the aidof alaserbeamcan be agreatdeal more reliable,not to say more accurate, than measuring it with the help of a meterstick. The final judgment on whether the experimenter is competentand theprocedure is reliable belongs to the scientific community.1

SinceIhavedecided toreplaceobservability byexperimental decid-ability, a rule of correspondence has to be defined to be a postulatethat relates the theoretical statements to the experimentally accessible(rather than observational) statesof affairs.

The switchfrom observability to experimentaldecidabilityhas somefar reaching consequences which in no way could be accommodatedto the orthodox empiricists' doctrine. Modern empiricisms, notablyvan Fraassen's constructiveempiricism, are much more flexible in thisrespect. Most importantly while the observationally accessible worldwasassumed tobe fixed onceand forever (to beobservable wasdefinedto be observable for any 'normal' observer, regardless of what couldbe his orherknowledge, training or past experience) the experimentalaccessibility of specific states of affairs or eventsdepends in the mostintimate way onboth the competence of theexperimenterandonher orhis technical possibilities.

2. PATRICK SUPPES'SET-THEORETICAL APPROACH

Even though someof the analysescarried outby the logical empiricistscan be viewed as semantical, the predominant orientation of logicalempiricism was syntactical. The investigations were carried out interms of sentences and deductive relations among them. For a logicalempiricist, the selection of the syntactical option was a matter of bothdoctrine andhis theoreticalpossibilities. Modern semanticsbegins withAlfred Tarski'scelebrated treatise on the notion of truth.2 But althoughthe semantic approach gained momentumrather quickly, it became aviable alternative to the syntactical one only after a series of furtherinvestigations which eventuallyresulted in transforming general ideasof logical semantics in a full-fledged mathematical discipline - thetheory ofmodels. This remark applies both tometamathematics andphilosophy of science. Suppes' research program for philosophy ofscience based on ideas drawn from the theory of models was the firstfully matureand viablealternative to the received view.3

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Unlikehispredecessorswhodefinedtheories to be deductive systemsformalized within the idiom of first order logic, Suppes argues that anempirical theory, in fact any theories whatsoever, should be viewedas being determined by the class of its realizations rather than that ofits valid sentences. Thus,if for the logicalempiricists theright way todefine anempirical theoryT wasto defineasetofaxioms from whichallthe other sentences valid in T are logically derivable, Suppes suggeststhat todefine T is todefine aset-theoreticalpredicate that denotes all theset-theoretical structures of which T is true in the Tarski sense. Recallthata realization(also referred to asa semanticmodelor justmodel) ofa theory is just a structure of which the theory is true.

The question of whether theories are classes of semantic models(non-statement view) or sets of sentencesattracted much attention andhasbeenbothwidely andvividly discussed.4Foranoutsider,theessenceof this discussion may not be clear. After all,according to an old butstill robustdoctrine, theories are neither sets of sentencesnor classesofset-theoretic structures. They are sets of propositions which one mayrepresenteither by the sentencesapplied to communicate them, or byclasses of possible worlds (thus semantic models) of which they aretrue. If one deals with formalized theories, i.e., ones for which boththe notion of truth and that of logical derivation are well defined,onecan equally well define a theory either to be the set of all sentencesderivable from a set of some initial principles (axioms), or to be theclass of all the realizations determined by exactly the same principles.As Suppeshimself states clearly (cf., Suppes, 1967, Ch. 2, p. 24) theessence of his set-theoretical approach "is to add axioms of set theoryto the framework of elementary logic,and then to axiomatize scientifictheories within this set theoretical framework." This by no meanspresupposes that the axiomatization should result in a non-statementreconstruction of the theory. One who sticks to the traditional linguisticoption can use exactly the same axioms by which the set-theoreticalpredicate has been defined to define the corresponding class of validsentences. What matters is the set-theoretical framework not thechoiceof either 'statement' or 'non-statement' option.

All these points granted, Suppes' set-theoretical approach was aradical turn inphilosophy of science. To think ofa theoryas anetworkofsentenceslinkedby therelation ofdeducibility or to thinkof itasasetofclaims that concern a specific class of structures is to chosebetweentwo dramatically different perspectives. The semantic shift originated

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bySuppeswas, toappeal to afamiliar Kuhnian idea,ashift ofparadigmwith all theconsequencesof such a step.

One mightexpect that the semantic turnoriginatedbySuppes wouldautomatically focus our attention on theproblem of the intended inter-pretation of empirical theories. Actually, Suppes' approach does notrequire any investigations into theissues characteristic ofsuch an inter-pretation. In order to define the class of set-theoretical structures ofwhicha theory is true- thus true relative these structuresbut notneces-sarily true in the absolute sense of the word-oneneednot know whatthe theory is intended tobe about. As was made explicitby Tarski, therelative notionof truthdoesnot coincide with theabsolute one. Onehastoknow theintendedinterpretation inorder to define thelatterbut nottheformer. Thus,regardlessof how surprisingit canbe,Suppes' semanticconsiderations are irrelevant to the problem of factual interpretation ofempirical theories.

One of the basic tenets of Suppes' analysis of empirical theorieswas to the effect thatboth empirical theories and mathematical onesaredeterminedby their realizations andnothingelse. According to Suppes(1967,Ch.2,p. 29)...there is no theoretic way of drawing a sharp distinctionbetween a piece of puremathematicsand a piece of theoreticalscience. The set-theoreticaldefinitions of thetheory of mechanics, the theory of thermodynamics, and the theory of learning, togive three rather disparate examples, are on all fours with the definitions of purelymathematicaltheories of groups, rings, fields, etc. From a philosophical standpointthereis no sharp distinctionbetweenpure and appliedmathematics, in spite of muchtalkto the contrary.

This standpoint calls for some comments. Tobegin with,it may notbe as radical as it looks like at the first glance. After all, Suppes doesnotmaintain that thereare nodifferences between empirical disciplinesand mathematics. What he denies is the existence of any systematicdifference between the two which can be grasped in rigorous terms.Even so,he is skepticalabout theexpediencyofanyattempt to separateempirical theories frompurely mathematical ones and to view them asbelonging to twodifferent areas of theoretical studies.

Suppes'position implies (andIdo not seehow onecould avoid thisconclusion) that philosophyof sciencereduces to selected problemsofmetamathematics. Indeed,ifwe arenotable toexaminein asystematic,rigorous manner the differences that separate empirical theories frompure mathematics, then any theorizingabout the former must collapseto theorizingabout their mathematical aspects, and thus,in fact,it must


collapse to metamathematics. The view thatphilosophy of science isimpossible as a separate discipline was explicitly held by Tarski (seethepreface tohisIntroduction toMathematical Logic).

The coincidence of Tarski's and Suppes' tenets is not accidental.Even though it departs from many dogmas of logical empiricism,Suppes' philosophy of science belongs to the logical trend originat-edby the logicalempiricists. Now, for logical empiricists, regardlessof the quarter they belonged to-ViennaCircle or, like Tarski, Lwow-Warsaw School, or any other- the only acceptable way of theorizingwas that which could be expressed in a fully precise language, andthe only precise language was that of logic and mathematics. On thispointneither logicalempiricists nor their successorshave beenready tocompromise.

For someone who restricted his interest to formal problems, theidea of a factual interpretation of an empirical theory appears as acertain vague heuristic notion which, escaping any formal treatment,escapes any theoretical investigations. Certainly, he would argue, weform empirical theories toaccountfor the empiricalphenomena,but ourintentions cannotbesubjectedtoany formal studies. What really countsis the product of our intentions not the intentions themselves. And thelatter are theories. Now, in order to characterize a theory adequately,one has to learn what are its realizations. Only after this is done mayone ask whether, besides abstract set-theoretical structures, the classof the theory's realizations contains also some empirical systems andthus whether one mayuse the theory to account for the statesof affairscharacteristic of the system. Thus,insteadofdealing with an enigmaticideaof a factual interpretationof a theory,oneis advised to look for theempiricalapplications the theory admits.

3. ADAMS' AND SNEED'SDOCTRINE OFINTENDED APPLICATIONS

In what follows the idea of a model of a theory which has some realobjects as its parts, andhence can be adomain of a factualapplicationof the theory, will be our special interest. The following passage fromSuppes (1960,p. 291-292) can serve as both a concise and instructiveintroduction to the topic.To define a model formally as a set-theoretical entity, which is of a certainkind oforderedtupleconsistingofaset of objects andrelationsandoperationsonthese objectsisnot to ruleout the kindof physical model whichis appealing tophysicists, for the

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physical model may besimply taken to definethe set of objects in theset-theoreticalmodel. ...We may axiomatizeclassicalparticlemechanics in terms of five primitivenotionsof a setPof particles,an intervalT of realnumberscorrespondingto elapsedtimes,a positionfunctionsdefined on theCartesianproduct of theset of particles andthe timeinterval, amass functionmdefinedonthe setof particles,andaforce functionf defined on theCartesianproduct ofthe set of particles,the timeinterval, andthe setof positiveintegers (the set of positive integers enters into thedefinitionof the forcesfunctionsimply inorder toprovidea methodofnamingforces). A possible realizationof theaxiomsof classicalparticlemechanics, that is,of classicalparticlemechanics, isthenanorderedquintupleP= (P,T,S, m,f). Amodelofclassicalparticlemechanicsis such anorderedquintuple. Forexample,in thecaseofthesolarsystem wesimplycantake theset of particles tobethe setof planetarybodies. Another slightlymore abstractpossibility is to take the set of particles to be the set of centers of mass of planetarybodies. This generallyexemplifies the situation.The set-theoreticalmodelof a theorywillhaveamongitspartsabasicset whichwillconsistsof theobjects originallythoughttoconstitute thephysical model.

Needless to say, as long as an empirical theory is not provided withany factualinterpretation, itremains merelyacertain formal system.Butinspite ofSuppes'skepticism onthematter,which wementionedbrieflyin theprevioussection,onemay wonder whether thedifferentiaspecificaallowing us to tell anempirical theory fromapiece ofpure mathematicsdoes not consist in the fact that the former has some intendedempiricalapplications. This is exactly the line of thought which was pursuedbyAdams (1959).

He suggestedthat while a purely mathematical theory can be ade-quately representedby the class of all its realizations,a theory meantto have a factual contentmust be representedby two elements. One ofthem is again the class of all the theory's realizations. Itforms, to usethe expression coinedby Sneed (1971) the formal core of the theory.Theother, whichis characteristic ofempirical theories only, is the classof all the intended applications of the theory which is meant to be theclass of all empirical structures (physical systems) of which the theoryisexpectedtobe true. Ingeneral,the theorymay fail theseexpectations;an intended applicationmay notbe arealization of the theory.

Adams' ideaof intendedapplications was modifiedbySneed. Whilefor Adams,anintendedapplicationofa theoryTisapossible realizationof this theory, i.e., it is a structure of the same set-theoretical type asany of the realizations of T, Sneed requires that no component of anintended application can be 'theoretic relative to T\ that is,one whoseuse presupposesappealing to some of the principles of the theory. Forexample, according to Sneed, mass is a theoretical conceptrelative to

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Newtonian mechanics,for in order tomeasure mass, onemust resort toNewton's SecondLaw.

Clearly, neither Adams nor Sneed have denied that the users ofan empirical theory can revise their views on which of the empiricalsystemsare legitimate intendedapplications of the theory. One can tryto improve an empirical theory both by changing its principles and bydefininganew its scopeof applicability.

Another idea relevant to the present discussion is Sneed's idea ofan empirical claim. Suppose the formal core M(T) of an empiricaltheory T is defined by a set of axioms. These conditions are someformulas (e.g., mathematical equations) which, as long as we abstractfrom the intentions of its users,are notstatements; they donot refer toany specific state of affairs. The situation changes if one claims thattheconditions whichdetermine M(T)are true of aparticular empiricalsystem cr, in other words one maintains that a is one of the structuresin M(T). An empirical claim is just a statement to this effect. Notethat an empiricalclaim need notconcern an intended application of anempirical theory; it may refer to anyphysical system whatsoever. But,clearly,defining a theory to be thecouple (M(T),J(T)) is tantamounttoaccepting all the empiricalclaims of the form 'cr is in M(T)' wherea is in /(T).

Onecanfind theAdams-Sneedamendment toSuppes' conceptionofthe formalization ofempirical theoriesboth self-explanatory anddesir-able. And, indeed, they were wholeheartedly accepted by the struc-turalists - philosophers who, grouped around Sneed and Stegmuller,undertook the task of scrupulous investigations into a 'formal archi-tectonic for science'.5 Still, lam going to argue that the amendment,as well as the structuralist research program based on them, divergerather substantially from themain ideas of the Suppes approach to theproblems ofphilosophy of science.

4. WHY NOT APPROXIMATETRUTH?

No scientific theory is just true about the empirical systems to whichit applies. The reason why it must be so is fairly obvious. Empiricalsystems are, to appeal toPeircean metaphor, 'nebular. They are nev-er fully separable from their environments; moreover they consist ofobjects whose properties are not uniquely determined and may not beexactly such as is required by the theory. Thereby, if a theory applies

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to any such system it applies toit in an 'approximate' way.Apparentlythen, what the scientists shouldattempt to reach isapproximate truthoftheir claims rather than truth in any plain senseof the word. The lattercanmerelybe viewed as an idealized counterpart of the former.

An immediate conclusion from the above observation seems to bethat inorder tomake the doctrineofintendedapplicationsmore realistic,the idea of an empirical claim to which it appeals should be defined interms ofapproximate truthrather thanplain truth. Exactly this step wastakenby the structuralists,who inmuch of their investigations devotedthemselves to the issue of approximate correspondence between therealizations and the intended applicationsof theory.

The idea of approximate truth may seem not tobe very involved.Actually it takes someeffort todefine approximate truth inasufficientlyrigorous way.6 Still,in the present discussion, we can safely restrictourselves toexaminingacommon sense version of this concept. Fromthe intuitive point of view a statement is approximately true, if thedifference between what it asserts and the actual state of affairs is'small. The question which immediately comes up is 'small underwhichrespect?';needless tosay, evenasmallerrormayyielddrasticallyfalseconclusions and thus fatal consequences.

On the other hand,it is a fact of life that we systematically base ouractivity onsome 'nearlytrue' hypotheses.Wecandothisbecause,inanyordinary circumstances, we are able to tell the acceptable conclusionswhich a nearly true sentence justifies from unacceptable ones. Whatreallymattersisnot the 'distance' ofahypothesisfrom truth but the factthat it allows for true conclusions. Even a patently false sentence canbe useful in this way.

But if so, wehad better giveup the misleadingidea of approximatetruth. The muchdebated fact that scientific theories dramatically fail tobe totally true of the states ofaffairs to which theyrefer,does notimplythat they are totally false. In other words, theories are partially true.More accurately, theyare true in certain selected respects; ifapplied intheright way, theyallow us toarrive at trueconclusions. This isnot thedeceptive notion of approximate truth,but rather thatof 'partial truth',orstill better 'relative truth', which weneedin order to account for howempirical theories refer toempirical phenomena. Inthe sections whichfollowIexamine the above idea in some detail.

THEORIES AND THEORETICAL MODELS 133

5. THEORETICAL MODELSOFEMPIRICAL PHENOMENA

An application of an empirical theory T, by itself or combined withsome auxiliary hypotheses, to an empirical phenomenonIIin order tosolvea specificproblem Q concerningnmay require the formation ofa theoreticalmodel (one mayprefer to say mathematicalmodel) of thephenomenon, thus itmay require definingan abstract systemM. meantto satisfy the following two conditions:(ml) M is a realization of T;(m2) M is a faithful representationofnunder all the respects that are

relevant toQ. Thus if A is the solution to thequestionQ, then Ais factually true, i.e., trueofnif andonly ifA is true ofM.

Note thephrase 'meant tosatisfy' whichappears in the abovedefinition.It may be desirable to state the definition of a theoretical model inpurely formal terms, thus avoiding any pragmatical references. But,in the context of the present discussion, the pragmatical option hasan immediate advantage - we need not discuss separately what thetheoreticalmodels are and what they are for.

If, instead of thequestionQ,one is interested in answeringanotherquestion Q' concerning the same phenomenon, it may turn out thatthe model M does not meet the faithfulness requirement with respectto the new application; it does not yield an adequate solution to Q',and therefore another modelM!of nshould be constructed. Thus, atheoretical model ofnis not somuch amodel ofn,but amodel ofanaspectU/Q,11/Q,...of thephenomenonor justamodelforaproblemQ,Q',...Also notethat theremay notbe any obvious way to associatenwith a specific singular empirical system.

An example can help clarify the above points. Suppose someonewants todetermine theposition of Mercury at a specific time t, say atsome fairly remotefuture time. Denote theproblem thus defined byQt.Actually,besides stating theproblem Qt,onehas to state the postulatedparameters ofaccuracyandreliability the solution totheproblem shouldsatisfy. But,for the timebeing, thisneednotbe ourworry;Ishallreturnto this point later.

One may rush to point out that the problem Qt concerns the solarsystem of which theplanet isapart. But,in fact, what weknow for sureis that Qt concerns Mercury's movement. Consequently, rather thanpresuppose in advance that Qt concerns a specific system of physicalbodies,one must take into accountall the factors whatsoever of which

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one has somerational reason tobelieve may affect the movementof theplanet. The proviso 'of whichone has some rational reason tobelieve'cannot be dropped out. We neither see the point of examining, norwe are able to examine,all theadhoc hypothesesonecan put forwardseekingto account for the movementof theplanet.

Suppose our knowledge of the matters relevant to Qt is roughly thesame as that of the nineteenth-centuryphilosophers of nature. Thus,we have some fairly sophisticated,even though inadequate,idea of thenatureofspace andtime. We are awareof variousphysical propertiesofcelestialbodiesand wehave someideaof their spatialconfiguration. Wehavegatheredalarge quantityof evidenceconcerning their movements,especially those of some planetsin our solar system. And on topof thatwehave somepowerful theories: Euclidean Geometry, EG, meanttobethe theory of physicalspace,Newton's Law of Universal Gravitation,LG, and Newton's Particle Mechanics, NM.This fairly large body ofbeliefs, combined with some ontological ideas on the structure of theuniverse,form the world view within whichour search for the solutionto Qt mustbe carried out. Still, the discussion which follows will beahistorical invarious respects.Most importantly,Ishallpresupposethattheconstruction of the theoretical model for Qt involves a fairly largeamountof statistical considerations based on some notions unfamiliarto the nineteenth-century scientists.

Tobegin, wenote that weare bound toaccept the default hypothesisto the effect that the only source ofMercury's movementsis the set ofgravitational forces. This is not to say that we are not able to figureout that besides gravitational forces there are other factors which arerelevant to the planet's motion, but we have no evidence which couldallow us to conjecture their nature and then state the conjecture in theform of a reasonablehypothesis.

Byasimilardefaultargumentwearriveattheconclusion thatonly thebodies in the solar system can influence themovementofMercury in adetectable way;the gravitationaleffects of allthe remainingonesare toosmall for the solutionof Qt to dependonthem. Moreover, the effects ofgravitational interaction between Mercury and various elements of thesolar system, say planetoids, satellites of Mars, and even most remoteplanets, can also benegligible small. Thus,we eventuallyarrive at theconclusion that in order to solve Qt, it suffices to take intoaccount thegravitational interactions amongafew bodies of the solar system whichare the most massive and nearest toMercury. Which of such bodies

135THEORIES AND THEORETICAL MODELS

have actually to be taken into account depends on how accurate andhow reliable the solution is postulated tobe.

The two requirements cannotbe separated from one another. Themore accurate the solution is postulated to be, the narrower is thelimitoferror weadmit for the solution and the lessreliable is the solution weareable to findout, i.e., thelower theprobability that the actualpositionof Mercury is within the admitted limit of error.7 Clearly, one is ableneither to experimentally findout the recentposition ofMercury,nor totheoretically predictany future onewithunlimited accuracy; the ideaofunlimited accuracy is justempirically senseless.

Ithas tobe noticedthat the two requirements are formal in the sensethat an evaluation of whether the solution satisfies them or not is amatter of statistical analysis of all the steps of the procedureby meansof which the solution has beenachieved. Fromtheactualpointof viewthe solution iseither factually adequate(thus true) or not.

Suppose, after defining the postulated parameters of accuracy andreliability for the solution to Qt,and after examining how strongly themovement of Mercury is affected by the bodies of the solar system,we have eventually arrived at the conclusion that the model for Qtcan be fairly simple: it can be defined to involve only the Sun a andMercury p, both being treated as 'particles' or, as we prefer to saytoday, 'mass points'. Since the gravitational interaction between thetwobodies is assumed to fully determine the movementofMercury, themotion equations should be derivable from the union N = NM + LGof the principles of Newtonian particle mechanics, NM,and the law ofuniversal gravitation, LG.

Let NM denote the equations of Mercury's motion derived from N.Observe that the formulas in N^ differ from those inNinthat theyreferto two specific 'masspoints', theSun andMercury, while theprinciplesofNare statedinthe formofageneral schemaofequationsapplicabletoany setofmasspoints whatsoever.Inorder for theequationsNM tohavea unique solution, the values of some of the parameters the equationsinvolve should be either established experimentally or deduced fromthe available experimental data with the help of the hypotheses weconsider tobe well confirmed, the laws of N included. Let E be theset ofconditions one has to add toN^ in order to assure theuniquenessof the solution; let us call them factual conditions. As will be clearsomewhat later, only a part of E states the initial conditions, thosewhichcharacterize the stateof the two bodies at agiven initial time to.


Tomakealong story short, wehavetodo two things. First, we mustempirically determine the motion of Mercury for some initial periodof time At, and then deduce the conditions E from the assumptionto the effect that this determined fragment of the planet's movementsatisfies theprinciplesN.Itshouldbe noticed that to some extentwearefree in choosing the factors which are tobe determined by the factualconditions E; all we have to be sure is that N^ +E admits only onesolution to Q^.

Suppose we selectE to define sixparameters: the mass, the 'initialposition' and the 'initial velocity' of both the Sun and Mercury. Infullaccordance with thenineteenth centuryworld view,twoof them,namelythe initialposition andthe initial velocity of theSun,canbe assumed tobe miliary. Actually, in order to make this assumption consistent withN, wemust view the Sun tobe 'almost' immobile, i.e., we mustassumethat themovementof theSun, whatever itmay be,doesnotresult in anyeffect relevant to the solution to Qt. Furthermore, note that instead ofcalculating both the massof the Sun and thatof Mercury,it will sufficethat we can calculate one of them, say that of the Sun, relative to theother. In this way we are left with three constants to find out via ananalysis of theinitial motion: the position, xo ± Ax,and the velocity,vo ± Ay,of Mercuryat the 'initial' time to and the relativemassof theSun,ma ± Am. The limitsof error Ax,Ay andAmmust correspondtosomepresupposedreliability of the relevant hypotheses.Letus dwellon this issue for a while.

Note that the data whichcharacterize themotion of the planet withinthe examined initial period At cannot be just a direct inductive gen-eralization of astronomical observations. The reasonis fairly obvious.The frame of reference relative to which we observe the movement ofcelestial bodies is the Earth. Consequently,inorder to know how any ofthesebodies,Mercury inparticular, movesrelative to the Sun, wemustcalculate theirmotion with thehelpof therelevant transformations fromthe dataconcerning the apparentmovementof the Sun. Certainly,Iamnotgoing tosurvey theprocedure; the technical aspects of itare oflittleinterest for us. Rather,Ishall briefly discuss acertain methodologicalpoint.

Denoteby Qa* theproblemofdeterminingMercury'smotion withinthe period At. There seems to be somethingparadoxicalin that in orderto solve the problem Qt one has to solve a problem which actually isa variant of it. In fact, the procedure is by no means paradoxical. Itis


fairly obvious that in order to apply the laws thatgovern the dynamicsof an empirical phenomenon to a specific instance IIand a specificproblem Q concerning this instance, one must learn the conditionscharacteristic of11, i.e. ones whichmay serve to distinguishIIfrom theother instances of thephenomenon. In this way onearrives at anotherquestion Q' concerningn. Since the factual conditions whichare to bedetermined by solving Q' are relevant to Q, the two questions Q andQ' are interdependent, and often fall under the same general category.This is exactly the case of Qt and Q&t-

It is fairly obvious thatunless we are going to find ourselves in thesituationof regressumad infinitum,themethodology ofsolving theQAtcannotbe the same as that for Qt. When dealing with the former, wecannot apply the laws of N in any way which requires appealing tofactual conditionsE.In fact, this is theonly restriction to be observed;note that it does not imply that the laws of N cannot be useful indealing with Q&t in any way whatsoever. Actually, since we assumethat Mercury's motion satisfies N, we require the solution to Q&t tobe consistent with NM. This amounts to imposing a rather essentialconstraint of the conditions of adequacy of the solution to Q&t, thusnarrowingthe spectrumof all the acceptable solutions at whichonecanarrive by the statistical analysis of the empirical data. In what followsthe solution will be denoted by E& t \ it goes without saying that theequations of Mercury's motion for theperiod At of which it consistscanbe neither fully accuratenor fully reliable.

The reader familiar with Suppes' (1962, 1974) notion of model ofthe data, as well as his analyses concerning this notion,has certainlynoticed that the procedures discussedabove for solvingQ&t are typicalfor forming a model of the data. Themodel of the data thus produced isthe two-body system whoseelements are theSunandMercury,assumedto satisfy the conditions Q&t- One can use the following rather non-standard but self-explanatory notation

{{a,p.] :EAt)

tostand for this defined system. Thefact that formation ofa theoreticalmodel presupposes formation of a model of the data as well as thefact that formation of a model of the data can be controlled by therequirement of consistencyof the model with the corresponding theoryareofkeysignificancefor properunderstandingof theinterplaybetweenthe data and the theories,and thus for proper accounting for both the


corrigibility of the dataand falsifiability of the theoretical claims. Iamnot going to examine these issues,however; they gobeyond the scopeof thispaper.

Suppose the factual data Eare given. Thenthe model for Qt is thestructure Sdeterminedby the following two conditions:(i) all the elements of S are a and fi;(ii) S is a realization of theconditions N^,E.

In what follows,rather than explicitly stating conditions of the form (i)and(ii),Ishalluse the already familiar notation:

Needless tosay,({cr, //,} :N +E) isbut oneof numerous theoreticalmodels of the movement of Mercury one can form, and which havebeen formed. Recall that for a longperiod of time, the planet was ofspecial interestbecause of thepeculiarities of its perihelion movement.Seekingto explain them, scientistsnot merely analyzedmodels which,besides the Sun, involved some of the remaining large bodies of thesolar system, they alsoconstructed models presupposingthe existenceof some hypothetical factors affecting the movement of the planet.Thus, for example, Leverrier postulated a group of planets revolvinginside the orbit of Mercury. Another hypothesis related the observedirregularities of the planet's movement to the allegedoblateness of theSun.8 Incidentally, an oblateness of the Sun's mass distribution rulesout the possibility ofrepresenting theSun bya single mass point.IhopeIhave succeeded inpresenting theidea ofa theoretical model

in a sufficiently clear way. Still some further commentson it can beuseful. Tobegin with let me point out that theoretical models are notempirical systems, unless we consider them to be empirical systemsof some hypothetical or possible worlds. The reason is quite obvious.Even though the objectsa model involves are supposed to be real (suchas e.g., the Sun and Mercury in the model discussed above), still theseobjectsare postulated to satisfy certaintheoretical assumptions (suchas,e.g.,the lawsof NM)which theymay notactually satisfy. A theoreticalmodel is just a certain theoretical construct, an abstract entity,even ifsome of its parts appear to be parts ofphysical reality.

Since theaboveconsiderations were precededbya criticalappraisalof the Adams-Sneed doctrine of intended applications,Ishould men-tion that theoretical models are not the right candidates for the role of

S= {{cr,»}:N +E).


intended applications in the Adams-Sneed sense. It is not so muchbecause they are not empirical systems- they can certainly be viewedas some idealized representations of the latter -but because, by thevery definition, they are realizations of the theories intended to referto those systems. On the other hand, an intended application in theAdams-Sneed sense is postulated tobe a structure which may turnoutto be a realization of the theory in question. This possibility is the gistof the doctrine of intended applications, for it is a formal counterpartof the fact that an empirical theory may happen to fail to account ade-quately for empirical states of affairs within the scope of its intendedapplications.

6. THEORETICAL VS. SEMANTICAL MODELS

While the notion of a theoretical model cannotbe accommodated toideas of structuralist philosophy,it certainly fits directly and in a mostnatural way the Suppes research program. The passage from Suppes(1960)Iquoted at thebeginningofSection 3 canserve as astraightfor-wardpieceofevidence supporting thisclaim. Asonemay easilynotice,thephysicalmodels discussedinthe quotationare just theoreticalmod-els in the sense of this paper.

In this sectionIdigress from the main topic of this paper. Still,Ibelieve,Ishould not avoid discussing the problemIam going toaddress, for discussing it may contribute to a better expositionof somemethodologicalpoints relevant to Suppes'conception ofmodel. Iwishto comment upon Suppes' claim that of the two notions of a model,namely the set-theoretical and the physical one (or ifIam tostick to myterminology - the semantical and the theoretical one) the former canbe viewed as more fundamental than thelatter. One of the passages inwhich thispositionis expressedis the following (Suppes,1960,p.291).In the preceding paragraphs we have used the phrases 'set-theoreticalmodel' and'physical model. There would seem to be no use in arguing about whichuse ofthe word 'model' is primary and more appropriatein the empirical sciences. Myowncontention is that the set-theoreticalusageis the more fundamental. The highlyphysicallymindedor empiricallymindedscientists who may disagree with this thesisand who believe that the notion of a physicalmodel is themore important thing in agivenbranchof empiricalsciencemay still agreewithmy systematic remarks.

Rather thanchallenge the opinion that 'theconceptofmodelusedbymathematical logicians is the basic and fundamental conceptofamodel

RYSZARD WOJCICKI140

needed for exact statementin anybranchof empiricalscience' (Suppes,1960,p.294),Iamgoing toargue thatthe questionof which of the twoconceptsof model is more fundamental can be examined from quite adifferent angle than that takenby Suppes.

There is no question that, onmany occasions, the search for a the-oretical model for a question Q is the search for a semantical modelof a specific theory T such that this particular model will be the rightone to examine the question Q. This is the situation which obtainswhenever onebelieves that T is the theory which allows for solvingQ.If the formation ofa theoreticalmodel is from the very beginningmeantto consist in selecting the right semantical model then, of course, thenotion of a semanticalmodel is prior to that of a theoretical one. Thediscussion intheprevious sectionconcerned theabovementioned case;Newtonian Particle Mechanics was viewedas the right theory to solvethequestion Qt concerningMercury'smovement.

However, there are numerous situations in which the search for amodel for Q starts whenone doesnotconsider any theory tobedirectlyapplicable to Q. Bohr's model of the atom, whichin fact was a modelfor some specific problems on the dynamics of the particles of whichthe atom was believed tobe composed, was certainly not meant to bea semantic model of any official theory; rather it was meant to be acertain tentative theory of the phenomenon. A long list of examplescan be produced to support the above observation. And, of course,there is a rather substantial difference between the situation when amodel is, as the scientists would say, 'derived from' a theory and thesituation in which a model is a step towards formation of a theory.In the latter case there is no theory (among the theories known at agiventime) from which the theoretical model in question isderived, sothe model cannotbe intended to be a semantic model of any officialtheory. The construction ofa theoreticalmodel precedes any semanticconsiderations andin this senseit is prior to the latter.

The above discussion can be summarized as follows. The idea of amodeliscentral for two, inasense,oppositeactivities: applyingtheoriesand forming them. The interplay between the two is what largelydetermines the dynamic of science, thus its successive transformationsin time. Now, while the conceptof semantic model is central for theformer, the theoretical model iscentral for the latter.

Suppes seems tobelieve thatthe triadic relation 'theoretical model-theory-semanticmodel' canbereduced to the dyadic relation 'theory-

THEORIES ANDTHEORETICAL MODELS 141

semanticmodel. Thisis whatIdoubt,andIdoubtit because the 'archi-tectonic of science' touse theexpressioncoinedby the structuralists,byno means can be adequately accounted for in terms of formalized theo-ries and formal relations between their semantic models. The structureofscience isin acontinuous processof formations,and thusanyattempttoaccountadequately for it mustbebased onananalysis of the functionwhichspecificunits are supposedtoplay,rather thanon any taxonomypresupposed in advance. If we take this perspective, the question ofwhether a theoretical model has been intended to be a semantic modelof a theory, or has been intended to be an embryo of a theory to beformed does really matter.

The present discussion provides a good occasion to note that themodel for a singular concrete problem concerning singular concretephenomenon is but oneof many kindsof models one encountersin sci-ence. Questions onemay seek tosolvebyconstructingamodel for themcan be both singularand general,both factual andpurely hypothetical.Another thing tobe noticedis the distinctionbetween theoreticalmodelsand theories may notbe as sharp as it is suggestedby some tentativedefinitions offered in the course of the above considerations. Let usdwellon the last point.

Consider themodel ({<7,fi} :N^),one which results from themodel({cr,n} :NM +E) for thequestion Qt byremovingthe initialconditionsE.Themodel thus formedisamodel for the two-bodyproblem. Clearly,theconditions imposedon this model define it merely partially; all thequestions concerning it which cannot be solved without resorting tothe factual conditions E are left open. The question Qt we have beendiscussing is but one of this kind.

A structure defined by the conditions which do not determine thebehavior of its elements is apartially defined structure, and hence itactually is the class of all those structures which satisfy theconditionsin question. As such,it determines a theory just in the set-theoreticalsense proposed by Suppes. What is more, the theoretical model canusually be identified with the conditions by means of whichit has beendefined. Consequentlyitcanbe viewedasatheory inthe linguistic senseof the word. Itis worthnoticing that thispossibility is fully exploitedinscientific discourse. Inall scientific disciplines in which the conceptoftheoretical modelis in use,it is customaryto shift freely from the set-theoreticaluse of the term 'model' toits linguistic counterpartand viceversa. Thus,dependingon circumstances,a model is viewed as either

RYSZARD WOJCICKI142

acertain mathematical entity (e.g., a curveinEuclidean spacemeant tobe a model of motion of aphysicalbody) or theclass of correspondingmathematical equations. This explains why the term 'model' is oftenused interchangeably with 'theory. Thus, for example, the physicistwillconsider of minor significance whether oneprefers to speakabout,sayBoyle'stheory ofideal gasor ratheraboutBoyle'smodelofan idealgas.

One may consider this stateof affairs deplorable, for certainly fromthe logicalpointofview asetofsentencesandaset-theoretical structurethat the sentencesare supposedto characterize are of entirely differentlogical type. But the difference between them matters only in somerather specialcontexts which are of interests for the logicians,but theyare oflittle interest for working scientists.

7. SOMECONCLUDINGREMARKS

Theproblemofhow empirical theoriesare related toempiricalphenom-enahas twoaspects. Only oneof themhasbeen discussedinthis paper.Iwanttomention the other in order to localize the ideasof thispaper inasomewhat moregeneral context.

An integral partof any empirical theory is its factualinterpretationdetermined by the relevant world view. One cannot understand anempiricaltheory and thus onecan know neither what the theoryis aboutnorhow to form a theoretical model for specific problems relevant tothe theory unless one has some idea what is the part (or aspect) of theworld to which the theory refers, how this part is related to the others,which is the ontology of all these parts, and how both the claims ofthe theory and theempirical data on which it is based arerelated to theentities whoseexistence theontologypresupposes.

Itshould beclear that the factual interpretation asunderstoodaboveis global, in the sense that it relates the theory to the universe as awhole. As such,it differs radically from any local interpretation,whichisdefined to be a specific theoreticalmodel formed to be amodel of thetheory.

The fact that theories are not autonomousunits of science,and thusthe fact that both to understand and apply them one must conceiveof them as parts of some larger setting, has been fully appreciatedbycontemporaryphilosophyofscience. Thisisnot to say thatphilosophersof science are of the same opinion onhow this larger setting is to be


defined. For Kuhn (1970) this is aparadigm, for Lakatos (1978) thisis a researchprogram,Laudan (1977) calls it a research tradition, tomention only a few.

Thispaper is not the right occasion to discuss the idea of the globalinterpretation ofempirical theories. Ihavementioned this topic mainlybecause ofits obvious andimmediate relevance to themain issueof thispaper. Still,partofmy concern wasalsoSuppes'claim to theeffect thatthere is no systematic difference between theoretical science and puremathematics. Such a difference,Ibelieve, cannotbe grasped in termsof the local interpretations of the theories. If it can be grasped at all, itcanbe graspedby resorting to theidea of factual interpretation.

Needless to say, the factual interpretation of a theory determinesits factual applications. On the other hand, in learning the factualapplications of a theory (or rather some paradigmatical examples ofthem) one acquiressome knowledge of its factual interpretation. Thusthe twocharacteristics of anempirical theory are related to oneanotherin adirect and strong way. Still, in this paperIhave dealt with factualapplications only. In the concluding remarks that follow,Icommentonthose few notions of factualapplication whichhavebeenmy specialconcern.

For logical empiricists afactual application of a theory consistedinderiving someobservational hypothesesfrom theprinciplesof a theory,the relevant rules of correspondence, and the available observation-al evidence. Nobody has ever attempted to reconstruct in a full andsystematic manner the rules of correspondence which allegedly wereassociated with empirical theories. Or rather, every attempt to recon-struct these rules resulted in a successive modification of the idea ofempirical meaningfulness,and thus theidea of what these rules shouldlook like. The swan song of logical empiricism was Carnap's TheMethodologicalCharacter of Theoretical Concepts.

Theenormous work done by the structuralists resulted in some mostvaluable case studies of the formal structure of scientific theories andformal aspects of interrelations between them. But, asIhave beenarguing, the doctrineof intendedapplicationsunderlying the structural-ist approach cannot work. It provides not merely an oversimplifiedaccountof the relation between theories and empirical phenomenabutit is based on some ideas which are substantially wrong. At this junc-ture,Ihavetomention thatexactly the samecriticism shouldbe directedagainstmy Topicsin theFormalMethodologyofEmpiricalSciences,the


monograph in whichIexamined some ideas based juston the doctrineof intended applications. Investigations into the theory of approximatetruth were a partof thismonograph. Thus the viewIpresent andadvo-cate in this paper is a radical departure from some ideas to whichIsubscribed earlier.9

A concise, though somewhat metaphorical, exposition of Suppes'position on how theories are related to empirical states of affairs iscontained in the following quotation (Suppes,1967, Ch.1,p.12):The concrete experiencethat scientistslabel an experiment cannot itselfbeconnectedto a theory in any completesense. The experiencemust be put througha conceptualgrinder thatinmanycases is excessivelycoarse.Once theexperienceispassed throughthegrinder,whatemergesare theexperimentaldatain canonicalform. These canonicaldataconstituteamodelof experiment,anddirectcoordinatingdefinitions areprovidedwith thismodelof theexperimentrather foramodelof a theory.Itis alsocharacteristicthat themodelof the experimentis of relativelydifferentlogical type fromthat of themodel of the theory. It is common for the modelsof a theory to contain continuousfunctions orinfinitesequences,butfor themodelof theexperimenttobehighlydiscreteand finitisticin character.

The assessment of the relation between the model of the experiment and somedesignated modelof the theory is the characteristic fundamentalproblemof modernstatisticalmethodology.

IbelieveIhave succeeded inmaking itclear that theidea of a theo-retical model around which this essayhas beenorganizedcorrespondsdirectly to 'somedesignatedmodel ofthetheory' from thelastparagraphof the abovecitation.

The last remark is personal. There hardly is any philosopher ofscience of whomIcould say that he has inspired my own work sodeeply as it has been done byPat. Inspite of all the reservationsabouthis viewsIkept expressing, thispaper,Ido hope, is in full agreementwith the main ideas of Pat's research program. Itcertainly was meantto be acontribution to it.

NOTES

1For adiscussion of theseissues seee.g., Woodward(1989).2 The originalversion, Tarski (1933) was publishedinPolish. For an English versionof it,seeTarski (1956).3 Somealternativeideas wereput forwardbyEverthBeth(1949) (see also vanFraassen,1970) and RomanSuszko (1957, 1968).4 For anaccount of this discussion thereaderis referredtoPearce(1987).5 1allude here to the titleof themost recentmonograph (Balzer,Moulines, andSneed,

145THEORIES ANDTHEORETICAL MODELS

1987) on the structuralist approachinphilosophy of science.6 Cf. Wojcicki (1979).7 It goes without saying that thispaper isnot the rightplace to discuss fairly sophisti-catedtechnicalcounterparts ofthe ideaof accuracy andthatof reliability.8 For a discussion of methodologicalaspects of various hypotheses concerning Mer-cury's movement seeGriinbaum(1971).9 The first publicationinwhichIhaverevisedmy earlierviews wasWojcicki (1990).

REFERENCES

Adams,E. W.: 1959, 'TheFoundations ofRigidBody Mechanics and the Derivationof Its Laws fromThoseofParticleMechanics', in: L.Henkin,P. Suppes, and A.Tarski (Eds.),The AxiomaticMethod,Amsterdam:NorthHolland.

Balzer,W., Moulines,C. W., andSneed,J.D.: 1987, AnArchitectonicforScience; TheStructuralistProgram,Dordrecht: D.Reidel.

Beth,E.: 1949, 'Towards an Up-to-DatePhilosophy ofNatural Sciences',Methodos,1, 178-185.

Carnap, R.: 1956, 'The MethodologicalCharacter of Theoretical Concepts', in: H.Feigl andM. Scriven (Eds.), Minnesota Studies inPhilosophyofScience, Vol.I,Minneapolis: University ofMinnesotaPress.

Griinbaum, A.: 1971,'CanWe AscertaintheFalsityofaScientificHypothesis?',in:M.Mandelbaum(Ed.), ObservationandTheory inScience,Baltimore:JohnsHopkinsPress.

Kuhn, T. S.: 1970, The Structure ofScientific Revolutions (second edition), Chicago:University of Chicago Press.

Lakatos, I.: 1978, TheMethodologyof Scientific Research Programmes, Cambridge:Cambridge University Press.

Laudan,L.: 1977,ProgressandItsProblems,London: Routledge andKeganPaul.Pearce,D.: 1987,Roads to Commensurability,Dordrecht: D.Reidel.Sneed, J. D.: 1971, The LogicalStructure ofMathematicalPhysics, Dordrecht: D.

Reidel.Suppes,P.: 1960, 'AComparisonof theMeaningand Uses ofModels inMetamathe-

maticsandtheEmpirical Sciences',Synthese,2/3,287-301.Suppes,P.: 1962, 'ModelsofData',in:E. Nagel,P.Suppes, andA.Tarski(Eds.),Logic,

MethodologyandPhilosophyofScience: Proceedingsof the 1960InternationalCongress,Stanford,CA: StanfordUniversity Press, pp.253-261.

Suppes, P.: 1967, Set-TheoreticalStructures inScience, Stanford,CA: StanfordUni-versityPress.

Suppes, P.: 1974, 'The Structureof Theories and the Analysis ofData', in: F. Suppe(Ed.), TheStructureofScientific Theories,Urbana,IL:University ofIllinoisPress.

Suszko,R.: 1957, Logikaformalna a niektore zagadnienia teoriipoznania ('FormalLogic and SomeIssuesof Epistemology',inPolish), MyslFilozoficzna,2 and3.

Suszko, R.: 1968, 'Formal Logic and the Evolution ofKnowledge',in: L. Lakatos(Ed.),Problems in thePhilosophy ofScience, Amsterdam:NorthHolland.


Tarski,A.: 1933,Poj§cie prawdy wjezykachnauk dedukcyjnych('TheNotionof Truthin the Languages of Deductive Sciences', in Polish), Warszawa: TowarzystwoNaukoweWarszawskie.

Tarski, A.: 1956, Logic, Semantics, Metamathematics; Papers from 1923 to 1938,Oxford: ClarendonPress.

VanFraassen, B.C: 1970, 'On theExtensionofBeth'sSemanticsofPhysicalTheories',PhilosophyofScience,37,325-339.

VanFraassen,B.C: 1980, TheScientific Image,Oxford: ClarendonPress.Wojcicki, R.: 1979, Topics in the FormalMethodologyofEmpirical Sciences, Dor-

drecht:D. Reidel andWroclaw:Ossolineum.Wojcicki, R.: 1990, TeoriewNauce ('Theories inScience',inPolish), InstytutFilozofii

iSociologii,PAN, Warszawa.Woodward, J.: 1989, 'Data andPhenomena', Synthese,79, 393^172.

COMMENTS BYPATRICK SUPPES

It is not surprising thatIlike Wojcicki 's sympathetic treatmentof myviews on theories, set-theoretical models and experiments. The onlyminor point on whichIdirectly disagree is his statement in Section 2thatphilosophy ofscience,onmy view,is aspecialbranchof metamath-ematics. It is exactly thepoint of usingset-theoretical methods that wethrow things intoa standardmathematical contextrather than the meta-mathematical contextfamiliar inmuch of logic. My point was that weclarify scientific theories the way we clarify mathematical theories byusingappropriate set-theoreticalmethods whichare notmetamathemat-ical in character. This is not however a major pointandIturnto somecomments thatarenotreally indisagreement with Wojcicki's but eithersupplementormodify in certain ways the views he sets forth.

Theoretical Versus Semantical Models. Although in Section 6 ofhispaperWojcicki's remarks about the contrastbetween theoretical andsemanticalmodelsof theories aremeanttoproposecertainmodificationsof my views,Ifindmyselfmostly in agreementwith what hehas to say.The important point is that he is not introducing theoretical modelsas a separate class of formal models as opposed to semantic models.The emphasis is rather on at what stage one is creating a model as amethod for finding anew theoryor asatestof agiven theory,theformerconstituting a theoretical model and the latter a semantical model.Icertainly would agree thatmodels areusedinmany different ways.

Moreover,Ialsoagree,andIhave indicated this in various priorpubli-

147THEORIES ANDTHEORETICAL MODELS

cations, that scientistsuse the term 'model' in a much looser way thando logicians,but without serious harm.

Something thatIhave omitted inmy own discussionsof these mat-ters and that is not referred to by Wojcicki explicitly is the importantrole playedby the derivation of differential equations in allof science,but especially inphysics. Derivinga differential equationis not at alla matter ofsetting boundary valueconditions as,for example, inhis ormy earlier discussions of the two-body problem. Here it is a questionof understanding how to conceptualize the forces at work that createchange. A differential equation can cover, as we know from modernstudies,agreat varietyofphenomena,but thegivendifferential equationisnotordinarily thoughtofaspartoffundamental theorybut one derivedfor a class of phenomena that broadly fall under some general theory.There is from a logical standpoint,Isuppose, a kind of hierarchy oftheories involved. The important point is that the differential equationin complex situations will be such that its class ofmodels, i.e., the classof objects characterizing solutions of the differential equation, willnotat all be understood. We can have a very poor feeling for the class ofsolutions,especially as theclass ofboundaryconditions becomes com-plicated,but we can have a very detailed scientific knowledge aboutmany features of thedifferential equation. We can,of course,formalizethe differential equation as a set-theoretical object,but this is not veryinteresting. The realpoint is that such differential equations representan important way of characterizing classes of models and therefore,ifyouwill,subtheories,but theyhavenotbeenasmuchrecognizedas theyshouldhavebeenbymeorothers workingin the set-theoretical tradition.

PureMathematics VersusTheoreticalScience. Wojcicki quotespas-sages in whichIindicate that from a set-theoretical standpoint there isnot a significant difference between pure mathematics and theoreticalscience.Istill stand by those passages butIhavealso come to empha-size themany ways in which scienceand mathematics are different. Astheoretical physicists whoare goodfriends of mine like to say "If thereare not real conceptual differences between mathematics and physicsthen we theoreticalphysicists are just tobe thoughtofasprimitive math-ematicians." Those of you who know the elanof theoreticalphysicistsknow how unlikely they are to accept this idea.

Thisis not the place to expoundin detail themany ways thatIthinkmathematics and theoreticalphysicsdiffer.Ido just wantto mention one


or two salientpoints. Successfulphysicistsneed to haveanenormouslygood intuitive feeling for physical phenomena, to know just whichproperties to reach for in derivingnew results. Moreover,almost everyprobleminphysics is now too intractable to be solvedby really simplemethods. The wise choice of just what to take account of and whatto try to compute in a serious way is a largepart of being successful.Unlikepure mathematics,a very large percentageofphysicalproblemsnow require extensive computation. Most mathematicians still avoidexplicit numerical computations like the plague. Physicists may notlike the waycomputershave takenovertheoretical work but itcertainlyis inevitable and it ishere to stay.

Aneven morecritical point concerning the difference is thatphysi-cists puttogetherjusttherightmixofassumptionstomake an analysisofthe given physicalphenomenawork. From amathematical standpointwe might say that they cleverly choose just the right axioms. Mathe-maticians on the other hand get kudos for doing clever things with afixed set of axioms, that is, clever derivations,clever arguments, etc.Of course,mathematicians alsogetcredit for introducingnew concepts,but still this is a working difference thatIthink is of great importancein contrasting the two disciplines.

Finally,for better or worse,theoreticalphysicistsare drivenbyexper-iments. Theydo notperform them,but theyeagerlyawait thelatest wordfrom the currentrun. A really significantempirical result inphysicscanbe followed in a matter of afew monthsby severalhundred theoreticalpapers. Nothing like this concern for empirical matters is tobe foundanywherein puremathematics.

HierarchyofModels for Experiments.Iwant toamplify my ownideasaboutexperimentsandthemodels relevant to them, whichWojcicki dis-cusses briefly. The moreIthink about scientific practice andreflect onhow to give an accurate account of the complicated processes that gointo experimentation, the moreIam persuaded that there are a largenumber ofdistinctions needed to describe experimentation thoroughly,especially as dataare purified for quantitative, and even more statisti-cal,analysis. Itis along way fromrunningaround the laboratorydoingone thing and then another, tohaving a set of data as printout or on acomputer screen ready for analysis. Thatprocess stillneedsmuch morethorough attention than it hasreceived. Much of the current interest inthephilosophyof scienceindiscussingexperimentshas shiedaway from

149THEORIES ANDTHEORETICALMODELS

the gruesomedetails of exactly how dataare purified and selected foranalysis, not to speakof details of how they are generated,whichitselfmay involve,as equipment becomes increasingly complicated, manydifferent independenttestsof reliability and accuracyof theequipment.Itis acommon complaint of engineers inhigh-tech laboratories thatthescientists do not understand well enough theactual performance char-acteristics of the equipment they use. We will have come a long wayin the philosophy of science when a really thorough analysis of theseimportant phenomenaappears for some particular discipline.

Unlimitability of Ordinary Informal Language. Once we turn toexperimentsit is importanttorecognize that wearenot inany meaning-ful waygoing toeliminateordinary languageusedinits informalnaturalfashion. The attempt to formalize all the way down what happens inexperiments is an enterprise that can only be pursued in vain. It is animportant point, in fact, in recognizingand constructing formal modelsof experiments in termsof thepurificationofdata for statisticalanalysisthat theanalysis doesnotgoin anyformal wayall the way down.Itisinthiskindof respect thatit seems to me thatQuine'seliminativeprogramfor theuse oflanguage is mistaken. Innoserious way will intentional,causaland otherkinds of ordinary idioms be successfully eliminated indescribing the rich activities that makeup actual experiments. It is notjust ordinary experience, but scientific experience that cannotbe sub-ject to any serious reductionist program inall respects. Thoseparts ofscience that are reducible are thin,austerepiecesof theory,farremovedfrom the rich experimentaljunglein whichany true sciencenecessarilylives.

This doesnotmean thatIamnotin favor ofreduction whenpossible.What wedonotneed,however,arereduction romances, stories toooftentold byphilosophers without recognition of their fairy talequalities.

N. C. A. DA COSTA AND F. A.DORIA

SUPPES PREDICATES ANDTHECONSTRUCTIONOFUNSOLVABLE PROBLEMSINTHE AXIOMATIZEDSCIENCES

ABSTRACT. We firstreviewourpreviousworkonSuppespredicatesandtheaxiomati-zationof theempiricalsciences. We thenstate someundecidability andincompletenessresults in classical analysis that lead to the explicit construction of expressions forcharacteristicfunctions inall complete arithmeticaldegrees. Out of thoseresults weshow that for any degree there are corresponding 'meaningful' unsolvable problemsin any axiomatizedtheory that includes the language of classical analysis. Moreoverwealso show that withinour formalizationthereare 'natural' unsolvableproblemsandundecidablesentences whichare harder than any arithmeticproblem. As applicationswediscuss a 1974HilbertSymposiumproblemby Arnold on the existence of algo-rithmsfor the decisionof properties of polynomialdynamical systems over Z, provethe incompletenessofthe theory of finishNash games,anddelveonrelatedquestions.Neither forcingnor diagonalizations are used in thoseconstructions.

1. INTRODUCTION

Suppespredicates were the starting point in the recent developmentof a technique for the construction of both algorithmically undecid-able sets of objects inphysics and undecidable 'meaningful' sentencesaboutphysicalobjects. (See for details (da Costa,1988; Suppes,1967,1988).) That technique allowed us to prove the undecidability andincompletenessof mostof classicalandquantum physics,providedthatthey are givena first-order axiomatization (through Suppespredicates)that includes the language of classical elementary analysis (da Costa,1991a,1991b, to appear,1994a;Stewart,1991). Actual examplesdealtwith theproofof the incompleteness of chaos theory (daCosta, 1991a)and with the related question of the existence of problems in dynam-ical systems theory whose solution is equivalent to solving very hardDiophantine problems, such as Fermat's Conjecture (daCosta, 1994a).The same techniquereachedbeyondphysics andled to theproofof theincompleteness of the theory of Hamiltonian models for thedynamicsof economical systems (Lewis,1991b), whileproviding a partial resultrelated to the recentproofbyLewis of the noncomputability of Arrow-Debreu equilibria (da Costa, 1992d; Lewis, 1991a). We can still list

151

P. Humphreys (cd.), PatrickSuppes: Scientific Philosopher, Vol. 2, 151-193.@ 1994 KluwerAcademic Publishers. Printedin theNetherlands.

152 N.C. A. DA COSTA AND F. A. DORIA

among its consequencestwoexamples that are discussedin the presentpaper: a resultonone ofArnolds problemsconcerningalgorithms forpropertiesofpolynomial dynamicalsystems over the integers (Arnold,1976a) and theproofof the incompletenessof the theoryof finite gameswith Nashequilibria (da Costa,1992d).

Our main theorems depend in an essential way on a lemma ofRichardson (Richardson,1968);noncomputabilityinaxiomatizedphys-ical theories was anticipated by Scarpellini (Scarpellini, 1963) andbyKreisel (Kreisel,1976). Also, we wish to emphasize that no forcingisneeded for our independenceresults.

We obtain here a whole plethora of new intractable questions inSuppes-axiomatizedtheories. Thepresent resultsare new in thefollow-ingsense: allourpreviousexamples for undecidabilityand incomplete-ness withinaxiomatizedphysicscanbe formally reduced toelementaryarithmetic problems. However, that reduction cannot always be madein the present case, as some of our new examples are not elementarynumber-theoretic problems in disguise; they stand beyond the pale ofarithmetic.

There are even weirder situations: we obtain formal expressionsthat describe physical systems such that nothing but trivialities can beprovedabout them. For wecanexplicitlyconstructundecidable familiesofobjects within aclassical first-order languageLt such that:

No nontrivial properties of those families can be algorithmicallydecided.No assertion about the system can be reduced to an arithmeticassertion,thatis tosay, the systemlies fully outside thearithmeticalhierarchyand belongs to the nonarithmetical portion of set theory(if weare working,say, withinZermelo-Fraenkel set theory).Those results are consequencesof general incompleteness theo-rems that apply to anynontrivialpropertyPin the theoryT; thosetheorems extend a previous one (Proposition 3.28 in (da Costa,1991a) that originated in a suggestedbySuppes.

Againwehavea correspondingincompleteness theoremas thereareformal expressions for systemsall of whose nontrivial properties mustbe formulated as undecidable sentences. (Again no propertyof thosesystemscanbereduced to anarithmetic property.) Thoseundecidabilityand incompleteness results are to be found below in Propositions andCorollaries 3.28, 3.30,3.37, 3.41,3.47,3.49, 4.1.

153SUPPESPREDICATESANDUNSOLVABLEPROBLEMS

Section 2 of this paperreviews the theory of Suppes predicates forempirical theories in the da Costa-Chuaqui version (da Costa, 1988,1990b, 1992a). Section 3 deals with the undecidability and incom-pleteness of classical analysis in its several aspects; the main pointsare theexplicitconstruction of expressions for characteristic functionsof any complete degree in the arithmetical hierarchy, and the gener-alincompleteness theorems about expressionsofelementary functions.Section 4discusses ourexamples,whileSection5commentson possibleimplications of our results.

Preliminary Concepts andNotation

We are here interested in formal languages strong enough to representall the usual mathematical theories. For a review of the main ideassee (da Costa, 1990b, 1991a). Those formal languages are built outof a finite alphabet, and its sentences- 'meaningful assertions' -arefinite sequencesof letters from thebasic alphabet. We therefore reduceeverything to finite sequencesof letters. (As an example,an intuitivelyinfinite set suchasastraight lineontheCartesianplaneR2isrepresentedby the finite sequenceof symbols {(x,y) € R2 :y =2x 4- I}, abbre-viations such as R for the set of realsbeing allowed as their definitionsare reducible to finite sequencesof letters from the theory's alphabet.)Different formal expressions may represent the same 'intuitive' mathe-matical object;however in most everydaysituations ourformal systemswill not be strong enoughto decide,given twoexpressions,whether ornot they represent the same object, even if they are strong enough toproveallusual mathematical results.

To be more specific, we suppose that our theories are formalizedwithina first-order classical predicatecalculus with equality. (It is alsoconvenient to suppose that our formal theories T include Russell's isymbol; in that case the extended theory is a conservative extensionofthe theory without that particular variable-binding termoperator.)

We follow the notationof (daCosta, 1991a) with afew changes thatare explicitly indicated; in particular v will denote the set of naturalnumbers, Z is the set of integers, and R are the real numbers. LetTbe a first-order axiomatic theory that contains formalized arithmeticN and such that T is strong enough to include the concept of set andclassical elementary analysis. (We can simply take T= ZFC, whereZFC is Zermelo-Fraenkel set theory with the axiom of choice.) IfLt


is the formal language of T, we suppose that we can form within Ta recursive coding for Lt so that it becomes a setLt G T of formalexpressions in T. Objects in T will be denoted by lower case italicletters, such as x,y, z, f,g, a,...Predicates in T will be noted P,Q,. ..The use of Greek letters and more particular notational features(suchas p, qfor polynomial functional symbols) will be clearfrom thecontext. (Predicates are openone-variable formulae in the language ofT.)

Fromtime totime we willplay with thedistinctionbetween anobjectand the expression in Lt that represents it. If x,y are objects in anintended interpretationof thetheoryT,theyareingeneralnotedby term-expressions £, £ thatbelongto the formal languageLt ofT. Ingeneralthere is no 1-1correspondencebetween objects and expressions; thuswemayhavedifferent expressionsfor the same functions: 'cos 7r' and'0' are both expressions for the constant function 0. If £ is a set in anintended interpretationofT, wenoteby \x] a setof expressionsfor theelements in x. We allow the following abuseof language: predicates Psometimes apply to objectsin Tandsometimes apply to expressionsinLt(P(OY> meaning will be clear from context.

Inparticular we notice that since our theory Tincludes formalizedarithmetic N, we will sometimes need the distinction between a partialrecursive function and thealgorithm that computes it. For any compu-tation 4>(x), (f>(x)Imeans that the computation converges (stops andproduces an output), while (f>(x)|means that thecomputation diverges(entersa never-endingloop).

We emphasize that proofs in T are algorithmically defined ways ofhandling the objects ofLt', for the conceptof algorithm see (da Costa,1991a;Rogers,1967).

2. SUPPES PREDICATES

Suppespredicates (or Bourbaki species of structures) were first usedby Suppesin the 'fifties as a way of directly axiomatizing any mathe-matically-based theory. Suppes'smain contention is that"to axiomatizea theory is to define a set-theoretic predicate." A formal treatment wasthengivenbydaCostaandChuaquiin1988,andimmediate applicationsensued. (For details, applications and references see (Bourbaki, 1957,1968; daCosta 1988, 1990b, 1992a;Suppes, 1967, 1988). Everything

SUPPES PREDICATES ANDUNSOLVABLEPROBLEMS 155

here is supposed tohappen within our arithmetically consistent theoryT.

Structures andPredicatesA mathematical structure w is afinite orderedcollection ofsets (whichmay be particularized to relations and functions) of finite rank over theunion of the ranges of two finite sequencesof sets, xj,xi,...,xm andy\,yii ■ " " 2/n» wherem>0andn > 0.If wearedoingourconstructionswithin ZFC, w is thus aZFC set. Thexs and the ys are called thebasesetsofw; the xs are theprincipalbasesets,while the ysare the auxiliarybasesets.

The auxiliarybase sets canbe seen as previously defined structures,while the principal base sets are 'bare' sets; for example, if we aredescribing a real vector space, the set of vectors is the only principalbase set, while the set of scalars,R, is the auxiliary base set.

ASuppespredicateoraspeciesofstructuresin the senseofBourbakiisa formula of set theory whose only free variables are thoseexplicitlyshown:

P defines it; as a mathematical structure on the principal base setsx\,...xm,withtheauxiliarybase sets y\,...,yn,subject to restrictionsimposed on w by the axioms we want our objects to obey. As theprincipal sets x\, ...vary over a class of sets in the set-theoreticaluniverse,we get the structures of speciesP,orP-structures.

The Suppespredicate is a conjunction of two parts: one specifiesthe set-theoretic process of construction of the P-structures,while theother imposesconditions that mustbe satisfiedby the P-structures. Thissecond piece contains the axioms for thespecies of structures P.

We write theSuppespredicate for ageneralw as follows:

The auxiliary sets are seen as parameters in the definition of w. Allofeverydaystandard 'professionals' mathematics canbe formalizedalongthoselines.

P(W, Xi,X 2, "" ",Km,Xm,2/1,2/2,---, 2/n)-

W(q) <-> 3xi3x2...3xmP(iy,xi,X2,...,xm,yi,...,2/n).

156 N. C. A.DA COSTA AND F. A.DORIA

DeducedandDerivedStructuresGiven a structure wof species P(w,x\,... ,xm,2/1,... ,yn), let 21,... ,zp,bep(p> 0) sets of finite rank over the union of rangesof thesequences

5" " ",%mi y\i

- - " yn,

also let v\,...,vq (q > 0)be qarbitrary sets. If the Suppespredicate

defines w* as a structure on the principal base sets z\, ... with the v\,...as auxiliary base sets, we say that the structure w* of speciesP* isdeducedfrom the structure w of speciesP.

Wecan obtain new structures out of (sets of) already defined struc-tures by the means of two basicprocedures:

1.With the help of set-theoretic operations, such as Cartesian prod-ucts, passages to the quotient, and the above-described operationof deductionof structures;

2. Through the imposition of new axioms to already existing set-theoretic structures.

Therefore we can introduce the notion of derived structure. Whenwe define anew structure w fromaset s ofother structures with the helpof the twoprocedures described above, we say that w is derivedfromthe structures s. The Suppespredicate of w canbe expressedin termsof the Suppespredicates of theelements of s. The conceptofdeductionof structures is aparticular case ofderivation of structures.

The set s is the set of groundstructures for w.Finally, let w and w1be twostructures of species Pand P', respec-

tively. We suppose that P andP'differ only in connection with theirsets of axioms,but that the conjunction of the axioms of P' implieseach axiom of P, with quantifiers restricted to sets of finite rank overtheunion of the rangesof the base sets for w. If that is the case,we saythat the P'-structure is richer than the P-structure (or thatP' is richerthan P). For instance, the species of commutative groups is richer thanthespeciesof groups.

The Q'-structure g' is then derived from the Q-structure g if Q' isricher thanQ,or Q'canbeobtained from Qin the way we havealreadydescribed above. The above ideas can also be extended to the conceptofpartial structures introduced by (da Costa, 1990a).

P*{w*,zi,...,zp,vi,...,vq)

SUPPES PREDICATES AND UNSOLVABLEPROBLEMS 157

TheAxiomatics ofEmpirical TheoriesAs a first approximation we seeempirical theories as triples

where (i) Mis aSuppes-Bourbakispecies of mathematical structures;(ii) A is the theory's 'domain of definition',and(iii) pgives the 'inter-pretation rules' or 'characteristic examples' that relate Mand A. Wecan be more specific about (ii) and (iii), however,as we did elsewhere(da Costa, 1992c); in any case we consider A to be a set-theoreticconstruct. (In that case p in general contains nonrecursive aspects (daCosta,1991a, toappear, 1992a).)

AnExample: SuppesPredicatesfor Classical FieldTheoriesinPhysics

We follow the usual mathematical notation in this subsection. In par-ticular,Suppespredicates are written in a more familiar but essentiallyequivalent way. Therefore somesymbols willhave a different meaningthanin theremainingportions of thepaper.

The species of structures of essentially all physical theories canbe formulated as particular dynamical systems derived.from the P =(X,G,fi), where X is a topological space, G is a topological group,andp is a measure on a setof finite rank over XUG. Thus we can saythat the mathematical structures of physicsarise outof the geometryofa topological space X: physical objects are those that exhibit invari-ance properties with respect to the action of G and the main speciesof structures in 'classical' theories can be obtained out of two objects,a differentiable finite-dimensional real Hausdorff manifold Mand afinite-dimensional Lie group G.

DEFINITION 2.1. The species of structures of a classical physicaltheory is givenby the9-tuple

1. The GroundStructures. (M,G), where Mis a finite-dimensionalreal differentiablemanifold andGisafinite-dimensional Liegroup.

A=(M,A,p),

E= (M,G,P,T,A,I,G,B,V<p = l)which is thus described:

158 N.C. A.DACOSTA AND F. A. DORIA

2. The Intermediate Sets. A fixedprincipal fiber bundle P(M,G)over M with G as its fiber plus several associated tensor andexteriorbundles.

3. TheDerivedFieldSpaces.Potential spaceA,field spaceTand thecurrentor source spaceX. A,TandTarespaces (in general,man-ifolds) of cross-sectionsof thebundles that appear as intermediatesets in our construction.

4. AxiomaticRestrictions on theFields. Thedynamicalrule Vy? = t,the relation ip = d(a)a between a field <p c T and its potentiala c A, together with the corresponding boundary conditions B.Here d(a) denotes a covariant exterior derivative with respect totheconnection form a,and V a covariantDirac-like operator.

5. The Symmetry Group. Q C Diff(M) ® Q', where Diff(M) isthe group of diffeomorphisms of M and Q' the group of gaugetransformations of the principalbundle P.

6. The Space ofPhysicallyDistinguishableFields. If /C is one of theT, A or Z field manifolds, then the space of physically distinctfields is

K./G.

(In more sophisticated analyses wemust replace our conceptof theoryfor a more refined one. Actually in the theory of science we proceedas in the practice of science itself by the means of better and betterapproximations. However for the goalsof thepresentpaperour conceptofempirical theory isenough.)

We show elsewhere (da Costa, 1992a) that what one understandsas the classical portion ofphysics fits easily into the previous scheme.We discussindetail two examples,Maxwellian theory andHamiltonianmechanics.

Maxwell's Electromagnetic Theory

Let M= R4,with its standard differentiable structure. Let us endowMwith the Cartesian coordination induced from its product structure,and let 7/ = diag(— l,+l,+l,+l) be the symmetric constant metricMinkowskian tensoronM. If the F^v {x) are components of a differ-


entiable covariant2-tensor field onM,p,v = 0, 1,2,3, thenMaxwell'sequations are:

Thecontravariant vector fieldwhosecomponentsaregivenby the setoffour smooth functions j^(x) onMis the current thatserves as sourcefor Maxwell's fieldF^. (We allow piecewise differentiable functionsto account for shock-wave-like solutions.)

Itisknown thatMaxwell's equations areequivalent to the Dirac-likeset

and

(where the {7^ :p — 0,1,2,3} are the Dirac gamma matrices withrespect to 77). Those equation systems are tobe understood togetherwith boundary conditions that specify aparticular field tensorF^ 'outof the source jv (Doria, 1977).

The symmetrygroup of theMaxwell field equations is theLorentz-Poincare group that actsonMinkowski spaceMand inan induced wayonobjectsdefined overM.However,since weare interestedin complexsolutions for the Maxwell system, we must find a reasonable way ofintroducingcomplexobjects inour formulation. One may formalize theMaxwellian system as agauge field. We sketch the usual formulation:again we start fromM= (R4,rj), andconstruct the trivial circlebundleP — M x Sl over M, since Maxwell's field is the gauge field of thecirclegroup 51(usually writtenin thatrespectasU(1)). We form the set£ of bundles associated to P whosefibers are finite-dimensional vectorspaces. Theset of physical fields in our theory is obtained out of some

<v^=r,

d^p+ dpFay+dvFpu= 0

V(/? = L,

where

tp = (1/2)^7^,

t-Jul,fi,

V =ypdp,

160 N.C. A.DACOSTA AND F.A.DORIA

of the bundles inS: the set of electromagnetic field tensors is a set ofcross-sectionsof thebundle F =A20s1(M)of all 2-formsonM, where s1 is the group'sLie algebra. To be more precise, the setof all electromagnetic fields is T C Ck(F),if we are dealing with Ck

cross-sections (actually a submanifold in the usual Ck topology due tothe closurecondition dP =0).

Finally we have twogroup actions onT: the first oneis theLorentz-Poincare action L which is part of the action of diffeomorphisms ofM; then we have the (here trivial) action of the group Q1 of gaugetransformations of P when acting on the field manifold T. As it iswellknown,its action is not trivial in thenon-Abelian case. Anyway,it always has a nontrivial actionon the space A ofall gauge potentialsfor the fields in T. Therefore we take as our symmetry group Q theproduct L ® Q' of the (allowed) symmetries ofMand the symmetriesof theprincipal bundle P.Formathematical details see (Doria, 1981).

We must also add the spaces A of potentials and of currents, X,as structures derived from M and Sl. Both spaces have the sameunderlying topologicalstructure; they differ in the way the group Q' ofgauge transformations acts upon them. We obtainI=A1® sl(M) andA=1=Ck(I). Notice thatXjQ' =X while A/Q' A.

Therefore we cansay that the 9-tuple

whereMis aMinkowski space,and B is a set ofboundary conditionsfor our field equations V</? = t, represents the species of mathematicalstructures of aMaxwellian electromagnetic field,where P,TandQarederived fromMand Sl.TheDirac-like equation

should beseen as an axiomatic restriction on our objects;theboundaryconditions B are (i) aset of derived speciesof structures from MandS\ since, as we are dealing with Cauchy conditions,we must specifya local or global spacelike hypersurface C in Mto which (ii) we addsentencesof the form Vx G Cf(x) = /o(x), where /o is a setof (fixed)functions and the / areadequate restrictions of the field functions andequations toC.

{M,SI,P,T,A,g,X,B,V<p = i)

V<£> =L

SUPPES PREDICATESAND UNSOLVABLEPROBLEMS 161

HamiltonianMechanicsHamiltonian mechanics is the dynamics of the 'Hamiltonian fluid'(Arnold, 1976b). Ourgroundspecies ofstructuresarea2n-dimensionalrealsmooth manifold,and the real symplectic group Sp(2n,R). Phasespaces in Hamiltonian mechanics are symplectic manifolds: even-dimensionalmanifolds likeMendowed with asymplectic form, that is,a nondegenerateclosed 2-form Q, on M. The imposition of that formcan be seen as the choice of a reduction of the linear bundle L(M)to a fixed principal bundle P(M,Sp(2n,R));however givenone suchreduction it does not automatically follow that the induced 2-form onMis a closedform.

All other objects are constructed in about the same way as in thepreceding example. However, we must show that we still have here aDirac-like equation as the dynamicalaxiomfor the species ofstructuresof mechanics. Hamilton's equations are

where %x denotes theinterior product with respect to the vector field XoverM, andhis theHamiltonian function. Thatequationis (locally,atleast) equivalent to:

or

whereLxis theLie derivative withrespectto X.Theconditiondip = 0,with (f = ix&>, is the degenerateDirac-like equation for Hamiltonianmechanics. We do not get a full Dirac-like operator V d becauseM, seen as a symplectic manifold,does not havea canonical metricalstructure,so that wecannotdefine (through the Hodgedual) acanonicaldivergence 6 dual to d. The group that acts onM with its symplecticform is the group of canonical transformations; it is a subgroup of thegroup of diffeomorphisms ofM so that symplectic forms are mappedontosymplectic forms underacanonical transformation. We can take as'potential space' thespace of allHamiltonians onM(which is arathersimple function space),andas 'fieldspace' thespaceofall 'Hamiltonianfields' of the form ixsl-

ix& — —dh,

LxSl =0,

d{ixQ) =0,

162 N. C. A. DA COSTA ANDF A. DORIA

The constructionofSuppespredicates for gravitation theory(generalrelativity), classical gauge fields, Kaluza-Klein unified field theories,and Dirac'selectron theory seen asa classical field theory can be foundin (da Costa, 1992a). We notice that Dirac-like dynamical equationshavebeenobtained for all those (Doria, 1975, 1986). Themathematicalbackground is in (Cho,1975;Dell, 1979;Kobayashi, 1963).

3. UNDECIDABILITY ANDINCOMPLETENESS

Wenow review previousmaterial andobtainnew resultson theundecid-ability and incompleteness of classical analysis; the chief new resultsare the construction of several intractable problems and undecidablesentencesin Twhichcannotbe reduced(in T) toarithmetical problems.

DEFINITION3.1. Tis arithmetically consistent if andonly if the stan-dardmodel Nfor N is a model for the arithmetic sentencesofT. ■

Now let \V~\ be the algebra of polynomial expressions on a finitenumber of unknowns over the integers Z; we identify \V\ with theset of expressions for Diophantine polynomials in T. Let \£~\ be theset of expressions for real elementary functions on a finite number ofunknowns,while [JF] is the set of expressions for real-valuedelemen-tary functions on asingle variable (daCosta, 1991a).

Givenapolynomialexpressionp(x\,...,xm,yi,...,2/n),letrm(xi,,xm) be the function that effectively codes m-tuples of natural

numbers (xi, ..., xm) by a single natural number (Rogers, 1967,

Let us inductively construct out of a polynomial qm{x\,...,xn) anRn-defined and R-valued function aqm given through the followingsteps:

InitialStep. Suppose that we are given the expressions qm as below:

"Ifqm(x\,...,xn) =c, where cis a constant, then we put aqm=

|c|+2;"If qm{x\,"" ",xn) = xi, then aqm =x]+2.

p. 63). Letr =rm((...». We abbreviatep(xi,.. .,xm,2/i,...,y„) =Pr(yu---,yn)-


Induction Step. We suppose that qm is given as indicated. We thenobtain the corresponding aqm as follows:

Supposenow that weisolate someof the variables in ourpolynomi-als as parameters. Then

DEFINITION3.2. The mapa : \V] -+ \£~\, givenby:

p(xi,...,xm,2/i,...,2/n) "->" ap(xi,...,xm,2/i,...,2/n)

isRichardson 'sFirstMap.

COROLLARY3.3. Givenapolynomialexpressionpm-

£ Lt, there isan algorithm thatallows us to obtainan expressionapm GLt for theimageofpm under Richardson'sFirstMap.

Proof. Immediate,from the definition of a.

Wenowdefine:

Given a set ofreal variables x\,. xn,we define the following maps:

(whereg is composedn — 2 times),and

"Ifqm =Sm ± tm,then aqm= asm + atfm.

"Ifqm = smtm, then agm = aSmOLtm.We then write ki(m,x\,...,xn) = adiqm(x\,...,xn), where<%

d/dxi.

= (n+1)V(xi,...,xm,2/i,...,2/n) +n+ XXsin27TXi)kj(xi,...,Km,Xm,2/1, "" ",2/n),

i=l

h(x) = xsinx,g(x) — xsinx3.

xi = h(x),x 2 = hog(x),X 3 = hogog(x),

Xn_i= hogo... og(x),

xn=gogo... og(x).

164 N. C. A. DACOSTA AND F. A.DORIA

Hereg is composedn times.Given apolynomial expressionpm(x\,...,xn) 6 \V~\,we define:

DEFINITION 3.4. The maps i' : \V\ -+ \T] and i" : \V] -+ \T],givenby:

whereais Richardson's FirstMap;and

areRichardson 'sSecondMapof the first (i1)and second {i")kinds. ■

COROLLARY 3.5. Given apolynomial expressionpm G Lt, thereis an algorithm that allows us to obtain expressions t!pm £Lt andi"Pm €Lt for the imagesofpmunder Richardson 'sSecondMap.

Proof. Immediate, from the definition of i'and i".We assert:

PROPOSITION3.6 (Richardson's Functor). Letpm(xux2,...,xn) =0be afamily ofexpressionsforDiophantine equationsparametrizedbythepositive integer m in an arithmetically consistent theory T. Thenthere is an algorithmic procedure a : \V~\ — * \£~\ such that out ofpm €V we canobtain an expression

fm €£, such that fm =0ifandonly iffm <1ifandonlyifthere arepositive integersx\,x 2

,... xnsuch thatpm(x\,...,xn) =0.Moreover, therearealgorithmic proceduresi',i":P — * Tsuch that

we can obtain out ofanexpressionpm two other expressionsfor one-variable functions,gm(x) = i'pm(x\,...)andhm(x) = i"pm(x\,...)such that there arepositive integers x\,...withpm(x\,...) =0ifandonly ifgm{x) =0andhm(x) < 1,forall real-valued x.

(1) pm(xi,...,xn) i-> i\prn{x\,...,xn)]{x)= apm(h(x),hog(x),..

gogo...og(x)),

(2) Pm(xi,...,Xn) »-> t"[pm(xu... ,X„)](x)= t /[pm(xi,...,X„)](x) - \ ,

Jm\X\,X 2, ..",Xn) —Q!Pm \XliX 2, "" " ,Xn),


Proof See (da Costa, 1991a).

PROPOSITION3.7. IfTis arithmetically consistent,and ifweadd theabsolute valuefunction \x\to \T] andcloseit to obtainanextendedsetofexpressions \T*~\, we have:

1. (Undecidability) We canalgorithmically constructin Tadenumer-able family ofexpressionsfor real-valued,positive-definite func-tions km(x) > 0 so that there is no generalalgorithm to decidewhether onehas,forall realx,km(x) =0.

2. (Incompleteness) ForamodelMsuch that Tbecomes arithmeti-cally consistent, there is an expressionfora real-valued functionk(x) such thatM |= Vx <E Rfc(x) =0 while TV- Vx € Rfc(x) =0andTY3x G Rfc(x) 0.

Proof. See (da Costa, 1991a).

If km (as inProposition 3.7)results out ofpm,we writekm= Xpm-

EqualityIs Undecidable inLtCOROLLARY 3.8. IfTisarithmetically consistent then for anarbi-trary real-defined and real-valued function f there is an expressionf G Lr such thatM f= £ = /,whileTY^= fandT V- -.(£ = /).

TheHaltingFunctionandExpressionsfor CompleteDegreesin theArithmeticalHierarchy

Now let Mn(q) be the Turingmachine of index n that acts upon thenatural number q (Rogers, 1967). Let 9(n,q) be the halting functionfor Mn(q), that is,0(n,q) =1if and only ifMn{q) stops over q, and0(n,q) =0ifand only ifMn(q) does not stop overq.

We need adefinition and a lemma.

DEFINITION3.9. For the followingreal-defined andreal-valued func-tions:

(i) m={-*:*<2:

Proof. Put £ =/+k(x), for fc(x) as inProposition 3.7.

166 N.C. A. DACOSTA ANDF. A. DORIA

LEMMA 3.10. IfT is (arithmetically) consistent, then each of thefollowing operationsgenerates the others within T:

(1) +,x,|

(2) +, X,7/(...).

Proof. Immediate.

Let pn,q(xi,X2,...,xn) be a universal polynomial (Jones, 1982).Since \T*~\ has an expression for |x| (informally one might have|x| =+Vr),ithasanexpression for thesign functiona(x). Thereforewe can algorithmically build within the languageof analysis(where wecan express quotients and integrations) an expression for the haltingfunction 6{n,q):

PROPOSITION 3.11 (The Halting Function). IfT is arithmeticallyconsistent, then:

/-n / \ fl,a;>0,(2) "M-10,x< 0.

/ox" f x — y, x — y >0,(3) x-y= { r,

y' ~nv J y [0, x—2/ <0.

( +1, x > 0(4) a{x) = < 0, x=0

[ -1, x<o

(3) +,x,(...-...).

(4) +,X,<7(...).

o(n,q) = a(Gnjq),

+00 2

a - f Cn.i(x>ce~xdxG"'""J 1+Cn,,(i)"*'

SUPPESPREDICATES AND UNSOLVABLE PROBLEMS 167

Proof See (da Costa,1991a).

There follows:

PROPOSITION 3.12. IfT is arithmetically consistent then we canexplicitly and algorithmically construct in Lt an expression for thecharacteristic functionofasubsetofujofdegree0".

Remark 3.13. That expression dependsonrecursive functions definedon ujandon elementary real-defined andreal-valued functions plus theabsolute value function,aquotient and an integration, as in the caseofthe9 function givenby Proposition3.11. ■

Proof. Wecouldsimply use Theorem9-IIin (Rogers, 1967,p.132).However,for the sakeofclarity wegiveadetailed,albeit informalproof.Actually the degree of the set described by the characteristic functionwhoseexpressionweare going to obtain willdependonthe fixed oracleset A; soour construction is ageneral one.

LetA C ujbe a fixedinfinite subsetof the integers:

DEFINITION3.14. The jumpofA is written A';A1= {x :<j>£{x) |},where 4>^ is the A-partial recursive algorithm of index x. ■

1. An oracle Turing machine (f>^ with oracle A can be visualizedas a two-tape machine where tape 1is the usual computationaltape, while tape 2 contains a listing of A. When the machineenters the oracle state so, it searches tape 2 for an answer to aquestionof theform 'is w G AT Onlyfinitely manysuch questionsare asked during a converging computation; we can separate thepositive andnegative answers into two disjoint finite sets DU(A)and D*(A) with (respectively) the positive andnegative answersto those questions; notice that Dv C A, while D* Cuj— A. Wecan view these sets asorderedk- and A;*-pies;vand v arerecursivecodings for them (Rogers, 1967). The DU(A) and D*(A) setscan be coded as follows: only finitely many elements of A arequeriedduring an actual convergingcomputation with input y; if

) — ,...,Xr ).

168 N. C. A.DA COSTA AND F. A.DORIA

k' is thehighestinteger queriedduringone suchcomputation, andif dA C ca is an initial segmentof the characteristic function ca,we take as a standby for D and D* the initial segment dA wherethe length l(dA) =k'+1.We can effectively listall oraclemachines with respect to a fixedA, so that,givena particular machine we can computeits index (orGodel number) x,and given x we can recover the correspondingmachine.

2. Given an A-partial recursive function <ft£, we form the oracleTuring machine that computes it. We then do the computation4>x(y) = z tnat outputs z. The initial segment dy tA is obtainedduring the computation.

3. The oracle machine is equivalent to an ordinary two-tape Turingmachine that takes as input (y,dy^)',2/ ls written on tape 1whiledVyA is written on tape 2. When thisnew machine enters state so itproceedsas theoracle machine. (For an ordinary computation, noconvergingcomputation enters so, anddViA is empty.)

4. The two-tapeTuringmachine canbemadeequivalentto aone-tapemachine,where some adequatecodingplaces onthesingle tapeallthe information about {y,dy^)- When this thirdmachine enters soit scansdy>A-

5. We can finally use the standard map r that codesn-ples 1-1 ontoujandadd to theprecedingmachine aTuring machine that decodesthe single natural number r((y,dy,A)) into its components beforeproceedingto thecomputation.

Let w be the index for that lastmachine; we noteit <j>w.If x is the index for 4>£, wenote w =p{x), where pis the effective

1-1procedure described above that maps indices for oracle machinesinto indices for Turingmachines. Therefore,

Now let us note the universal polynomial p(n,q,X\,...,xn). Wecan define the jump of A as follows:

<f>£(y) = 4>p(x)((y,dy,A))'

A1= {p(z) :3xi,...,xn € vp(p(z),(z,dZjA),xv ...,xn) =o}.


With the helpof the Amap definedfollowing Proposition3.7, wecannow form a function modelled after the 9 function inProposition 3.11;it is the desired characteristic function:

(Actually we haveprovedmore; we have obtained

We recall (Rogers,1967):

DEFINITION3.15. The complete Turingdegrees,0,0',...,o^\p < uj, are Turing equivalence classes generated by the sets 0, o', 0",

Now let 0(n)be thenth complete Turing degrees in the arithmeticalhierarchy. Letr{n,q) =mbe thepairing function in recursive functiontheory (Rogers,1967). For 9(m)= 9(r(n,q)), we have:

COROLLARY3.16 (Complete Degrees).IfTisarithmetically consis-tent, forallp £ uj the expressions9p(m) explicitly constructedbelowrepresentcharacteristic functions in the complete degrees0^p\

for ca as inProposition 3.12.

Incompleteness TheoremsWe now stateand prove several incompleteness results about N anditsextension T;they will be needed when weconsider ourmain examples.Werecall that

'— ' is aprimitive recursive operationon uj.

CO/(x) =0(p(x),(x,4jO/».

ca'(x) = 9{p(x),{x,dX)A )),

with reference to anarbitrary A Cuj.)We write 9^2\x) =cr(x).

0<p>...

Proof. FromProposition 3.12,'0(°) = c0(m) =O,

< 0O)(m) =cy(m) =0(m),0(n)(m),

170 N.C. A. DACOSTA ANDF A. DORIA

The starting point is the following consequence of a well-knownresult:

PROPOSITION 3.17. IfTisarithmetically consistent, then we canalgorithmically constructapolynomial expression g(xi,...,xn) overZ such thatM(=Vxi,...,xn G ujq(x\,...,xn) >0,but

Proof. Let(GItbe an undecidable sentenceobtained for T withthehelp of Godel's diagonalization; letn^ be its Godelnumber andletrriTbe theGodelcodingofproof techniquesinT(of theTuringmachinethat enumerates all the theorems of T). For an universal polynomialp(m,q,x\,...,xn) we have:

COROLLARY 3.18. IfH is arithmetically consistent then we canfindwithin itapolynomialp asinProposition 3.17 ■

Wecan also stateand prove a weaker versionof Proposition 3.17:

PROPOSITION3.19. IfTis arithmetically consistent, there is apoly-nomial expressionover Z.p(x\,...,xn) such thatMf= Vxj,.. .,xn Gujp(xi,...,xn) > 0, while

and

Proof. See (Davis, 1982): if p(m,x\, ..., xn), m = r(q,r),is a universal polynomial (r is Cantor's pairing function (Rogers,

TV- Vxi,...,x„ G wg(xi,... ,xn) >0

and

TV- 3xi,"" ",xn G ujq(x\,...,xn) =0.

g(xi,...,xn) = (p(mT,n^xv... ,xn))2.

TFVxi,...,x„ G wp(xi,...,xn) >0

TP3xi," " ",xn G ujp(x\,... ,xn) =0.

SUPPES PREDICATES ANDUNSOLVABLE PROBLEMS 171

1967), then {m :3xi... G ujp(m,x\,...) =0} is recursively enu-merable but not recursive. Therefore there is an mo such that Vxi...uj(p(m0,xi,...))2 > 0. ■

Predicates orproperties in T are representedby formulae with onefree variable in Lt.

DEFINITION3.20. Apredicate PinLt is nontrivial if there are term-expressions(,CG LT such that ThP(£) andT h -iP(C). ■

Then:

PROPOSITION 3.21. IfH is arithmetically consistent and ifP isnontrivial then there is a term-expression(G Ln such thatN |= P(C)whileNV P(()andNV -.P(C).

Prao/ Put (=£ 4- r(xi,...,xn)v, forr=1-erg2,gas inPropo-sition 3.17 (or asp in 3.19). ■

Remark 3.22. Therefore everynontrivial arithmetical propertyP intheories from arithmetic upwards turns out to be undecidable. We cangeneralizethat result to encompass other theories T that include arith-metic; seebelow. ■

We now give alternative proofs for well-known results about thearithmetical hierarchy that will lead to other incompletenessresults:

DEFINITION3.23. The sentences £, £ G Lt are demonstrably equiv-alent if andonly ifTh £ <-» (. ■

DEFINITION3.24. The sentence£ G Lt is arithmetically expressibleif and only if there is an arithmetic sentence£ such that Th^<->(. ■

Then

172 N. C. A. DACOSTA ANDF. A. DORIA

PROPOSITION3.25. IfT is arithmetically consistent, thenfor everym G uj there is asentence £ G T such thatM |= £ while forno k < nis there aEfc sentence in Ndemonstrablyequivalent to £.

Proof. The usual proof for N is given by Rogers (Rogers, 1967,p. 321). However, we give here a slightly modified argument thatimitates Proposition3.19. First notice that

is recursively enumerable but not recursive in 0(m). Therefore 0(m+Uis not recursively enumerable in 0(m), but contains a proper 0(m)-

recursively enumerable set. Let us take a closer look at those sets.We first need a lemma: form the theory T m̂+l^ whose axioms are

those for T plus a denumerably infinite set of statementsof the form'n0 G 0(m)', 'ni G 0(m)',...,that describe 0<m). Then,

LEMMA 3.26. IfT n̂+l^ is arithmetically consistent, then 4>xm (x) J.ifandonly if

Proof. Similar to theproofin thenonrelativized case; see (Machtey,1979,pp. 126ff). ■

Therefore wehave that the oraclemachines (f>xm (x) [ ifandonly if

However, since 0(m+1) is not recursively enumerable in 0(m) thenthere will be an indexmo(0 (-m^) = (p(z),(z,d 0(m> )) such that

while it cannotbe provednor disproved within J'(m+1) - itis thereforedemonstrably equivalent to anm+i assertion. ■

Now let g(mo(0(m) ),xi..)= p(rao(0(m) ),xv ■ " -))2 be asinPro-position 3.25. Then:

0("+" ={x : <mV)}

T(m+l) |_ 3xU...,Xn eUJp(p(z),(z,dy0(m)),xi,...,xn)=0

T(m+l) |_ 3xU...,Xn eUJp(p(z), (Z, dy>o(m)), xi,...,xn) =0.

M(= Vxi,...,xn[p(m0,xi,...,xn)]2 >0,

SUPPES PREDICATESANDUNSOLVABLEPROBLEMS 173

COROLLARY 3.27. IfTisarithmetically consistent, thenfor:

Then,

COROLLARY3.28. IfTisarithmeticallyconsistentandifLT containsexpressionsfor the9^ functionsasgiveninProposition3.16, thenforanynontrivialpredicatePinN there isaC, GLt such that theassertionP(C) isT-demonstrably equivalent to andT-arithmetically expressibleas a nm+i assertion, but not equivalent to and expressible as anyassertion with a lower rank in the arithmetic hierarchy.

Proof. As in theproofofProposition 3.21, we write:

wherep{...) is as inProposition 3.25.

Remark 3.29. Rogers discusses the rank within the arithmetical hierar-chyof well-known openmathematical problems (Rogers, 1967,p.322),suchas Fermat's Conjecture-whichin its usual formulation is demon-strablyequivalent toaniproblem,orRiemann'sHypothesis,alsostatedas a111problem. Rogersconjectures thatourmathematical imaginationcannothandle more than four or five alternations of quantifiers. How-ever theprecedingresult shows thatany arithmetical nontrivialpropertywithin Tcan give rise to intractable problemsof arbitrarily high rank.

We obviously need the extensionT D N, since otherwise we wouldnot be able to find an expression for the characteristic function of aset

/?(m+l ) =a(C(mo(0 (n))),

G(mo(0("))) =+[°C(rno(^),x)e-° 2

C(mo(0(n)),x) = Ag(mo(0(n)),xi,...,x r),

M |= /3(m+l) = 0 butfor alln < m+ 1, T^ V flm+^ = 0 andT{n) y. _,^(m+l) _

o). ■

C =S + [1- ff(p(mo(0m),x 1,...,xn))>,

174 N. C.A.DACOSTA AND F. A. DORIA

with a high rank in the arithmetical hierarchy within our formal lan-guage. ■

An extension of theprecedingresult is:COROLLARY 3.30. IfT is arithmetically consistent then, for anynontrivialpropertyP there is a £ G Lt such that the assertion P(()is arithmetically expressible, M |= P(Q but it is only demonstrablyequivalent to a Tln+\ assertion andnot to a lower one in the hierarchy.

Proof. Put

where oneuses Corollary 3.27.

Undecidable Sentences OutsideArithmeticWerecall:

DEFINITION3.31.

for x,y G uj.

Then:

DEFINITION3.32.

where c^{u)(m) is obtainedas inProposition 3.12.

Still,

DEFINITION3.33.

C =

0^= {(x,t/) :xG0(y)},

#H(m) = C0(w)(m),

o(w+i) _ (oHy_


COROLLARY3.34. O 1̂ ) is thedegreeo/0(w+l).

COROLLARY 3.35. 9û+l\m) is the characteristic function ofanonarithmetic subsetofujofdegree0û+l\ ■

COROLLARY 3.36. IfTis arithmetically consistent, thenfor:

PROPOSITION 3.37 (Nonarithmetic Incompleteness). IfT is arith-metically consistent then givenanynontrivialpropertyP:(1) There is a family ofexpressions (m GLt such that there is no

generalalgorithm to check, for every m G uj, whether or notP((m).

(2) There is an expression( G Lt such thatM |= P(C) while T VP(()andTY-^P(().

(3) Neither £ mnor £ are arithmetically expressible.

Proof. We take:

(3) Neither 9ÛJ+l\m) nor /3W+l) are arithmetically expressible. ■Remark 3.38. We have thus produced out of every nontrivial predi-cate in T intractable problems that cannotbe reduced to arithmeticalproblems. Actually there are infinitely many such problems for everyordinal a, as we ascend the set of infinite ordinals in T. Also, thegeneral nonarithmetic undecidable statement P(() has been obtainedwithout the help of anykind of forcing construction. ■

/^+1) =(7(G(mo(0(v;) )),

m («M« Tc(m0(»(")),x)e-'1G(mo(0 ))=il+ C(ra„(oH),z)^

C(mo(0 )̂ ),x) = A,(m0(0 )̂ ),xi,...,x r),

M(= /?^+1) =0butTV /?^+1) =OandTY -,(p(v+l) =0). ■

(1) Cm =x9^+l\m)+ (1-9^+l\m))y.(2) (= x+yP^+l\

176 N. C. A. DA COSTA ANDF. A.DORIA

AllNontrivialPropertiesAre Undecidable

We motivateour nextintractability results with aconcreteexample fromHamiltonian mechanics.

Suppose that we have axiomatized Hamiltonian mechanics over afixedphase spaceM with the help of aSuppes predicate within a first-order theory T (which we can take to be ZFC) as done in Section 2.We then have a predicate H(£) inLt that asserts, 'the expression£ isa Hamiltonian function. We can enumerateall other predicatesP& inLt,k anatural number,and wecan alsoenumerateall the theorems inT. We start such anenumeration,and select theorems of Twhich havethe form:(l)For&eLr,Tr-#(&).(2) For fc,Zj,i/j,G Lr,ThPfc(&) andTh -Pfc fe).

Outof that we list allnontrivial predicatesPfc thatapply to Hamilto-nian functions.

We have proved:

PROPOSITION3.39. IfTis arithmetically consistent, we can obtaina recursive enumerationofpairs ofexpressions £ 2i,&i+h ianaturalnumber, that representdifferentfunctions andsuch that:

where theP{arenontrivialpredicates(relative to theHamiltoniansoverM) in T that rangeoverHamiltonian functions. ■

Remark 3.40. The previous result allows us to obtain, out of an enu-meration of the theorems in Tand of allpredicates Pi in the languageof T, an enumeration of different expressions £2*. £2^+l* ia naturalnumber,for different Hamiltonians thatsatisfy (and donot satisfy)eachpredicatePi. ■

Wecan state:

PROPOSITION3.41. IfTis arithmetically consistent, then there is acountable undecidablefamily Cm ofexpressionsforHamiltonians in the

(1) Both ThHfoi) andTY H(&i+l);(2) T hPifoO o/k/T h -P;(6i+i)/


languageofTsuch that there isnogeneralalgorithm to decide,foranynontrivialpredicatePk relative to the Hamiltonians over M, whetherthatexpressionsatisfies (ordoes notsatisfy) Pk in T.

Moreover, there is noproblem in the arithmetic hierarchy that canbe made equivalent to the (algorithmetically unsolvable) decisionpro-ceduresfor Cm-

Sketch of the proof. We proceed in a stepwise manner. Let £1,...,be an infinite countable sequenceof mutually independentGodelsentences in T. Let m\, ...,be the corresponding Godel numbers.Form the0{

= 6(mi). Put: eJ=1-9j,c) =9j,alljG uj.

Let mrn be a variable that ranges over all 2nbinary sequences oflengthn. Code thoseby orderedn-tuples of O's and l'sand establishamap/ between those n-bit binary sequencesandalln-factor products£f e2 ... £"" ', so that in the 7-th position (0)1 ■-» ej(e|). Given m,rn,the associated product is /(rn). Given a specific model for T, for aprescribed length, all such sequencesequal0but for a single one, thatequals 1.

Order thepredicates Pi,P2,...;we write thatTh P»(C2x-i) whileTY^P^).

Now list all finite binary sequences and select from those an infi-nite set of mutually incompatible sequences such as 1, 01,001,0001,...Note the incompatible set {t\,t2,.. .};if Tj is any sequence, noteTj>,Tjii,..., its extensions. (Ageneralextension is notedr^.)

The expression we require is:

(s2* denotes sumoverall extensionsof equallengthin the factors c.) ■

Remark 3.42. Notice that, as each9^k\m) is either 0 or1, C willequalasingle oneamong theQ,despite the fact that itis anexpressionwith acountably infinite recursively defined number of symbols. Also,sincetheactual value of C can only be determined if we solve an intractableproblem in the arithmetic hierarchy, and since it contains (possiblyrepeated)representative expressions ofallproperties inT, we will onlybe able to check for any property if we first solve a complete problemof degree> o'.

C = Ci[(l/2)/(ri)+ (1/2)2 £ */(rr ) +...] ++ C2[(l/2)/(r2)+ (1/2)2 ]T7(r2' +..J +

178 N. C. A. DA COSTA ANDF. A. DORIA

Moreover, since every arithmetical statement has a finite, bound-ed degree in the hierarchy, nontrivial properties of C cannot be madeequivalent to anything in the hierarchy.

Finally, when we (informally, butin an actual computation) 'openup' theexpression for C, weobtain an infinite formal sum whose termsare products of iterated integrals (the expressions for the 9^); suchexpressionsare certainly uncommon in classical mechanics,but theyare the standard staple of quantumand particle physicists. So, nothingout of the mainstream here. ■

SCHOLIUM3.43. The lowest degree in(m maybe arbitrarily high.Proof. For anyp>owecan substitute9p+k for 9k in theexpression

for Cm-

COROLLARY 3.44. The decision problem within an arithmeticallyconsistentTforCm cannotbe madeequivalent to adecisionprobleminthe arithmetic hierarchy.

Proof. From Remark 3.42.

Remark 3.45. Rogers gives the proofby a diagonal argument (Rogers,1967, Section 14.8, Theorem XIII) of the following assertion: if theaxiomatization for ZFC is consistent then there is a sentencein ZFCthat cannotbe madeequivalent to an arithmetic sentencewith the toolsof set theory. As in most proofsby diagonalization the counterexampleobtainedisalegitimate assertioninZFCbut itasmeaninglessasGodel'soriginal undecidable sentencein arithmetic. However,out of theprevi-ousresult (Proposition3.41) wecan immediately obtainanonarithmeticsentencein set theory. ■

Sincethe set of theorems of T(supposed [arithmetically] consistent)is a creative set (Rogers, 1967), its complement is productive. There-fore, we can add theaxiom 'There isno solution foxpo{x\,...,xn) — 0over thenaturalnumbers' toanew polynomialp\(x\,...,xr) such thatthe Diophantine equationp\ = 0has no solution in M,but such thatagain wecannotprove that fact from T. Again, due toproductivenesswe obtain athirdpolynomial,p2,whichleads to a thirdundecidable sen-tence and to another extended theory T". We thus generatea sequence

SUPPESPREDICATESAND UNSOLVABLEPROBLEMS 179

of unsolvable Diophantine equations po —o,p\ =0,...,pj =0,...,which lead to undecidable sentences in the theories T(°) =T,rpl rp(i)j. .., j. ...

That construction is equivalent to obtaining a recursively enumer-able set out of a productive set (Rogers, 1967); no such constructionwillexhaust the productive set,and severalalternative recursively enu-merable subsets of theproductive set can be found.

We now use those pj as follows: if A is the map defined followingProposition3.7, we form the sequenceki(x)= Xpi,iG uj,and obtain:

PROPOSITION3.46. IfTis arithmetically consistent, thenM |= /% =0, alli,where the j3i aregivenby:

However, T^V Pi =0, allk < i.Proof. See (da Costa, 1991a).

Then,

PROPOSITION3.47. IfT is arithmetically consistent, thenfor everyrecursively enumerable extension of theaxioms ofT there is aHamil-tonian C all whose nontrivialproperties cannot be proved within thatextension.

Proof. The expression we require is:

(J2* denotes sum over all extensions of equal length in the factors c.)Here the #'s in the r'sare substituted for /3's. ■

Theseexampleslead to the immediate proofofageneralundecidabil-ity and incompleteness theorem that deals with those systems for which

Pi =v(Gi),

/ fci(s)e

C = Ci[(l/2)/(tI)+(1/2)2 £*/(t,.) + ---] ++C2[(l/2)/(r2)+(l/2)2 £7(T2' + ...] +

180 N.C. A. DACOSTA ANDF. A.DORIA

no nontrivialpropertycan be algorithmetically decided orproved. LetQ{x, ai,a 2

,.. ",an)be a Suppespredicate on the fixedparameters a\,Suppose given an enumeration of the predicatesPk in T. Again

we suppose that:WFor&GLr.ThQte).(2) For&,&,** j,cLT,TY Pfc(&) andTh -Pfc fe).(3) Out of that we list all nontrivial predicates Pk that apply to Q-

defined objects.Out of that weobtain:

PROPOSITION3.48. IfT is arithmetically consistent, we can obtaina recursive enumeration ofpairs ofexpressions 2i, * anaturalnumber, that representdifferent objectsandsuch that:

where the Pi arenontrivialpredicates (relative to Q) in T that rangeoverQ-objects. ■

Then:

PROPOSITION3.49. IfTisarithmetically consistent then:Undecidability. There isa countablefamily (m ofexpressionsforQ-objects in T such that there is no generalalgorithm to decide,for any nontrivial Q-property Pk in T whether that expressionsatisfies (or doesnotsatisfy) Pk.Incompleteness. There is a Q-objectfor which none of the non-trivial Q-propertiescan beprovedwithin T. ■

So, incompleteness of the nastiestkindis to be expectedeverywherein the axiomatized sciences. For density theorems related to thoseintractable problems see (da Costa, 1993a).

4. TWO APPLICATIONS

We discusshere twoproblems thathavebeen recently handled with thehelp of the present methods. Arnolds 1974 questions on polynomial

(1) Both Th Q(i2i)andTY Q(&i+i).(2) T hP*fe)«^T h -Pifei+i),


dynamical systemsand the incompleteness of the theoryof finite Nashgames. Full details will appearin references (da Costa, 1992d,1992e).

PolynomialDynamicalSystems Are Undecidable andIncomplete

In the 1974 AMS Symposium on the Hilbert Problems, V.I.Arnoldsuggested the following questions for the list of 'Problems of PresentDayMathematics' that wascompiled at that symposium:

Is thestability problemforstationarypointsalgorithmically decid-able? The well-known Lyapunov theorem solves the problem inthe absence of eigenvalues with zero real parts. In more compli-cated cases, where the stability depends onhigher order terms inthe Taylor series, there existsno algebraiccriterion.

Let a vectorfieldbe given bypolynomials ofa fixed degree, withrationalcoefficients. Doesanalgorithm existallowing us todecidewhether the stationarypoint is stable?

A similarproblem: Doesthereexistanalgorithm todecide whethera planepolynomial vector field has a limit cycle?

See(Arnold,1976a).We notice that Arnoldsays nothingaboutbound-ary values,despite the fact that the system's behavior may vary wildlyas a function of the initial vales; he also imposes noconditions on thegeometry of the manifolds where those dynamical systems are to befond. Therefore we may offer as a counterexample a specific poly-nomial dynamical system over Z with fixed initial values such that noalgorithm in the sense ofArnold exists for therequiredproperties.

LetPbeanontrivialpredicate (relative to theadequate vector fields)in T. We assert:

PROPOSITION4.1. IfTis arithmetically consistent, then:(1) There is anexpressionfora vectorfield v over Rn in T such that

M\='v is a smooth vectorfieldon Rn,n>2 withpropertyP',andTV -> 'v is a smooth vectorfieldon Rn withpropertyP.

(2) There is an expressionfor a vector field v in T such that thesentence 'v is a smooth vectorfieldon Rn,n> 2 with propertyP'is T-arithmetically expressibleasann+i sentence,m>\,butsuch thatforno m it willbe equivalent toa Tim sentence.

182 N. C. A. DA COSTA AND F. A.DORIA

(3) There is an expressionfor a vector field v in T such that thesentence 'v is asmooth vectorfieldon Rn,n> 2 withpropertyP

'cannotbe taken to be T-demonstrably equivalent to any sentencein the arithmetical hierarchy.

(4) We canexplicitly findanexpressionfor apolynomial vectorfields over7.on Rm,mfixed, such that,for agivenP,M \= P(s),butTV P(s)andTY^P(s).

HereMis amodel whereTis arithmetically consistent.Proof. The first three assertions are immediately proved with the

methods presented in (da Costa, 1991a) and expandedin the previoussection of this paper. Actually they generalizeourprevious results ondynamical systems (daCosta, 1991a).

The lastassertion is the negative solution (in an obvious sense) to aproblem related to Arnolds problem; for details on Arnolds see (daCosta, 1992e). We make two remarks:

" We write x= (xi,...,Xj),where j=dimx." We notice that if the polynomialp(x) is notidentically zero, thengiven the expressionbelow for a smoothelementary function,

if rG R, then for s(x,y) =u(x,y) —r, R2-7 — s_1(0,0) is openin R2-7,where vis taken as areal-valued function onR2j. (See onvaboveDefinition 3.2.)

The first lemma we requireis:

LEMMA 4.2. IfTis arithmetically consistent, then:

together with

(1) «(x, y) =(j+1)V(x)+ £ 0&4(*)1.i=l

(2) tt(x,y) =(j+ l)4b2(x)+J2yM(*)l

(3) yi—

sin7TXi, wi = cos7TXj,

SUPPESPREDICATES AND UNSOLVABLE PROBLEMS 183

is the unique solution of the followingpolynomial dynamical systemwith coefficients in Z over R3j+l endowed with the usual Cartesiancoordinates andthe correspondingEuclideanmetric tensor:

dxi dvdt dxi

for i=l,2, 3,...,j,plus the boundary condition u(0,0) =p(0) (withan obvious abuseoflanguage);

whereagaini= 1,...,j,withboundary conditions 7/i(0) =0,Wi(0) —1. Andfinally,

withboundarycondition z(0) = 7r.

Proof. Equations (4) are immediately integrated (since they aregradientequations)toEquation (2). Equation(4.2) trivially implies thatz = 7r, and therefore (also from Equations (4)) wehave:

(no sumoni). Thatsystemhas the solutions yi = sin7TXi,Wi =cos7TXf.Since the du/dxi arepolynomials overZ whicharenot identically zero,the lemma is proved. ■

We need a secondlemma: here we allow v torange over R — {I}.Therefore for eachxiandfor each yi atmostacountably infinitenumberofpointsare tobe deletedfrom thecorrespondingdomains,thosepointsthat are solutions of the (fixed) equation v — 1= 0 over the reals. We

,« dyi idu(5) Id= W Zdx~- '

dwi i dvdt dxi

" t=°

dyi— = irwi,dxi

dwi-T-

= -*yi,dxi


then state:

LEMMA 4.3. IfTisarithmetically consistent, then

is the uniquesolution to thepolynomial equations on R — {I},

wherec=—l, v G (— oo, 1) andc=o,v G (1,+oo);and

together with theconditions v(0) =1— ti(0), v G (— oo,1), andv=0/oru G (I,+co).

Proof. Immediate.

We conclude the proof with still another lemma: let S, Tbe poly-nomial vector fields over Z on a A;-dimensional real smooth manifoldMsuch thatT h P(S) andT h + vT). Let us form the vectorfields S = (x, 0)and T =(0,y). Then:

LEMMA 4.4. IfTis arithmetically consistent, then thereis an expres-sion for thepolynomial vectorfieldover Z of theform:

such thatM \= $ =E,butTV $ /E. (AgainMisamodel where Tis arithmetically consistent.)

Proof. It suffices to take thepolynomialp(x)asinPropositions 3.17or 3.19,and getv as inEquations (2) and (3), and finally obtain v outofit. ■

This lemma concludes ourproof.

Let M'be themanifold where the expression for $ is defined. Then

(8) v =l-v

dc(9) Tt

=°'dTj dv(10) Tt= c*

(11) $= £ +7jT


SCHOLIUM 4.5. //Tis arithmetically consistent, then T h 'Thedimensions ofMl,m<2k +3(j'+ 1)'. ■

SCHOLIUM 4.6. IfT is arithmetically consistent, then M |= Thedimension ofMl,m= k', while

TV 'The dimension ofM',m =k\and

TV -> 'Thedimension ofM',m =k '.

So we might end up dealing with a planar vector field despite thefact that T will notrecognize it!

COROLLARY 4.7. IfT is arithmetically consistent, then it has anexpression(inM)foraplanarpolynomial autonomousvectorfieldwithcoefficients in Z such that we cannotprove (from the axioms ofT) thatthatfield isplanar. ■

With thehelp of Proposition 3.37 we can state:

COROLLARY 4.8. IfT is arithmetically consistent, then it has inMan expressionforaplanar polynomial autonomousvector field xwith coefficients in Z such that the assertion 'x is planar' cannot bemadedemonstrablyequivalent withinTtoanyarithmeticalassertion. ■

Other consequencesof theprevious resultsareeasily stated andproved.

Remark 4.9. Our exampleofapolynomial vector field withundecidableproperties has either ahighapparent dimension or an apparently veryhigh degree. If we take k = 2 and the original Diophantine universalpolynomialp with 11variables,m=46 and the degree is ridiculouslyhigh. For degree 16 we will have that m = 250. A discussion onthepossibility of reducing those values can be found in (da Costa,1992e). ■

186 N. C. A. DA COSTA ANDF. A. DORIA

IncompletenessofFiniteNash GamesThe results givenhere are presentedin full detail in (daCosta, 1992d).It is intuitively obvious that, for a finite Nash game, we can algorith-mically check for equilibria in it. However, things turnout to be muchmore delicate when we work within aformalization for the theory offinite Nash games. We start from:

DEFINITION4.10. A noncooperativegame is givenby the yonNeu-mann triple T = (N,Si,Ui), with i= 1,2,...,N, where N is thenumber of players, Si is the strategy set of player % and Ui is the real-valued utility functionUi :Yli Si -+ R. ■

Then:

PROPOSITION4.11. IfTisarithmetically consistent, then there is anoncooperativegameTwhereeachstrategysetSi isfinitebut such thatwe cannotcompute its Nashequilibria.

Proof. Let V andT" be two different games with the same numberof players but with different strategy sets and different equilibria. IfP = /3(w+l) in Corollary 3.36, wecan form theutility functions:

Therefore the game T = (N,Si,Ui) does not have a decidable set ofequilibria. ■

SCHOLIUM 4.12. Determining the equilibrium setofT is anonarith-meticproblem. ■

We cantake Tbarely beyondformalized arithmetic N: let N* be thetheory whoseaxioms are N plus acompatible version of the separationaxiom. Therefore we can give an 'implicit' definition for sets of num-bers. We put Ui : Y\iSi —+ uj. We now use q as in Proposition 3.21and form the utility functions Ui = u\ +qu". The remainder of theargumentgoes as usual. Thus:

'I Q II

Ui =Ui+ fJUi .


PROPOSITION4.13. 7/N* is arithmetically consistent, then there isagame f inN* such thatthe assertion 'u{ =u\ 'is undecidable m N*.■

5. CONCLUSION

In 1987 the authors started a research program whose goals weretwofold: we wished to axiomatize as much as possible of physics inorder to searchfor physically (i.e. empirically)meaningful undecidablesentencesand physically meaningful unsolvable problems within ourformalizations. The axiomatization program was fulfilled in part withour construction of Suppes predicates for classical physics out of anunified framework that reminds one of ideas and tools from categorytheory (da Costa, 1992a). The incompleteness portion of the programwas certainly much more difficult to pursue: we had two candidatesfor undecidable problemsinphysics,and we hopedthatthose problemsmight lead us to thedesired incompletenessproof. Thecandidates wereclassification schemes in generalrelativity andHirsch'sproblem ontheexistence of algorithms tocheck for chaos in dynamical systems. (Seethe references in (daCosta,1990b, 1991a).

At first webelieved that forcingmodels might provide the indepen-denceproofs we were looking for (daCosta, 1990b,1992b), since forc-ing is an obvious source of so many mathematically relevant undecid-able statements in ZFC.Therefore weexploredour Suppes-formalizedversion of generalrelativity and the corresponding sets of noncompactspace-times, but the results on incompleteness were meager at best.Then,due to a suggestionby Suppes, we turned to Richardson's 1968undecidability results in analysis. It was immediately apparent to usthat those resultsentaileda full-fledged incompleteness theorem, whichweextended,after some failedefforts, to the explicitconstructionof thehalting function for Turing machines within classical elementary realanalysis.

Wethenstarted to turnoutseveralundecidability andincompletenessresults indynamical systemsandrelatedquestions,as wellas counterex-amples to theexistenceofalgorithms of the kind thatHirsch wasaskingfor in chaos theory. Again Suppespointed out to us that there wassome sort of a very generalundecidability and incompleteness theoremat work here; the first version of that theoremis Proposition 3.29 in (daCosta, 1991a).

188 N. C. A. DA COSTA ANDF A. DORIA

That theorem is so general (it reminds one of Rice's theorem incomputer science (Machtey, 1979; Rogers, 1967), from which it canbe derived) that for a moment we felt it might announce some kindof imminent disaster, as every nontrivial property in the language ofclassicalanalysiscanbemadeintoanundecidability andincompletenesstheorem. So, in generalnothing interesting can be decided and verylittle can be proved. (This point wasalso emphasizedby Stewart inhiscommenton our results (Stewart, 1991).) At the same time we receivednotice thatour results were at first met with disbelief by researchers indynamical systems, precisely due to that very general incompletenessstatement;it was thenarguedthatthoseresults werecorrectbut 'strange'due to some undetected conceptual flaw in the current view on thefoundations ofmathematics.

Yet our assertions have a very clear, letus say,practical meaning:if what we mean by 'proof in mathematics is algorithmic proof, thatis to say, something that can be simulated by a Turing machine, thenvery difficult problemsare tobe expectedeverywhereat the very heartof everyday mathematical activity; more and more innocent-lookingquestions are tobe found intractable (as we now see in chaos theory)and(to repeat an example that we haveoffered before) systems willbeformulated that havea tangled,chaotic appearance when approximatedona computer screen,but such thatno proofof their chaotic propertieswill beoffered within reasonable axiomatic systems such as ZFC.

Godel incompleteness isnooutlandishphenomenon;itisanessentialpart of the way we conceive mathematics.

What can we make out of that? We do not think that there is anyessential flaw inthepresent-dayview aboutfoundational concepts;how-ever we think that our incompleteness theorems point out very clearlywhere the problem lies. Nothing will be gained by adding 'stronger'and 'stronger' axioms to our current axiomatizations. But wecertainlymust strengthen our current concept of mathematical proof. Turing-computable proofs are not enough, for Church-Turing computation isnot enough. We must lookbeyond.

6. ACKNOWLEDGMENTS

It certainly is a pleasure to dedicate this work to Pat Suppes on theoccasionof his 70th anniversary. We deem that dedication tobe espe-cially adequatesinceour workon theundecidabilityand incompleteness


of axiomatized theories has been inspired from its very beginnings bymany fruitful intuitions and suggestionsof his.

We also wish to acknowledge the constant cooperation andinterestof our coworkers: J.A. deBarros,A. F.Furtado do Amaral,D.Krause,andM.Tsuji. Thepresentideas werealsodiscussed withM.Corrada,R.Chuaqui,and D.Mundici,to whom we owe friendly remarks andcriti-cisms. A first presentationof our incompleteness results was carefullyreadbyM.Hirsch,to whom weare indebted for severalcorrections andimprovements.

The present paper was completed while the second author held avisiting professorship at the University of Sao Paulo's Institute forAdvanced Studies under aFAPESP scholarship program. The authorsalsoacknowledge support from the CNPq(Brazil) scholarship programinphilosophy and wish to thank the ResearchCenter on MathematicalTheories of Communication inRiode Janeiro (CETMC-UFRJ) for theuseof its computer facilities.

N.C. A. da Costa,ResearchGroup onLogic andFoundations,InstituteforAdvancedStudies,UniversityofSao Paulo,05655-010 SaoPauloSP,BrazilF.A. Doria,Research Center onMathematical Theories ofCommunication,SchoolofCommunications,Federal UniversityatRio deJaneiro,22295-900RioRJ, Brazil

REFERENCES

Arnold, V I.: 1976a, 'Problems of Present Day Mathematics', XVII (DynamicalSystems andDifferentialEquations),Proc. Symp. PureMath.,28, 59.

Arnold, V I.: 1976b, Les Methodes Mathematiques de la Mecanique Classique,Moscow: Mir.

Atiyah,M.F: 1979, Geometryof Yang-MillsFields,Pisa: LezioneFermiane.Bourbaki,N: 1957, Theoriedcs Ensembles,Paris:Hermann.Bourbaki,N.: 1968, TheoryofSets,Boston: Hermann andAddison-Wesley.Dalla Chiara,M.L.andToraldodi Francia, G.: 1981,Le TeorieFisiche, Boringhieri.Cho, Y.M.: 1975, J.Math. Phys.,18, 2029-2035.da Costa, N. C. A. andChuaqui, R.: 1988,Erkenntnis, 29,95-112.

190 N. C. A. DA COSTA AND F. A. DORIA

da Costa, N. C. A. and French, S.: 1990a, 'The Model-Theoretic Approach in thePhilosophy of Science',Philosophy ofScience, 57,248-265.

daCosta, N. C. A.,Doria,F. A., andde Barros, J.A.: 1990b, Int. J. Theor. Phys., 29,935-961.

daCosta,N. C.A. andDoria,F. A.: 1991a, Int.J. Theor. Phys., 30, 1041-1073.da Costa,N. C.A. and Doria,F. A.: 1991b, Found. Phys. Utters,4,363-373.da Costa,N. C. A. andDoria,F A.: 1992a, 'SuppesPredicatesforClassicalPhysics',

to appearin: A. Ibarraet al. (Eds.), The Space ofMathematics, De Gruyter.da Costa, N. C. A. and Doria, F. A.: 1992b, 'Structures, Suppes Predicates and

Boolean-ValuedModels in Physics', to appear in: P. Bystrov and J. Hintikka(Eds.),Festschrift inHonor ofV. I.SmirnovonHis 60thBirthday.

da Costa,N. C. A. andDoria,F. A.: 1992c, 'Jaskowski'sLogic', preprintCETMC-10.da Costa,N. C. A.,Doria,F. A., andTsuji, M.: 1992d, 'The Incompleteness ofFinite

NoncooperativeGames withNash Equilibria',preprint CETMC-17.da Costa,N. C. A., Doria,F A., Baeta Segundo, J. A., and Krause, D.: 1992e, 'On

Arnolds1974Hilbert Symposium Problems',inpreparation.da Costa,N. C. A.,Doria, F. A., and Furtadodo Amaral, A. F: 1993a, Int. J. Theor.

Phys. 32,2187-2206.daCosta,N. C.A.,Doria,F. A.,FurtadodoAmaral,A.F,andde Barros,J. A.: 1994a,

'Two Questionson the Geometry ofGauge Fields',Found. Phys., toappear.da Costa,N. C. A. andDoria,F A.: toappear, MetamathematicsofPhysics.Davis, M.: 1982, 'Hilbert's Tenth Problem Is Unsolvable', in: Computability and

Unsoh'ability,2ndedition,Dover.Dell,J. and Smolin,J.: 1979, Commun. Math. Phys.,65, 197-212Doria, F. A.: 1975, Let.NuovoCim.,14, 480^182.Doria,F. A.: 1977, J.Math. Phys.,18, 564-571.Doria,FA.: 1981, Commun.Math. Phys.,79,435-456.Doria,F A.,Abrahao,S.M., andFurtadodoAmaral,A.F: 1986,Progr. Theor. Phys.,

75, 1440-1446.Jones,J. P.: 1982, J.Symbol. Logic,47, 549-561.Kobayashi,S.andNomizu,X.: 1963/67, FoundationsofDifferential Geometry,Vols.I

and11, New York:Wiley.Krantz, D. H.,Luce, R. D., Suppes, P., and Tverski, A.: 1971, The Foundationsof

Measurement,Vol.I,New York: AcademicPress.Kreisel,G.: 1976, 'ANotion ofMechanistic Theory',in: P. Suppes (Ed.), Logic and

Probability inQuantumMechanics,Dordrecht: D. Reidel.Lewis, A. A.: 1991, 'On Turing Degreesof Walrasian Models anda GeneralImpossi-

bilityResult in theTheory ofDecision Making',preprint,University ofCaliforniaatIrvine, Schoolof Social Sciences.

Lewis, A. A. and Inagaki, V: 1991, 'On the EffectiveContent of Theories', preprintUniversity of California atIrvine, Schoolof Social Sciences.

Machtey,M.andYoung, P.: 1979,AnIntroduction totheGeneralTheory ofAlgorithms,Amsterdam: North-Holland.

Narens,L.andLuce,R.D.: 1986,Psych. Bull, 99, 166-170.Richardson,D.: 1968, J.Symbol. Logic,33, 514-520.Rogers, H., Jr.: 1967, Theory ofRecursive Functions and Effective Computability,

MacGraw-Hill.

SUPPES PREDICATES AND UNSOLVABLE PROBLEMS 191

Scarpellini,B.: 1963, ZeitschriftfurMath. Logik v.Grundl. derMath.,9, 265-289.Stegmuller, W.: 1970, Theorie undErfahrung,Vol.I,Berlin: Springer.Stegmiiller, W.: 1973, Theorie undErfahrung, Vol. 11,Berlin: Springer.Stegmuller,W.: 1979, TheStructuralist Viewof Theories: APossibleAnalogueof the

BourbakiProgramme inPhysical Sciences,Berlin: Springer.Sternberg, S.: 1964,Lectures onDifferential Geometry,Prentice-Hall.Stewart, I.: 1991, Nature,352,664-665.Suppes,P.: 1967,Set-TheoreticalStructures inScience, mimeo., StanfordUniversity.Suppes,P.: 1988, Scientific Structures andTheirRepresentation,preliminary version,

StanfordUniversity.Taubes,C.H.: 1987, J. Diff. Geometry,25, 363^112.

COMMENTSBYPATRICK SUPPES

Foroverhalfacenturysince Godel's famousresults, the incompletenessof any axiomatization of classical analysis has beenknown, but in thepast several decades other results have stressed how far we are fromhaving anything like a real algorithmic approach to any significantclass of mathematical problems. One example is the realization thatTarski's decision procedure for elementary algebra is not feasible, inthe technical sense that the computations grow exponentially in thelengthnof any formula whose truth is to be decided. So evenpositiveresults on decision procedures themselves do not guaranteepracticalapplicability. Itused tobe thoughtofaspartof the folklore,butofcoursenot in any sense proved, that 'most' problems in mechanics that werenot toocomplicated to formulate wouldhave relativelystraightforwardsolutions. Therecentdiscoveryofchaotic systemsinallsortsofdomainshas shownhow the problems that fill the textbooks of mechanics are acarefully selected group. That such difficulties were lurking about hasreallybeenknown since theintensiveworkonthe three-bodyproblem inthe nineteenth centuryand theculminating negativeresults ofPoincare.One way to put it is that any undergraduate in physics can derive thedifferential equations governing familiar cases of the restricted three-bodyproblem-therestrictedproblemis when the massof the third bodyis negligibleand therefore does not influence the regular motionof theother two bodies. But the problemof findingmathematical solutions ofthe differential equationsis whollyunmanageable for most cases.

Da Costa and Doria have embarked on a program, as indicated bythe many additional references in theirpaper, toshow how widespreadthepresenceof unsolvableproblems is inphysics andother sciences.

192 N. C. A. DACOSTA AND F A. DORIA

Thatdifficulties inalldirections can be found with alittle effort wasanticipatedin some senseby Richardson's 1968 results onundecidableproblems involving elementary functions of a real variable, but whatNewtonandChico have doneis extend thiskindofresult much further.It has probably been the feelingof most nonlogicians working in theseareas that results on incompleteness or undecidability were really notgoing to stand in the way of standard mathematical progress. Thesefoundational results were quirky oneson the edgesand couldnot haveanything to do with anything lying in the heartland of analysis andmathematical physics. Newton and Chico are certainly in the processof showinghow this is not thecase.Ihave learned a lot from both Newton and Chico over the years,

especially through theirlivelyparticipation inmy seminars atStanford. Iespeciallybenefittedfrom longconversations withChicoDoriaacoupleofyearsago whenhe spentayear atStanford. Infact thoseconversationsabout thefoundations ofphysics,and recentdiscussions with Acacio deBarros,his former student from Brazil, havecompletely rekindled myinterest in working onthe foundations of quantummechanics.

The kind of results that Newtonand Chico have obtained havealsobeen a motivation for the work thatRolando ChuaquiandIhave beendoing on constructive foundations of infinitesimal analysis, especiallyaimed atphysics. Ourobjective hasbeen to giveafree-variablepositivelogic formulation of infinitesimal analysis which is finitarily consistentand which is still strong enough to prove,in somewhat modified form,many of the standard theorems that underlie constructive methods intheoreticalphysics. Itis afeature of theoretical physics that it ismainlyelaborate computations, either of a symbolic or numerical kind from amathematical standpoint. Little is ever done in theoretical physics assuchaboutprovingexistence theorems. Here,of course,Iamdrawingadistinctionbetween theoretical andmathematical physics,a distinctionthatis nowrelatively wellestablished. Ofcourse,necessarily such weaksystems as Rolando andIhave been workingon cannot do everythingthatone wants. We do think thatmuchcanbedoneanditis important tounderstand where the boundaryexists. The important methodologicalpoint is thatby usinginfinitesimals,wecan give afree-variable formu-lation of such standard theorems ofcalculus as themean value theoremor Green's theorem. Ishould mention that these theorems are provedin approximate form, that is, equality is replaced by an equivalencerelation that means there is adifference that is infinitesimal.


Ido not mean to suggest that the system that Rolando andIhavebeen workingongetsaroundall theproblemsuncoveredbyNewtonandChico. What it does show is that weak constructive systems, demon-strablyconsistent,are sufficient for a greatdeal of the work.

REFERENCES

Chuaqui,Rolando and Suppes,Patrick: toappear, 'Free-Variable AxiomaticFounda-tionsofInfinitesimalAnalysis: AFragment withFinitaryConsistencyProof.

Suppes,PatrickandChuaqui,Rolando: inpress, 'AFinitarilyConsistentFree-VariablePositiveFragmentof InfinitesimalAnalysis',Proceedingsofthe9thLatinAmericanSymposiumonMathematicalLogic,heldatBahiaBlanca,Argentina, August 1992.

JOSEPHD. SNEED

STRUCTURALEXPLANATION

ABSTRACT. This paper sketches an account of scientific explanationbased on asemantic ormodel theoretic conceptionof scientific theories derivativefromthe workofSuppesandhis collaborators[8, 10, 14] andaBayesianaccount ofscientificreasoningsuggestedby Rosenkrantz [9], HowsonandUrbach [7] and Earman [3] amongothers.The model theoretic conception of scientific theories has been describedindetailbySneed [12], and Balzer, Moulines and Sneed [I]. A somewhatsimilar view - thesemantic view-has beendevelopedby vanFraassen [15], Suppe [13] andothers. Thediscussion in thispaperemploys the conceptualapparatusdevelopedin [I]. However,mostofwhatis saidcan beapplied toamoregeneral 'semantic' conceptionoftheories.The applicationofthisconceptionofscientific theoriestoscientificexplanationhasbeendevelopedfromsomewhatdifferentperspectivesbyForge[4,5]andSintonen [11]. Thispapergeneralizes (and somewhatsimplifies)the workofForge,indicates how it appliesto so-called 'functional explanations', and connects it with the Bayesianaccount ofscientificreasoning toprovidean account of howcompetingscientificexplanationsare(shouldbe) evaluated.

1. THEORIES AS MODELCLASSES

On the model theoretic view, the simplest kind of scientific theory(theory element in the vocabulary of [1]) is an orderedpairT — (K,I)consistingof aconceptual core X anda range of intended applicationsI.Very roughly, X defines a set of possible situations or ways thingscouldbe which wecall

Content(if).

The theory isused tosay something-make aclaim-about itsintendedapplications I. The claim is simply thatIis one of those situationscharacterized byK.Formally, thisclaim is just that

(1) Ic Content^).

Note that formally Content(if) is a set of sets (of something). Sothat it makes sense to say that a set / is a member of Content^).Roughly,we should think ofIas the totality of potentialdata the theory

195


196 JOSEPHD. SNEED

is supposed to account for. Individual members of/ are pieces of thatdata- individual applications of the theory. Content(if) characterizesentireconfigurations of data-not just single individualapplications.

In cases of genuine scientific interest, it appears thatIwill be anintentionally described, 'open-ended'class - for example the set of allparticle collisions - the set of all state-level societies. In these cases,though wemayhavegoodreasons tobelieveItobe finite,we willneverbe certain that we haveexamined all its members. Thus,however surewe may beof the truthof (1), there exists thepossibility of discoveringadditional members of / which would cause us to revise our epistemicattitude toward (1). Indeed,Ithink one mightpersuasively argue that anecessary condition on 'real' science is that / be 'open-ended' in thisway.

It is perhaps worth noting that (1) may be recastinto the form of a'general statement':

(1') For all X, if X is anIthenX is a Content^).

Inthis form theclaimof atheorylooks verymuchlikethe 'generallaws'that play a major role in traditional accounts of scientific explanation.Roughly, whatmakes claims like (1), (1') 'interesting' is that theypointout some feature that allmembers ofI-both those we have examinedand those we havenot-allegedly have. The major differences betweenthe account ofexplanationIwilloffer and the traditional accountis thatmy accountmakes explicit use of the set-theoretic properties of thingsthat are (in) / and things that are (in) Content(if). The 'features' thatmembers of / allegedly have are 'structural' features most naturallydescribedby the language of set theory.

Again, in cases of genuine scientific interest, Content(i^) has theformal property that, whenever V is in Content(i^), so is every non-void subset of Y. Intuitively, this means that inductive evidence for(1) may be provided by examining subsets of / and discovering themto be in Content(i^). Indeed, one, overly simple picture of 'theoryconfirmation' is that we justexamine more andmore members of /.Ifthe set of those we have examinedcontinues tobe in Content(if ), webecome more andmore confident that (1) is true.

Itis convenient to view thissimple picture of theoryconfirmation asa 'research program' carried out over time. Thus, at any time t duringthe course of this program there will be some subset Ft of / that is'believed with good reason' (by members of the scientific community

STRUCTURAL EXPLANATION 197

pursuing the research program) to be in Content(if). We may callFtthe set of 'firm' applications of Tat time t. Firm applications that arein fact in Content(if) we may call 'successful' applications. On thecustomary accountof 'knowledge', theusersof Tknow that successfulapplications areinContent(if). (Theseideas areelaborated indetail in[I], Ch.V.)One mayprovide an accountof explanationbasedeither on'firm' or 'successful'applications. Thedifference is justin theepistemicstrength of the notion of explanation. To make matters simple,Iwillopt for 'successful' andan epistemically strongnotion of explanation.

So far, very little has been said about theexplicitly model theoreticnatureof this conception of scientific theories. This becomes apparentonly when wecharacterize X inmore detail. Intuitively, X consists ofa vocabularyor conceptual apparatus characteristic of the theory (Mp)from which a non-theoretical (Mpp) may be distinguished, some laws(M) formulated in the full vocabularyand someconstraints (C)whichlimit the ways in which theoretical concepts may be applied acrossdifferent applications of the theory. Thus Xis an ordered 4-tuple:

What makes this conception 'model theoretic' is that each of the ele-mentsofXmaybe viewedasaclassofmodels-set theoretic structures.These model classes may sometimesbe characterized by sentences infirst-order logic. But,for theories of genuine scientific interest, char-acterizing them in the language of set theory by defining a set theoret-ic predicate is almost always easier and sometimes the only practicalmeans. The insight that classes of models characterized in this waycould serve to illuminate parts of interestingempirical science isdue toSuppesand hiscollaborators [8,10, 14].

These model classes hang together in the following way. Non-theoretical structures

-members ofMpp-areobtainable frommembers

of Mp simply by loppingoff their theoretical components. Formally,there isa 'forgetful functor' that mapsMp ontoMpp.Mis justasubsetofMp.Itrepresents scientific laws inthat this subset is acharacteristicway in which the 'values' of the components of single members ofMp are related. C is a subset of the power set ofMp (Pot(Mpp)). Itrepresentscharacteristic waysin which the values of thecomponentsindifferent members ofMp arerelated.

Content(iiT) consists of all the sets Nof non-theoretical structures(subsets of Mpp) that canbe filled out with theoretical components in a

(2) X= (Mp,Mpp,M,C).

198 JOSEPHD. SNEED

wayso that: (i)each individualmember ofAf is filledout toa memberofM(satisfies the laws);and(ii) the setof theoretical structuresproducedby filling out every member of N is a member of C (satisfies theconstraints).

The set of intended applications / is a set of non-theoretical struc-tures described in the manner sketched above. Thus the claims of thetheory (1) amounts to saying that / (and everysubset of/)can be filledout with theoretical components in some way that satisfies these condi-tions. That this renditionof theempirical claimof theories likeclassicalparticle mechanics,classical equilibrium thermodynamics andothers isplausible has beenarguedin detailin [12, I]. These arguments will notbe repeatedhere.

Somewhat more complicatedkinds of scientific theories can be rep-resentedas 'nets' of theories of the sort describedhere (theoryelements)and an appropriate concept or 'content' can be defined for these nets.Sintonen [11] hasexploitedthismore generalconceptofscientific theoryto explicate some contextual,pragmatic aspects of the concept of sci-entific explanation. This macroscopic,net-level account ofexplanationeffectively presupposes a microscopic, theory-element-level accountwhich is not explicitly provided. This paper complements the work ofSintonenby providing the microscopic accountit presupposes.

2. SCIENTIFICEXPLANATION

These ideas may be applied to scientific explanation in the followingway. Veryroughly, theoccurrence ofXisexplainedbyshowingittobeaspecificpartofsomethingelse thatregularly occursin a well-understoodway. For example, the occurrence of long-neck, brown bottles in therecyclingbin in front of my house is explainedby showing it to be apartofaregularlyoccurringpatternofbeverageconsumptionandrefusedisposal characteristic of myhousehold.

A bit more precisely, to explain some phenomenon X is to showthat X is a specific aspector part of some largerphenomenonsharingsome interesting features with (generally occurring in the same waysas) a wider class of phenomena. To recall Hempel's classic example,we explain the burstingofmypipes lastDecember 28 bynoting thatmypipes are part of a thermodynamic system and thermodynamic systemsbehave in certain general ways described in detail bya physical theorylike classical equilibrium thermodynamics. We explain the entry of


the 8-ball into the corner pocket by noting the context in which thisevent occurred - a collision sequence involving the 8-ball, the cue-ball and a third ball. Further, this sequence of collisions has somegeneral properties characterized by classical collision mechanics. Weexplain the presence of turquoise fragments on part of the surface ofa grooved stone by noting features of the context in which the stonewas found. It was in a stone enclosure on a river bank about a milefrom a turquoise outcropping. We conclude the stone was part of acertain kind of 'technological system' -perhaps a 'mineral extractivesystem. Further,weare able tosaysomethingingeneralabout 'mineralextractive systems'. That is, wehave a 'theory' about them.

Though theserough 'explanationsketches' donotmake itexplicit,itis worth noting the somewhat obviousfact that the intuitive plausibilityof these explanations depends on identifying the phenomenon to beexplained with a specific kind of part of the larger system. Though itis true, it is not enough to say the turquoise fragments were just anypartof a mineralextractive system. We must specifically identify themas debris resulting from the use of a specific kind of tool used to breakturquoisebearing rock.

This rough, intuitiveglossof 'explanation' doesnotsound toodiffer-ent (Ihope) from more traditional accounts. Surely, it should not. Myintention is tocapture the same intuitiveideaas thetraditional accounts.The difference between my accountand the traditional accounts lies inthe way these initial intuitions are made precise. Note too that theseexamples are prima facie muteon the traditionally controversial issueof whether (a description of) the explained phenomenon is logicallyentailed by theexplanation.

What all theseexampleshavein commonis thata phenomenonXisbrought within the scopeof someplausible theory. The epistemic valueofexplanationis that itallows us toview Xas anaspect of 'the familiar'- things we think we understand. Tobegin to make this accountmoreprecise, wemight say roughly this. Toexplain somephenomenonXisto show that X'is' (may be viewed as)apartofsomeknown,successfulapplications of a theory that can be taken seriously. On this account,providingan explanation for Xcan be analyzedinto three parts:(A) Showing X is a specific part of some A;(B) ShowingA are successful applications of some theory T;(C) ShowingTcanbe taken seriously.

200 JOSEPHD. SNEED

Let us postpone saying what it means to say a theory can be takenseriously.Iwill return to this aspectof explanation in Sections 4and5below. Rather,Iwill focus on (A) and (B) -explaining what it meansto say that X is brought within the scope of theory T.

The phenomenon X is explainedby bringing it within the scope oftheory T = (X,I). This simply means that weidentify X as a specificpart of some intended applications A of T that are in fact successfulapplications of T. Thus we explain why the 8-ball went in the cornerpocket by describing a collision of which its path to the corner pocketwas a part and noting that this collision is a successful application ofclassical collision mechanics - i.e. it conserves momentum withoutrequiringus to assign masses to the billiard balls that are incompatiblewith data about other collisions in which theyhave been observed. Inthis exampleXis shown tobe apart of a singlesuccessfulapplication.This is perhaps the typical case. However, the suggested formulationalso allows for the possibility that X is shown to be a part of a set ofsuccessful applications. For example,X might be data about the roleof a single particle-the cue ball-in several different collisions.

Note that 'part' here is tobe understood toinclude 'improper part.Thus,showing that X is (all of) some successful intendedapplicationcounts as explaining X. In particular, showing that X is an intendedapplication that can be added to the set of unsuccessfulapplications (attime t) without the resulting set falling outsideof Content(X) countsasexplainingX.This appears tobe the caseconsideredby Sintonen [11]for theory nets. This kind of explanation is 'progress' of the researchprogram associated with the theory. But thereare more mundane kindsof explanation -perhaps those involving technological applications ofthe theorythat are not cases of 'progress' in anyinteresting way. Theaccountofferedhere includes both cases.

Very roughly wehave somethinglike this:

(3) T = (X,I) explains X iff X is a specific part of a set of somesuccessful intendedapplications ofT.

Intuitively, a 'part of an applicationis just something that canbe filledout to a (full) application. For example,it mightbe a specification ofthe velocities of asubsetof theparticlesinvolved in acollision;itmightbe the specification of the velocities of all the particles before,but notafter, thecollision. We need to talk aboutparts of sets of structures andthis maybe done in roughly the following way.


(4) Xis partof A iffeach member ofX ispartofexactly onememberof Aand each member of Acontains exactly onemember of X.

Formally, a 'part of an individual set-theoretic structure is just asubstructure. A set-theoretic structurehaslotsofdifferent substructures- different parts of the structure. An explanation of X must specifywhich specific substructure is identified with X. For presentpurpose,Iwill use the notation 'part-i' to denote a specific kind of substructureof intended applications of T. In doing this, lam ignoring certaintechnical questions about how substructures of intended applicationsare specified. Thus,a slightly moreprecise version of(3) is:(3') T= {X,I)explainsXiff there are someisuch that X ispart-i of

a set of some successful applicationsof T.Wenow need tobe a littlemore explicit about thepragmatic aspects

of explanation. Successful applicationsof a theory (X,I)at time t aresimply intendedapplications that are knownby theusersof T toat timet tobe in Content(i^). Thus:(5) A is a set of successful applications of theory T = (X,I)at time

tiff:(i) A CI;(ii) A 6 Content^);(iii) at time t users of T have goodreason to believe that A c

Content^).Thus,we have the following accountor definition of 'explanation':

(6) (X,/,A) is an explanation for X at time t iff:(i) (X,I)is a theory element;

(ii) A CI;(iii) A 6 Content^);(iv) at time t users of T have good reason to believe that A €

Content (X).(v) Xis part-i of A.

Thisaccountofexplanationisnot free fromknownproblems.'

Theseturn on the conceptof 'part' that is appropriate here. Roughly, some'parts' ofstructures seem to be inappropriate candidates for explanationaccording to this model. For example, one might take a 'part' of aclassical collision mechanical structure to be simply a subset of the setof particles appearingin the structure. It seems counterintuitive to say

202 JOSEPHD. SNEED

that we 'explain' this set of particles by embedding it in a model forclassical collision mechanics. The obvious solution to this problem istodefine 'part' in sucha way that subsets ofnon-basic sets in structuresmust appear in 'parts' of these structures.

Anotherproblem appears in connection with theories (likeclassicalcollisionmechanics) whichhaveaconceptof 'temporalorder.Supposewe 'explain' agiven structure byidentifying itas thatpart ofaclassicalcollisionmechanical structure consisting of (someof) theparticles plustheir initial velocities. One might contend that doing this explainsnothing about 'why the particles have these velocities'. In contrast,identifying parts of the same given structure with the final velocitiesof some other models of collision mechanics does seem to provide anintuitively satisfactory explanation.

Roughly, the same data may be explained by classical collisionmechanics in several ways. Some of these ways appear to be intu-itively satisfactory. Some do not. The source of this difference clearlylies in our intuitions about temporal order. In the intuitively unsatis-factory explanations weappear to be explaining the given structure byshowing 'what happens next' rather than 'what happenedbefore. Wemight avoid this byplacingadditional restrictions onthe temporalorderof 'parts' appearing in explanations. Those who think that 'causes' arean essential feature of explanation would endorse this move. Thoughtthis might be a feasible move for theories with a conceptof temporalorder,itis hard to see what(ifanything) might correspond to this for the-ories (likeequilibrium thermodynamics) without aconceptof temporalorder.

Perhaps the accountof explanationoffered here should be viewedascharacterizingarather generalframework forexplanation whichcan,insome cases,beaugmentedwithfurther conditions to characterize 'causalexplanations'. However, it should be noted that this augmentationwill apparently usually involve adding something that is not naturallyregardedas a part of theconceptual apparatus of the theory providingthe explanation. That is, 'causality' -even when we can make sense ofit -is usuallynot going to appearexplicitly in theconceptual apparatusthat is capturedby the set-theoretic axiomatization.

Having said what Ican about known difficulties,Iwill exploreimplications of this account of explanation in two directions. First,Iwillconsider thetraditionalquestionabout the symmetryofexplanationand prediction in relation to the question of functional explanations.

STRUCTURALEXPLANATION 203

Second,Iwill consider how comparative evaluations of explanationsare made.

3. EXPLANATIONASARGUMENT

Traditional accounts of explanatioirhave tended to view anexplanationas akind of argument. Rough, the conclusion of the argument is thatwhich is explained while the premises are providedby the theory thatprovides the explanation, together with some 'collateral information'or 'initial conditions'. How can we compare the present account ofexplanation with this 'deductive model' ofexplanation?

The most obvious approach to comparison is trying to convert thevarious pieces of a model theoretic explanation into linguistic entitiesthat at leastcouldbe piecesof adeductive argument.For thetheoreticalpartof the explanation,the obvious candidate is simply theclaim of thetheoryT= (X,I) that

(1) Ic Content^).

It is also reasonably clear that A - the set of intended applicationsused in the model theoretic explanation- plays a role' correspondingroughly to that of the collateral information in the deductive model.A bit more precisely, what remains of the set-theoretic structures in Awhen the explanandum Xis deleted - this lA{— }X' is the collateralinformation. More intuitively, A is the total context within which welocate X for purposesof bringing T to bear on it. A{—}X is thatpartof the context which was not initially given,but rather added (perhapsdiscovered) in theprocess of providing the explanation for X.

Thus, toprovide additional premises for a deductive argument weneed to assert that thecollateral information A{— }X is,in fact, part ofan intended application ofT. This we may do with two propositions:

(7) ACI

and

(8) A{-}XispartofA.

With (8) we simply claim that the additional structure A{—}X whichwe have added to X is a part of the total context. This is, of course,

204 JOSEPHD. SNEED

trivially true. Non-trivialis theclaim (15) that the total context Ais anintended application ofT.

What is now the appropriate conclusion to thisargument?Isuggestthat we would like tobe able toconclude from (1),(7),and (8) that:

(9) X is part of A.

That is, we would like to conclude from the truth of the theory's claim;the fact that the theory applies to the total context A and the fact thatthe added contextA{—}X isa part of A that X is apart of A.

Thus,Isuggestthat themostnatural way toconvertamodel theoreticexplanationinto adeductive argumentyields the following argument:

(1) / <E Content^). [T is true.]

(7) ACL [T applies to A.]

(8) [collateralinformation]

(9) X is partof A [explanandum]

Theproblem with this argument form is thatit isnot generally valid.Whether it is valid dependson the nature of the theory Tused in theexplanation and upon the nature of the application A of the theory.The reason is roughly this. For applications A in Content(K), partsof A having the structure characteristic of yl{-}X will not always beassociated with unique parts of A having the structure characteristicof the part X. Thus simply knowing how the part isinstantiated doesnotgenerally tellushow theX-likepart is instantiated.

Tomake this clearer,consider the special casein which X containsno theoretical concepts and trivialconstraints. That is,

Inthis case,

Content^) =Pot(M)

Mpp =Mp; C=Pot(Mp).


andnothing islostby simply forgetting aboutmultiple applications andtaking

Suppose that theexplanandum £is asubstructure ofsomemp inMp.That is,xisobtained from mpbydeleting somepartsofmp. Intuitively,wemay think of systematicallygoingthrough all ofMp removing fromeachmember parts isomorphic to those we removed from mp to obtainx. Call the set of structures we obtain in this way '5X [MP]\ Providinga precise definition of '5X[MP]' is somewhat tedious soIomit it here.Intuitively,SX [MP] is the setof all x-typedatastructures. The structurex is justone 'value' or 'instantiation' of this datatype.

To explain x is simply to add the remaining structure required tomake it an a in Mp which is also a model for T, i.e. a in M. Thisremainingstructure is a{-}x. As before,wemay considerSa{_}x[Mp]- intuitively the set of alla{ — }x-type data structures.

Now, starting with

a{-}x inSa{_ }x[Mp]

we may generally add the remaining x-type data toproduce a modelfor T in a variety of ways. That is the emendations of a{ —}x in Mform a 'blob' rather than a point (see Figure 1). In turn, this blob ofemendations to modelsfor TprojectsdownintoSx[Mp] as ablob,ratherthana point. In this blob areall the possible waysof filling outa{ —}xwith x-typedata that yields amodel for T.

Thus,knowingonly thata{ — }x is apartofa model for T, the mostwecan infer about the value of the x-typepart of this model is that itis somewhere in the blobprojected into Sx[Mp].Insome special cases,this blob will be a singleton. But generally it will not be. Whether ornot there is aunique way of makinga{ — }x intoamodel for Tdependson thespecific natureof Tand a{— }x.

Thus structural explanations are, in general, what Hempel called'partial explanations' ([6], pp.16-17). They allow us to deduce somefeatures of the explanandum. But, theydo not, in general,allow us todeduce everything we know about it - not even everything we knowabout it that is expressible in the vocabulary of the explaining theo-ry. What the structural account ofexplanationallows us to see (whichis, at least, not readily apparent in linguistic accounts) is that partial

Content(if) =M.

206 JOSEPH D.SNEED

Fig. 1.

explanations occur in the 'hard' sciences as well as the 'soft. Phys-ical theory offers partial explanations that are, in their logical form,no different than the functional explanations appearing inbiology andcultural anthropology. The only difference is this. Insome cases, atleast,physical theories can provide 'complete' explanations. Itmightbe the case that the explaining theories used in sciences like biologyand anthropology are simply so weak that the functional explanationsthey providecan neverbe more than partial explanations. Indeed,oneaccountof functional explanation suggests that this is true [16]. Iftheo-ries supporting functional explanations are 'optimizationmodels' withmultiple local optima, then it is intuitively clear that the explanationswill alwaysbe partial.


4. BAYESIANRATIONALITY

The obvious way to applyaBayesianaccountofscientific reasoning toamodel theoretic conception ofscientific theoryisto consider conditionalprobabilities of the following form:

(10) P(Ic Content^) |E)

where Eis the 'totalevidence' available. Emight simply be aconjunc-tion of things like

Ii c Content(if)

not (Ijc Content(K))

and

P(li C I) ± 0.

But there is no reasonat thispoint to rule out otherkinds ofevidence.

5. EVALUATING EXPLANATIONS

It iscommon practice amongempirical scientists (andothers) to debatethe relative merits of different explanations offered for the same phe-nomenon. Do we best explain an apparent rise in temperature in apalladiumcell by the presenceof 'cold fusion' or by 'faulty instrumen-tation. Dowebestexplainthe constructionoflarge masonry structures,roads, etc. centered aroundChaco Canyon, NMin the 10th— 11th Cen-tury ADby some variantof 'indigenous development' or some variantof 'foreign influence. At least a half dozen alternative explanationsfor this phenomenonare evaluatedcomparatively in arecentdiscussion([l l],p.391-402). Supposing that thealternative explanations in theseexamplescouldbe mashed into the format suggestedabove, would thisilluminate the discussion of the alternatives?

Given that we have two distinct putative explanations for X, howshould we compare them? Intuitively, two considerations appear tobe relevant: (1) how good is the theory involved in the explanation?;(2)howprobableisit thatthe theoryinvolvedreallydoesexplainXIOn

208 JOSEPH D. SNEED

aBayesian accountof scientific reasoning this suggests that somethinglike:

(11) P(IcContent^) & (X,J, A) is an explanation for X|E)

could be taken as a 'figure of merit' for explanations. That is, alterna-tive explanationsof X,(X, /,A) and {Xl,Kl,I',A'),should be evaluatedaccording to the valuethey give for (11). Infact,it appears that inmanycases of interest this expressioncanbe considerably simplified.

First,it appears that in many cases of interest, the conjuncts in (11)will be probabilistic independentso that (11)reduces to:

(12) P(I G Content^) |E) xP({X,I,A) is an explanation for X \ E).

Clearly this independenceassumption will not generally be true. Insome cases that (X, /,A) explains X may provide evidence that / €Content(if ), that is

P(IE Content(X) {X,I,A) explains X &E)P(Ic Content(X) |E).>

Intuitively, this is because

ACIand A c Content^)provide evidence that

i" € Content^).

However, in thecase of theories with lots of evidence of this sort, thecontributionof the evidence from specific putative explanationmay benegligible. So theproduct form maybe acceptableasanapproximation.This would mostlikely be the case when the theories were wellestab-lished and supportedby substantialamountsof evidenceapart from theexplanatory contextin question.

The product form (12) has the nice intuitive feature of clearly sepa-ratingjudgementsabout themeritsof thetheory invokedin explanationfrom judgementsabout whether the theory explains thephenomenoninquestion.Focusingon thelatter judgement,someadditional simplifica-tion is possible. Considering(6) again,one might mostnaturally view(6-i), (6— ii), and (6— iii) as purely formal conditions so the probability


of each (and the conjunction) will always be either 1 or 0. We couldcomplicate things here by consideringnon-trivial probabilities for theformal conditions, as for example when we are uncertain about ourmathematical or calculational results. But this complication would addlittle to the present discussion. Thus in the case the formal conditionsare satisfied(P(6-i) & 6— ii) & 6— iii)) =1), the probability that we havean explanation at all is just the probability that the empirical condition(6-iv) is met. Thus, the probability that (X,/,A) explains Xis:

(13) P(Xpartof,4 \E).

Thusfar wehaveavoidedexplicit discussionofthe formalpropertiesof the explanandum X. In some cases X may be (described as) aset-theoretic structure in a fragment of the vocabulary of the theoryT = (X,I)used in the explanation. In this case, whether X is a partof A may be a purely formal question. In the more generalcase Xwill be described insome vocabulary disjoint from thatof T and itwillremain a 'contingent' question whether this description of X has beenappropriately 'translated' into the vocabulary of T.

Under thesesimplifying assumptions, the goodness of the theory ismeasured by the probability (given the available evidence E) of theempiricalclaims of the theory, i.e.

P(I <E Content^) |E).

Theprobability that the theoryexplains Xis:

P(Xpartof A),

and the 'figure of merit' for explanations is simply the product of theseprobabilities. Thus:

(14) If (X,I,A) and (X',/',A') are putativeexplanations for X then:

(X,/,A) is better than (K\I',A')

iff

P(I £ Content^) |E) x P(X partof A \ E)> P(l' G Content^') | E) x P(X partof A' \ E)

210 JOSEPHD. SNEED

Thus far we have assumed that the theories involved in alternativeexplanations are totally different. That is,both their vocabulary andlaws are different. This is not always so. Insome cases of interest,thealternative theories will share all orpart of their vocabulary. The mostinteresting of thesecases are thosein which the theories in questionare'competing' theories about the same data -thatis,whenMpp

—Mppand

1=I.In this case, the conditionalprobability of the theories' claims(10) do not always provide a good indication of the relative merit ofthe theories. The reason is roughly this: relatively weak theories mayhave claims with high probabilities. But, intuitively, stronger theorieswhoseclaimshavelower probabilitiesmightbepreferable. How shouldtheories be comparedin these cases? Andonce theories are compared,how shouldexplanationsbased onthem be compared?

Fromamodel theoreticpointofview,thecontentof theoriesprovidesan indication of their relative strength. Clearly,(15) T = {X,I) is stronger thanT' = (X',I)

if

Content^) C Content^')

whether the 'if in (15) should be 'if and only if is not entirely clear.More explicitly, it is not evidenthow we should compare the strengthof theories with identical classes of non-theoretical structures whosecontents intersectpartly or fail to intersect at all. Forpresent purposes,itappears that little is lostbysimply taking (15) to benecessary,as wellas asufficient,condition. We may, somewhat arbitrarily, say Tand T'areequally strong (equipotent) inallother cases. Thus:(16-i) T= (X,I) is stronger than T' = (X',I)

iff

Content^) C Content^')(16— ii) Tisequipotent with T

iff

T is stronger than T' andT' is not stronger than T.

Havinga way ofcomparing thestrength of theories allowsus to addressthe questionof how probability trades off againststrength inevaluating


theories. Isuggest that the trade off is made lexicographically withprobability dominating strength. That is,strictly more probable theoriesare alwayspreferredto strictly lessprobable. Only in thecaseofequallyprobable theories does strength play role. Then, stronger theories arepreferred toweaker. Thus:(17) T = (X,I) is better thanT' = (X',I)

iff

P(IG Content^) |E) >P(IG Content(A") | E)

or

and

Tis stronger thanT'This seems plausible for the following reason. Numerous weak the-ories with high probability are easy to construct. But most such the-ories receive short-lived attention because it is also relatively easy tostrengthen some of the theories without substantially diminishing theirprobability. The theories we consider seriously are those which retaintheir probability under strengthening. Having a 'better than' orderingfor theories allowsus toreaddress thequestionofa'better than' orderingfor explanations. Here the question is this. How does the probabilitythat the theory provides an explanation trade off against the quality ofthe theoryin evaluating thequality of theexplanation? Again,Isuggesta lexicographiccriterion with the quality of the theory dominating.

Thus:(18) If (X,/,A) and {K\ I,A'} are explanations for X then:

{X,I,A) better than (X',I,A')

iff

{X,I)strictly better than (K1,1)

or

(K,I) equivalent(K\l)

P(I G Content^) | E) =P(IG Content^') |E)

JOSEPHD. SNEED212

and

P(Xpa.rtofA) >P(X partof A').This 'better than' orderingfor explanations clearly emphasizes thequal-ity of the theory rather than the probability that the theory explains. Isthisplausible? The main argument for its plausibility is this. The pur-pose ofexplainingXis to showthat Xcanbe integrated intoour existingbody ofknowledge-without substantially revising that body. Explana-tion isreducingtheprima facieunfamiliar to the familiar. Amongotherthings, this makes us more comfortable with the already familiar. Wefeelcomfortable thatexistingknowledge isadequate-weneednot takethe trouble to learnsomething new. But, some things are more familiarthanothers. Integrating Xinto theframework ofa moreprobable theoryis just 'more satisfying.

DepartmentofHumanities andSocialSciences,ColoradoSchoolofMines,Golden,CO 80401, U.S.A.

NOTE

1Iam indebted to Mr. Michael Pierce forcallingmy attentionto these problems anddiscussing them withme.

REFERENCES

1. Balzer, W.,Moulines,C.U.,andSneed, J. D.: 1987,An Architectonicfor Science:TheStructuralistProgram,Dordrecht: D.Reidel.

2. Balzer, W.,Moulines, C.U., andSneed, J. D.: 1986, 'The Structure ofEmpiricalScience-LocalandGlobal', in:R.B.Marcus,G. J. W. Dorn,andP. Weingartner(Eds.), Amsterdam:North-Holland,pp.291-306.

3. Earman,J.: 1992,Bayesor Bust, Cambridge, MA:BradfordBooks,to appear.4. Forge,J.: 1980, 'The StructureofPhysicalExplanation',Philosophy ofScience,

47,203-226.5. Forge, J.: 1985, 'TheoreticalExplanationinPhysical Science',Erkenntnis, 23,

269-294.6. Hempel,C. G.: 1962, 'ExplanationinScienceandHistory', in: R. G. Colodny

(Ed.),Frontiers ofScience andPhilosophy, Pittsburgh: University ofPittsburghPress, pp. 9-33.

213STRUCTURAL EXPLANATION

7. Howson,C.andUrbach,P.: 1989,Scientific Reasoning: TheBayesianApproach,La Salle: Open Court.

8. McKinsey,J. C.C, Sugar,A.C,andSuppes,P.C: 1953, 'AxiomaticFoundationsof ClassicalParticleMechanics', JournalofRationalMechanicsandAnalysis,2,253-272.

9. Rosenkrantz,R.D.: 1977', Inference, MethodandDecision: Towarda BayesianPhilosophy ofScience,Dordrecht,D.Reidel.

10. Rubin, H. and Suppes, P.C: 1954, 'Transformationsof aSystem ofRelativisticParticleMechanics',Pacific JournalofMathematics,4, 563-601.

11. Sintonen,M.: 1989, 'Explanation:InSearchof theRationale',in: W. SalmonandP.Kitcher (Eds.),Scientific Explanation:Minnesotastudies, Vol. 13,Minneapolis:University ofMinnesotaPress.

12. Sneed,J. D.: 1979, TheLogicalStructureofMathematicalPhysics, (2nd edition),Dordrecht: D. Reidel.

13. Suppe,F: 1989, The Semantic Conception ofTheories andScientific Realism,Urbana: University of IllinoisPress.

14. Suppes,P. C: 1959,IntroductiontoLogic,New York: VanNostrand.15. vanFraassen,B.C: 1980, The Scientific Image,Oxford: ClarendonPress.16. vanParijs,P.: 1981, EvolutionaryExplanationin theSocialSciences: AnEmerg-

ingParadigm, Totowa:RowmanandLittlefield.17. Vivian,R. G: 1990, The ChacoanPrehistoryof theSanJuanBasin,New York:

Academic Press.


Iamsympathetic toJoe Sneed's generalstructural viewpointas well ashis Bayesian approach. The analysis of structure, the characterizationofscientific structures, and theanalysis ofthe natureof the structures,aswellas a strongBayesianbent,havebeen long-standing features of myown work in the philosophyof science. On the other hand,as the yearsgoby,Ifindmyself increasingly skepticalofvery generalphilosophicalaccounts of explanation or causality or even of the viability of verygeneralBayesianmethods. Myview of sciencehas movedincreasinglyfrom that of afoundationalist to the viewpoint that the conceptualcon-tent of science is bestanalyzedin terms of adiverse set of methods forsolvinga wide variety of problems. Schemes of great generality aboutscientific explanation, such as those of Sneed, can perhaps be usefulin providing some kind of general framework, but Ithink they missthe main part of what we intuitively want from good scientific expla-nations. Moreover, our sense of satisfaction with good explanationsis particularistic in natureand not in any natural way subsumed underthe general schemes proposedby Joe. Above all,Iam skeptical that a

214 JOSEPHD. SNEED

general notion ofpart, as he uses it, can do the work he would like tohave itdo.

For those raised in traditional philosophy or even traditional phi-losophy of science, with the search for generality and universality ofconceptual schemes so dominant, it is not easy to accept or even besympathetic with a view that is skeptical of the success of any of thegeneral schemesaimed atproviding a traditionalphilosophy ofscience.

The pictureof scientific activityIincreasingly favor myself is clos-er to that of apprenticeship than to the propositional organization ofknowledge. Perhaps only in mathematics do wehave the generality ofstructure andgenerality ofresult that would satisfy the hearts as wellastheminds of philosophers. Evenphysics, the mostsophisticatedof theempirical sciences, has only a pretense at great generality. Individualproblems must be tackled by individual methods. The methods thatareused to attack aparticular problem dependupon theexperienceandinsight of the investigator, not upon the sharp and exact codificationof theory and its range of application. It is only a myth engenderedby philosophers- even in the past to some extent by myself - thatthe deductive organizationof physics innice set-theoretical form is anachievable goal. A look at the chaos in the current literature in anypartof physics is enough to quickly dispel that illusion. This does notmean that set-theoretical work cannotbe done, it is just that its severelimitations must be recognized.

There are many ways of expanding upon the views Ihave justexpressed. A central one is the current realization of how few evensimplephysical systemscan be thoroughly understood,in the sense thatdetailed and precise predictions about the behavior of the system canbe successfully verified. In a way, this is a lesson that was ready forunderstanding already in the nineteenth century in terms just alone ofthe massive andunsuccessfuleffort to masterthe three-bodyproblem inclassical mechanics. But the moderndevelopments of chaoshavemadethe facts much more salient and very much increased the awareness ofhow difficult it is tomake successfulpredictions. Butonce the rarity ofsystems or structures whose behavior can be successfully predicted isrecognizedthen much of the older talk about general schemesof expla-nation seems unsatisfactory for dealing with the rich details of actualscience. This is not the place for a full-scale exposition of my views,but let me try to giveone or two examples.


Iwant to contrast our attitude toward various cases of failure ofprediction in classical mechanics. Let us begin with a case alreadymentioned, that of the three-bodyproblem. Our inability to solve thedifferential equations, even in principle, in a satisfactory analyticalform has been recognized for over 100 years. On the other hand, thederivation of the differential equations governing the motion of threebodies acted uponby the force of gravitation alone is an easy exercise.There is avery general belief that the differential equations accuratelyreflect many real situations toanextraordinarilyhighdegreeof accuracy.There is no need to say 'with complete accuracy' for no real systemsare sufficiently isolated, just tomention one central reason. What isimportant is thebelief that theequations hold to a very high degreeofaccuracy. But they areunmanageable from the standpoint ofsolutions.

A closely related example is that of a single body sliding or rollingdown an inclined surface of variable degrees of roughness. We canwrite adifferential equation including friction thatalso,because of thecomplicated natureof the surface, wecannot solveexceptnumerically.But in this case the status of the equation is quite different. We do notthink that it is possible to write down, to the samedegree of precisionat all,a differential equation governing the complicated physical phe-nomena that reflect theinteraction between the surface and the movingbody. Because our feelings of definiteness about these two cases seemsostrong,a theory ofstructural explanation should provideavery com-pellingaccountof their difference. But to provide this requiresenteringinto the details in a way that is not afeature of the current work.

Let uscontinue the same line of examples. A favorite oneof many,also usedby Sneed, is the way in which we accept the explanationofthe motion ofbilliard balls on a table based upon the mechanical lawsof collision. However, if we consider a somewhat more complicatedbilliard ball -1have inmind thekind studied by Ornstein et al,Sinai,and others -we enteran entirely new realm. The most striking theo-rems are that when the obstacles onour new and wonderfullydifferentbilliard table are convex in their shape then we cannot distinguish-nomatter how many observations we take -between the motion of thebilliard balls following the usual laws of mechanical collision,and themotion of theballbeing describedby aprobabilisticMarkov process. Itis oneof the great insights of modern mechanics, whosephilosophicalimportanceIhave tried to stress in severalpublications over the years,that the separationbetween deterministic mechanical systemsand ran-

216 JOSEPHD. SNEED

dom probabilistic ones is not at all what it was once thought to be.Structuralist views of this kind of example are as yetmissing from theliterature.

The examplesjust cited are in a way toomathematical in character.Theanalysis ofaphysicalproblem we hope to solve, think wecansolveand are willing to tackle,is more open-ended,less honed down to afewsets of variables than thekinds of examplesIhave just given. Here theapprenticeship of thephysicist seems tome of the greatestimportance.Toknow whatto countand whatto discardas unimportant inanalyzingthe givenphysical situation isnot something wherein noendof traininginmathematics andinthe solutionofdifferential equations willbe muchofhelp. As wemove from applications of theory to anew situation andthe organizationof experiments, whatIhave to say is even more true.Moreover,incomplicatedhigh-technologyexperiments involvinglargenumbers ofindividuals itis fair tosay thatnoonecommands anylongerall the details of the experiments. It is not just apprenticeship, buta collection of mature apprenticeships, that are required to organize,prepare and execute the experiments planned. Individual papers withmore than 100 authors are now notuncommon in high-energy physics,yet theobviously socialnatureof theseexperimentsandthe complicatedsets ofskillsrequired toexecutethemhaveas yetreceivedlittle attentionin the analysis of physics by philosophers of science. Inmaking theseremarks about the collective enterprise of runningexperiments inhigh-energyphysics,Iam not interested as such in the sociology of sciencebut more in the bewildering variety of patches of theory and patchesof skills required to put the whole thing together. Itis that it would bewonderful to have adetailed structural analysis of. Perhaps Sneed andhis energeticcollaborators canbe persuaded toenlightenus allon someof these detailed matters.

PART V

MEASUREMENT THEORY

R.DUNCAN LUCEAND LOUISNARENS

FIFTEENPROBLEMS CONCERNINGTHEREPRESENTATIONALTHEORY OFMEASUREMENT

1. INTRODUCTION

The typeof theoryofmeasurementwediscussis therepresentationalonethat is treated in greatdetail in the three volume treatise Foundations ofMeasurement.lWeattempt tomake thepresenttreatmentself-contained,including definitions of major concepts, but these volumes may beconsulted for greaterdetail.

Our aim is to discuss two classes of,probably difficult, issues thatneedclarification. The first type,covered in Section 3, has to do withprincipled arguments to justify therepresentationalperspectiveandpos-siblemodifications of it to deal with error and with limiting propertiesof passing from finite, through denumerable, to continuum represen-tations. These issues are, at least partially, philosophical in character.The second type, covered in Sections 5 and 6, is more technical andfocuses primarily onthe issues of relating theuseful,but abstract, ideas(to be defined below) of scale type, Archimedeaness,and Dedekindcompleteness to observableproperties ofqualitative structures.

Manyof the problems wedescribe areprimarily conceptual andtheirresolutionprobablyrequires somethingof anintellectual break-through.Once that is achieved,however,somemay wellprove to be technicallyeasy to solve. Others that we cite are well defined within existingmathematical ideas,and some of these appearto be difficult.

2. THEREPRESENTATIONAL THEORY OF MEASUREMENT

2.1.HistoricalBackgroundMathematicians and scientists, among them Helmholtz (1887), Holder(1901) and yon Neumann (in yonNeumann and Morgenstern, 1947),

219


220 R. DUNCAN LUCEAND LOUISNARENS

clearly had the idea of axiomatizing ordered qualitative structures aspossible models of empirical attributes and in the latter two cases ofestablishing,as mathematical theorems,theexistenceanduniquenessofnumerical representations. The powerand generality of this approach,however, wasnot widelyappreciatedby empirical scientistsorphiloso-phersof science during the first half of the 20th century,as evidencedby the discussions of measurementin, for example, Bridgman (1922,1931), Campbell (1920, 1928), Cohen and Nagel (1934), and Ellis(1966). More than anyone else, Suppes brought to the attention ofnon-mathematicians this axiomatic style of studying the measurementofattributes. Particularly important wereScottandSuppes (1958),Sup-pes (1951), and Suppes and Zinnes (1963). The latter was especiallyinfluentialamongmathematical psychologists, andit was the forerunnerof the Foundations ofMeasurement.

For over30years,representationanduniqueness theorems2 were themode of research. New ordered mathematical structures that appearedto haverelevance to themeasurementofcertainattributes were isolated,and two theorems were established: the existence of a representationinto or onto some prescribed numerical structure, and the uniquenessof that representation in the sense of formulating the class of trans-formations relating equally good representations into or onto the samenumerical structure. This has come tobe called the RepresentationalTheoryof Measurement,abbreviatedRTM.

ManykeyconceptsandmethodsofRTM,asitisnow formulated,hadanaloguesin 19thcentury geometry,particularly in(i) theassignmentofcoordinate systemsto qualitative geometries(e.g.,Hilbert,1899),(ii) theidea that invariants of structure preserving mappings had geometricalsignificance(Klein,1893),and (iii) the idea that the geometric structureof space is the same at each point, i.e., that space is homogeneous(Helmholtz, 1868;Lie, 1886).

2.2. The UnderlyingPrinciples

Theformal part ofRTMcan be reduced to five main ideas:1. Aqualitative situation isspecifiedby a (usually ordered)relational

structure X consisting ofa domain X,finitely many relations onX andfinitely many specialelements of X.

REPRESENTATIONAL THEORY OFMEASUREMENT 221

Theserelations,subsets,andelements are called theprimitives ofX.Measurement axiomsare thenstated in terms ofprimitives ofX.Theseaxioms areintended to be true statements about X for some empiricalidentification andareintended to captureimportant empiricalpropertiesof X, usually ones that prove useful in constructing measurementsofits domain,X.

2. The representationaltheory requires that theprimitives ofX canbe givenan empirical identification.

In particular, if R is an n-ary primitive relation on X, then it isrequired that the truth or falsity of R(x\,...,xn), for any particularchoice of the n-tuples X{,be empirically decidable.

3. As much as possible, measurementaxioms, statedin terms of theempirically identifiedprimitives, shouldbe empirically testable.

Next,a numerically-based,representingstructureMis selected,andfor eachstructure Xthat satisfies themeasurementaxioms,thesetS(X)of homomorphisms ofX intoMisconsidered.

4. Measurement ofXis said to takeplace ifandonlyifthefollowingtwo theorems canbe shown:(i) (Existence Theorem). <S(X) is non-empty for each X that

satisfies the measurementaxioms.(ii) (Uniqueness Theorem). An explicit description is provided

about how the elements ofS(X) relate to one another. Inpractice thisdescriptionusually consistsofspecifyingagroupoffunctions G such thatforeach <f> inS{X)

where * denotesfunction composition.A concept of meaningfulness similar to that proposed by Stevens

(1946) is used for judging the empirical significance of quantitativestatements: ann-ary relation Son the domain of the numerically basedstructure J\f is said to be meaningful about X if and only if for alliri,...,xnandall </> and ip in S{X),

FollowingSuppes andZinnes (1963)

S{X) = {9*<t> :^isinC},

S[(f)(xi),...,(f)(xn)] iff S[if>(xi),...,il>(xn)].

222 R. DUNCANLUCEANDLOUIS NARENS

5. RTMidentifies empiricalsignificance with meaningfulness.

The conceptof meaningfulness with respect to S(X) is easily extend-ed to numerical statements involving measurementsof elements of thedomain: meaningfulstatements are those whose truth value is unaffect-edby the particular representationinS(X) used tomeasure X.

It should be noted that the concepts of empirical significance andmeaningfulness as embodied in RTM are independent of truth; bothtrue and false statementscanbe meaningful andempirically significant.

Although the following two features are not part of the formal the-ory, they hold inmany important measurementsituations with infinitedomains:

6. Using the measurementaxioms, sequences ofequal spacedele-ments- standard sequences-areconstructed.

These standard sequencesare then used to establish constructivelythe setofhomomorphisms of X into Af. Toconstruct aspecificone,oneproceeds as follows: a first standard sequences\,...,s\,...is chosento produce the first level of approximation 01 of a homomorphism (j).For each element xinX, find the n such that s\ <x < s ln+l,where <denotes the primitive of X that orders X. Set

A secondlevelofapproximation iscarriedoutbychoosing thesequence5\,..., 52,...,s 2,...,where for eachpositive integeri,s\{ =s}. The secondapproximation to </> is obtainedby setting

wherekis such thats\ <x < s|+1.Continuing in this way,asequenceofapproximations (f)m isconstructed,andfor many measurementaxiomsystems it canbe shown that as m— > 00, the limit of </>m exists and isthe desiredhomomorphism 0 of X into Af.

Beginning with a different standard sequence willusually yield adifferent homomorphism.

Theaccuracyofmeasurement,inpractice, iscontrolledbyhow manyterms in thisapproximation are used.

The ancient geometerEudoxesused standardsequencesin much thesame way as inRTM.They alsoplayeda critical role in the approachestaken by Helmholtz (1887) and Holder (1901). Later Luce and Tukey

<j>i(x) =ra/1.

(f)2(x) =k/2,

223REPRESENTATIONAL THEORY OFMEASUREMENT

(1964) used related kinds of standard sequences to obtain their repre-sentation and uniqueness results for additive conjoint measurement.3Krantz (1964) and, in a moreuseful way,Holman (1971) recognizedthat the standard sequenceapproach of Luce andTukey closelyrelatedto thealgebraic structure describedby Holder for physicalmeasurement-an Archimedean,4 totally orderedgroup. Basically, Krantz and Hol-man defined an operationon one component that capturedcompletelythe trade-off structurebetween the components. Under the axioms, theoperation was shown tobe associativeand so the problem wasreducedto an application of Holder's theory.

MostofFMIisdevoted to recastingvarious measurementsituations,often using the ideas inherent in Krantz andHolman's approach, so thatArchimedean ordered groups (or large semigroup portions of them)come into play. It has been remarked, with some justice, that thetheory of measurementis largely an application of Holder's theorem,i.e.,of standard sequences in the guiseofArchimedean orderedgroups.Moreover, as we shall see in Section 4.2, this continues to be the caseevenfor non-additive structures.

7. The representational theory offers anabstract theoryof the kindsofwell-behaved scales that one encountersin science.

This theory is based on a conceptof homogeneity that abstractly ismuch like the geometric one mentioned above and provides qualita-tive andempirical criteria for empirical structures to be homogeneous.(Issues involving homogeneity are discussed more fully in Sections 4and 5.)

Until the 1980s, the major emphasis of RTM has been on produc-ing existence and uniqueness theorems for various kinds of empiricalsituations. Scott andSuppes (1958) remarked:A primary aim of measurement is to provide a means of convenient computation.Practical control or predictionof empirical phenomenarequires that unified, widelyapplicablemethods ofanalyzingthe importantrelationshipsbetweenthephenomenabedeveloped.Imbedding thediscoveredrelationsinvariousnumerical relationalsystemsis the most important such unifying methodthathas yetbeenfound(pp. 116-117).

224 R.DUNCAN LUCE ANDLOUIS NARENS

3. PROBLEMS CONCERNINGTHEFORMULATIONOFTHEREPRESENTATIONTHEORY

Althoughmostcritics accept thatRTM,viaitsaxiomatizations ofimpor-tant empirical situations and the various representationand uniquenesstheorems,is amajor contribution to ourunderstandingof measurementin science, they criticize itas a theory of measurement. The criticismsare of four major types: (1) questions about the definition of measure-mentin terms of scalesofhomomorphisms, (2) objections to the lack ofa theory of error for RTM, (3) concerns about exactly what invarianceconcepts have to do with the conceptofmeaningfulness,and (4) doubtsabout the heavy use of infinite structures and continuum mathematicswhen, after all, the universe is, according to contemporaryphysics,composedof only finitely many particles. A number of these concernsare expressedby Kyburg (1984) and in papers in Savage and Ehrlich(1992).

3.1. HomomorphismDefinition ofMeasurement

The criticisms of the homomorphism definition of measurementare oftwotypes: those arguingthat thedefinition is toonarrow in the sense thatit does notaccountfor certainkindsof measurement;and thosearguingthat itis toobroadinthat itpermits toomanykinds ofmeasurement.Theformer usually contain elements of the following criticism of Adams(1966):Itseems to me that in characterizing measurement as the assignment of numbers toobjects according to rule, the proponentsof therepresentational theory have fastenedonsomething whichisundoubtedly of greatimportance inmodernscience, but whichis not by any means an essential feature of measurement. What is important is thatthe real numbers provide a very sophisticatedand convenient conceptual frameworkwhichcanbeemployed indescribing the results of makingmeasurements: but, whatcan be conveniently described with numbers can be less conveniently describedinother ways, and thesealternative descriptions no less 'give the measure' of a thingthan do the numericaldescriptions.... Note, too, that theancientGreeks didnot haveour concepts of rational,muchlessrealnumbers, yet it seems absurdto say that theycouldnot measure because they didnot assign numbers to objects. In sum,Iwouldsay that the employment of numbers in describing the results of measurement is notessentially differentfromtheiremploymentinothernumerical descriptions,andthatthisemployment isneither a necessarynor a sufficient conditionformakingor describingmeasurements (pp. 129-130).

Evengrantingtheassumptionthatmeasurementnecessarilyinvolvesassigningnum-bers,it seems tome tobe far fromtrue thatinmaking theseassignments it is always the


case thatmathematicaloperationsandrelationsaremade tocorrespondtoor representempiricalrelationsand operations....The situationis worsewith most ofthe widelyused measures in the behavioral sciences, likeI.Q.s andaptitude test scores. It maybe claimed, of course, that theseare not reallymeasurementsatall,but to justify this[claim], some argument wouldhave tobe given,unless the theoryof representationalmeasurement is not to degenerateinto a mere definition(I.Q.s are not measurementsbecause they do not establishnumericalrepresentationsof empirical operationsandrelations)(p. 130).

Niederee (1987, 1992) and others criticizeRTMas being toobroad.Henotes that requiring a structure tobe numerically representable intostructures based on the real numbers places little restriction on thestructure beyondcardinality and even that has not been justified. Thiskind of restriction is too liberal because... itdoes not involveany conceptof measurement whatsoever. Indeedmeasurementtheorists wouldhardly bepreparedtoaccept [it] as asufficient criterionfor a structureto be called representablein terms of fundamental measurement. What seems to belacking hereis an analysisof whatit should mean thata 'number' expresses an idealvalueof measurement (Niederee, 1992,p. 245).

The above criticisms ofRTMsuggest the following problem:

PROBLEM 1. Justify inaphilosophicallyprincipledfashionRTM(ora largepartof it)as ageneraltheory ofmeasurementwithout severelyrestricting itspositive uses.5

A few comments are in order. First, theories of ability measure-mentalluded to by Adams,althoughbased onquantitative data,are notin principle ruled out by RTM.Empirical structures can be based onnumerical information justas long as those numbers arise fromempiri-calmeans-whichis the case for ability measurement.What is lackingfrom the RTMperspectiveare axiomatic theories for the various kindsof ability measurement.We do not seeany inprinciple impediments tothe development of such axiomatic theories,although inpractice nonehas been devisedand it does notappear to be easy to doso.

Second, either directly or indirectly standard sequencesare used toestablish scalesinalmostall6of themajorresultsofFM.Thus,since theprocess of measurementthrough standard sequencesisusually taken asparadigmatic of 'measurement processes', almost all of the represen-tational results ofFMare valid not only from the RTM viewpoint butfrom a number of different perspectivesabout whatmeasurementis.

226 R.DUNCAN LUCEAND LOUIS NARENS

Third, some attempts at solving Problem 1 already exist, includingMichell (1990)and Niederee (1987, 1992). Thepresentauthors are notyetpersuaded that their attempts are useful beyond the additive cases.Inparticular, as we shall seein Sections 4and5,the greatestprogress todate inunderstandingnon-additive structures involvesmapping certaingeneral types of structures into a particular subgroupof their automor-phisms, showing that this subgroup is Archimedean ordered,and thenusing Holder's theorem to map these into the multiplicative positivereal numbers. Because we do not know, in general,direct structuraldescriptions of individualautomorphisms, itis notobvious thatthepro-cedure proposed byNiederee for constructinga suitable representationwill capture the known results. Considerably more work is neededto establish how these approaches lead to these generalmeasurementrepresentations.

Fourth,much ofFM is concerned with issues that are not limitedto measurement per se, but are closely related to measurement. Themost prominent of these is the testing of scientific theories through theuse of RTM. An example of this is subjective expectedutility (SEU)theory, which holds that a subject's preference orderingover lotteriesis explainable as if the person is trying to maximize a numerical SEU,whichis computedin theobvious wayusinga simultaneouslyconstruct-edsubjectiveprobability overthe family ofuncertaineventsandautilityover the set of consequences. One way to test this scientific theory isto formulate it as aquantitative model,gather data, and test the modelusing standard statistical methods to determine the degree to which themodel accounts formost of the variance in the data. Thisapproachofteninvolves approximate construction of standard sequences. A differentway is to formulate the scientific theory as a qualitative, axiomaticmeasurement model that through RTM is equivalent to the quantita-tive SEU, and then test the axioms' qualitative theory through directempirical observationsof samples of the stimuli.

Three additional examples of theories that are closely related tomeasurementissues are the axioms for distributive triples, which canbe used to construct the space of physical quantities (see Section 6.1);a variety of psychophysical models involving more than one sensoryattribute (one of which is described in Section 6.2); and a theory ofcertainty equivalents to uncertain monetarygambles (Section 6.3).

REPRESENTATIONALTHEORY OFMEASUREMENT 227

3.2. ErrorinMeasurement Theory

Thesecondclassof criticisms concern error andhow itshouldbe treatedvis-a-vis measurementmodels. While 'error' considerations for mea-surement give rise tomany different kinds of problems, we cite threetreatmentsof error thatare intimately connected withkey concepts ofRTM.The first is basedonthe idea thatprobabilitiesunderlie the obser-vations that are made. Although some progress has been made on thisgeneralapproach(Falmagne,1979, 1980;FalmagneandIverson,1979;Iversonand Falmagne,1985;Michell,1986), littleeffective use has yetbeenmade of it to test measurementaxiomatizations in detail.

PROBLEM 2. Specify aprobabilistic version ofmeasurement theoryand the related statisticalmethodsforevaluating whether ornotadatasetsupportsor refutes specific measurementaxioms.

One approach,taken by a number of people,is to assume the data baseconsists of probabilities of binary or more complex choices, with themajor questionbeing the conditions on these probabilities that corre-spond to theexistenceof an underlyingrandom variable representationin which the largest value observeddetermines the choice. A survey ofsuch work is found in Chapter 17 of FM2. One of the most interestingrecent contributions is Heyer and Niederee (1992) who study when aprobabilistic structure, e.g., ofchoices,can beconsidered to arise froma probability distribution over a family of conventional measurementstructures that allsatisfy the same axioms.

From afundamental measurementperspective, this approach is notfully satisfactory because it assumesas primitive anumerical structureof probabilities and thus places the description of randomness at anumerical, rather than qualitative, level. One would like to formulatethe qualitativeprimitives soas simultaneously tocapture ataqualitativelevel both the structural and the random qualities of the situation.

The following is a reasonablyconcrete instance of what we have inmind. Consider the typicalextensivesituation whereboth judgments oforder can be made and entities can be concatenated to form new onesexhibiting the same attribute. One would like axioms that lead to arandom variable representationthatspecifies the random variables withrespectboth to their structuralrelations witheachother andto thenatureof their distributions. For example,if © denotes concatenation within

228 R. DUNCAN LUCE AND LOUISNARENS

the extensive structure, it would be interesting to arrive at qualitativeaxiomssufficient to guaranteethat therepresenting family canbe takento be the gamma family. Mathematically, the problem undoubtedlyentails finding a functional equation characterization of the gammafamily which, toourknowledge,has neverbeengiven.More generally,one would expect the distribution of therandom variable associated tox © y to be the convolution of the distributions of those associated to xand to y. With such arepresentation, the expectedvalue would behaveas a traditional measurementrepresentation,inparticular

Thus,we state the problem as:

PROBLEM3. Extend the qualitativeprimitives ofRTMin sucha waythat the objects of the domain are represented by random variables(insteadofby numbers = constantrandom variables).

Theideathatrandomness canbecapturedqualitatively within arelation-al structure is not totally idle because,in a certain sense,that is exactlywhat has been done in theories of subjective utility theory (Savage,1954,andmuch subsequentliterature). Insuch theories,choices amonguncertainalternatives are used to infer arandom variable -utility -andprobability distributions over families of eventsunderlying the uncer-tain alternatives. The problem is to doit in contextsmore analogous toextensive and conjoint measurement, not preference among uncertainalternatives.

Progress onProblem 3 hasbeen made for the special case of finite,ordinal empirical structures, i.e., there is just one primitive, an order-ing relation (Cohen and Falmagne, 1990). Suppes andZanotti (1992)have followed a different tack in attempting to axiomatize qualitativemoment information and touse that to characterize a random variablerepresentation.

Alternative approaches to random variable ideas of error may beappropriate for extensions of RTM.Inparticular, applicationsof Bool-ean-valued andother multi-valued logics and fuzzy logics seempoten-tially interesting.

x®y =z iff E[4>{x)}+E[qb(y)]=E[(t)(z)].


PROBLEM4. Extendthe RTMapproach toinclude axiomsformulatedin termsofmulti-valued logics.

Efforts inthis direction shouldbecarried outsoas toproduceeither newkinds ofresults -not just translations ofknown random variable repre-sentation results- or new insights into measurement through conceptsnot available in the standard approaches to random variables. Someprogresshasbeen madeby HeyerandNiederee (1989),but much moreresearch onthe topic is needed.7

3.3. Meaningfulness

The meaningfulnesspartofRTMhasnotreceived asmuch attention astheexistence and uniquenessparts and so it is less fully developed. Inparticular, as with the definition of a measurementscale, the criterionfor meaningfulness invokedin RTMhas notbeen adequately justified.Although FM3 describes methods for linking qualitative correlates tomeaningful quantitative relations,no justification is provided for whythesequalitative correlates are indeed empirical.

Narens (1988) showed that thesequalitative correlates are definablein terms of the primitives througha very powerfulhigher-orderlogicallanguage that includes individual constant symbols for purely mathe-maticalentities. Becauseempiricaldefinitions requireonly much weak-er logical languages,Narens' resultsestablish that the qualitative corre-lates of non-meaningful qualitative relations cannotbe defined empir-ically in terms of the primitives of the empirical structure. Thus, thecorrelates of non-meaningful relations are non-empirical with respectto the primitives. These results can also be used to show that thereexistqualitative correlates of meaningful quantitative relations that arenecessarilynon-empirical. The conclusion to be drawn from this is thattheRTMconceptof meaningfulnessgivesanecessary but notsufficientcondition for empirical significance. It should be remarked that mostof theapplications of themeaningfulness concept,such as dimensionalanalysis (Bridgman, 1922, 1931; Luce, 1971, 1978), use it only as anecessarycondition for empirical significance,i.e.,use itas aconditionfor eliminating from consideration non-meaningfulrelationships.

230 R. DUNCANLUCEANDLOUIS NARENS

PROBLEM5. AmendRTM'sconceptofmeaningfulness so that it cap-turesinamoreappropriatefashion the conceptofempiricalsignificance(with respect to the qualitative structure). Obviously, amajor partofthisproblemisgivingacoherentformulationof'empiricalsignificance.It may well be the case that the solutions to Problems 1 and 5 areintimately connected.

3.4. ContinuumRepresentationsAlthough there are someresultsonfinite measurementstructures, someof whichare quiteusefulin applications,andondenumerable structures,the general consensus is that the strongest, most elegant results areabout structures that map onto a continuum. Examples were cited inSection 2.1 and additional onesaregivenbelow.

The results about structures on finite sets are of two distinct types.One in essence axiomatizes a finite standard sequence, which leadsin the usual way to an integer representationand a simple uniquenesstheorem. Theother establishes asetofinequalities thatmustbe satisfiedand the existence of a numerical representation is established but weusually are unable togive a compact anduseful characterization of itsuniqueness.

Assuming auniverse of finitely many particles, which manybelieveis implied by current physical knowledge, why does one ever needto look at denumerable let alone continuum results; yet it is these,especially the latter, that seem to be of greatest import for scientificmeasurement. The problem divides naturally into two phases: fromfinite to denumerable and from denumerable to continuous. One canenvisage some sort of theory concerning a nesting of finite systemsthat leads, asymptotically, to a denumerable structure from which thefinite systems can be thoughtof as samples. But it is well known thatthe step from denumerable systems,even the rational numbers, to thecontinuum is delicate. For example, the results on scale type to bedescribed in Section 4becomevastly morecomplex in the denumerablecase (Cameron, 1989). Considerable research exists on the Dedekindcompletions8 of certain classesof structures (seeNarens, 1985; Ch. 19of FM3), but this has not yet been given an adequatejustification forusingcontinuum models.


PROBLEM 6. Provide aprincipledaccount of why it is scientificallyuseful to replacefinite structures by continuum ones. In particular,it is important to make clear just what limiting processesgive rise toDedekindcompletenessorArchimedeaness in the continuum,and alsowhatgives rise to thepropertiesofhomogeneity andfinite uniquenessthatare discussednext.

This problemmay well be closed related toProblems 1and5.

4. CLASSIFICATIONOFSTRUCTURES BY SCALE TYPE

Coexisting with therepresentational approach has been another themewhich, beginning in 1981, began to flourish as a major alternative.During theearlierdebates over the existenceofpsychologicalmeasuresof any sort,as distinctfrom physical ones,Stevens (1946, 1951) placedgreat emphasis on the uniquenessof representations. His list of scaletypes - ordinal, interval, ratio, and absolute - is famous, and mostempiricalexamples fell within it.But not all. For example, Narens andLuce (1976) showed thatonecouldsimply drop the associativity axiomfrom classical extensive measurementand still show the existence ofan (inherently non-additive) numerical representation. But they failedto provide a satisfactory description of its uniqueness beyond the factthat specifying a single point rendered it unique. The reason for theirfailure did not become apparent until the work of Cohen and Narens(1979)in which the uniquenessproblem was first treated asessentiallyequivalent tounderstandingthegroupofsymmetries (=automorphisms)of the structure, i.e., isomorphisms of the structure onto itself. Theyshowed that for non-associative concatenation structures the group ofautomorphisms issurprisingly simple: an Archimedean ordered group.Holder (1901)hadcompletelycharacterized suchstructuresbyshowingthat each is isomorphic to some subgroup of the multiplicative positivereal numbers (MPRN). Ratio scales onto the positive reals are thosecases where the automorphism groupis isomorphic to theentireMPRNgroup.

4.1. Homogeneity, Finite Uniqueness, andScale Type

The results just described were a precursor to Narens' (1981a,b) pro-posedclassification of the automorphism groups of ordered relational

232 R. DUNCANLUCE AND LOUISNARENS

structures in terms of two major properties,called the degreeof homo-geneityanddegreeofuniqueness. Theautomorphism groupissaidtobeM-point homogeneousif for any two ordered sequencesofMdistinctelements, thereexistsan automorphism that takes the first sequenceintothe second. That group is said to be N-point unique if whenever twoautomorphisms agree atNdistinctpoints, theyarenecessarilyidentical.For anM-point homogeneous structure with more than Mpoints, it iseasy to see that M < mmN. The scale type of the structure is theorderedpair ofnumbers (maxM,mmN). Ratio scale structures areoftype (1,1); interval scales ones,of type (2,2).

The question posedby Narens was: what scale types are possible?A number of general,but still partial, answers are known for (simply)orderedrelational structures inwhich theprimitiverelations are of finiteorder.

4.2. A Recipefor ConstructingNumericalRepresentations

The major results rest onproperties of a subset of the automorphismscalled translations: the identity map together with all automorphismsthat do not have any fixedpoints. Three major questions about themare:

(i) Dothe translationsform amathematical group? Theonlyrealprob-lem is in showing that they are closedunder function compositionwhich is equivalent to showing that theyare1-pointunique,

(ii) Under the orderingof the translations naturally induced from theorder of the givenstructure,9are the translations Archimedean?10

(iii) Are the translations 1-point homogeneous?Once these facts are established,one can construct a numerical repre-sentation in whichthe translations appearasmultiplication byconstants(i.e., the similarity group) as follows: using homogeneity, map thestructure isomorphically onto the group of translations;usingHolder'stheorem, map the translation group, and so the structure, into MPRN(Alper, 1987;Luce,1986, 1987). Thatleaves unanswered the questionabout the rest of the automorphisms. A simple answer is known whenthe underlying ordered relational structure is order dense:11 the auto-morphisms are a subgroupof the power group x— ► rxs,which meansthatmmN<2 (see FM3,Theorem20.7,Corollary 2).


4.3. Some Structures with TranslationsSatisfying the RecipeSo a key problem is to try to understand what structural propertiesgive rise to these three properties of translations. So far, no one hasderived any necessary structural properties from these three propertiesof translations. All that is known are certain sufficient conditions. Wecite two partialresults.

First, if the ordered relational structure is Dedekind complete andorder dense, then (i) implies (ii) (see Theorem 20.6, FM3). Second,and this is the mostgeneralresultknown at present,if theordered rela-tional structure canbe mapped onto the real numbers,is homogeneous(maxM> 1),and is finitely unique (mm N< oo), then the threecon-ditions are satisfiedand thepossible scale typesare (1,1),(1,2)and (2,2), with the first beinga ratio scale, the thirdan interval scale,and the(1, 2) case falling between the two. An example of the (1, 2) case isthe group of real transformations x— > knx +s,where k>o is fixed,n ranges over all integers, and s ranges overall real numbers. Narens(1981b)proved thepart of this result for which maxM= mmN,andAlper(1987),usingaverydifferent approach, proved theresult withoutthatrestriction. Alper'smethods first made veryclear the importanceofthe threeproperties of the translations. Surprisingly, the most difficultproperty toestablish is that the translations form agroup.

4.4. StructuralEquivalents to the KeyProperties ofTranslations

Given the results justmentioned, it is clear that we still have much tolearnabout the conditions under which the translations forma homoge-neous,Archimedean-ordered group.

PROBLEM 7. What structural propertiesare implied by each of thetranslation properties-group, Archimedean,and homogeneity -sep-arately or together? As none are currentlyknown, a simpler problemmay be: what structuralproperties are sufficientfor eachof the threeproperties of translations, either separately or jointly? Ideally, onewould like tofinddistinct structuralfeatures thatcorrespondseparatelyto eachproperty,although that may wellprove infeasible.

For any homogeneous and finitely-unique ordered structure having abinary,monotonic operation, not necessarily on a continuum,one can


prove directly that minN < 2 (Luce and Narens, 1985). Little isknown in generalabout the automorphism groups in this case, except,as was noted above, when the operation is also positive,solvable,andArchimedean, one can prove, without assuming homogeneity, that theautomorphism group is Archimedean ordered. We do not know ofcomparable results in the remaining homogeneous case in which theoperation is necessarily idempotent.12 In particular, we do not knowof conditions that result in the translations forming an Archimedeanordered group.

PROBLEM 8. What algebraic properties on homogeneous, finitelyunique relational structures are just sufficient' to prove mm A^ < 2?The twoknown results shouldbe specialcases. When mmN< 2canbeproved, whatcanbe saidabout the setof translations?

(It strikesus asunlikely thatareasonable setofnecessary andsufficientconditions willbe found, andso the criteria for 'just sufficient' shouldbe considered to be flexible.)

5. SCALE TYPEINSPECIFIC STRUCTURALCONTEXTS

5.1. CombiningScale TypeandStructural ConditionsOnce theresultsonscale typesbegan to be discovered,thepossibility ofapplying them to specific measurementproblems beganto be explored.Basically the strategyhas been to combine the homogeneity and finiteuniquenessconditions with more specific structural assumptionsand tocharacterize in greaterdetail theclassesofstructures that canarise. Oneexample of this is found in Luce and Narens (1985) in which homo-geneous, finitely unique concatenation structures onthe continuum aredescribed quite fully. Closely related is the fact that for the class ofconcatenation structures that are positive, solvable, Archimedean, andDedekindcomplete,if the ordered automorphism group is order dense,thenit isalsohomogeneous. A thirdexampleis continuous semiorders;theyhavea (1,oo) group of automorphisms with asubgroup of transla-tions that is homogeneousandcan be ordered so thatit isArchimedean.We do not know of other cases where the numerical representation ofastructural axiomatization has been studied by proving that its transla-

REPRESENTATIONALTHEORY OF MEASUREMENT 235

tions form ahomogeneous,Archimedean ordered group.

PROBLEM 9. Explore the possibility of using the recipe describedabove to construct numerical representations ofparticular orderedstructures that do notfall under the scope of the theorem of AlperandNarensmentioned in Section 4.3.

5.2. Structures ThatAre Not Finitely Unique

A structure may fail either homogeneity or finite uniqueness; the twofailures are very different.

Ordinally scalable structures are not finitelyunique; indeed,it takesat least an order dense subset of the domain to fix an automorphism.Moreover, they are also M-pointhomogeneous for each finite M. Wesay they are of scale type (oo,oo). Roberts and Rosenbaum (1985)established thatfor an orderedrelational structure that isM-pointhomo-geneous, ifMdoes not exceed the cardinality of the domain andif theorder of each defining relation of the structure is not greater thanM,then the automorphism groupof that structure is identical to thatof justthe ordereddomain. Thus,if these conditions are met and the ordereddomain is isomorphic to the ordered real numbers, then the structure isof scale type (00,oo). Droste (1987) has characterized in aconvenientform theautomorphism groups of such structures.

Thisleaves the (M,oo) cases,about which relatively little isknown.Althoughit hasbeenshown thatcontinuous semiorders areexamplesof(1,oo) structures (Narens,1994), weknow of noexplicit measurementstructures with 1 < M < 00. Thus, many questions remain unan-swered.

PROBLEM 10. What canbe saidabout structures and their automor-phism groupsofscalar type (M,oo) with1<M< oo?

5.3. Structures ThatAre Finitely Unique ButNot Homogeneous

For generalfinitely unique,non-homogeneous structureson the contin-uum,Alper (1987) providedadescription of thepossibleautomorphismgroups, but so far his classification has not been used successfully todescribe the corresponding structures. Luce (1992a) followed a far

236 R. DUNCANLUCEAND LOUISNARENS

more restricted tack,but one that seems highly relevant to some mea-surement applications. He defined a point in a structure to be singularif it remains fixed(or invariant) undereveryautomorphism of the struc-ture. The conceptofa translationis generalizedtobe either the identityoranyautomorphismwhoseonly fixedpointsaresingular ones. Clearly,if the structure is finitely unique,then ithas only finitelymany singularpointsandsoitismeaningful tospeakofsuch astructure asbeingtrans-lationalhomogeneousbetween adjacent singularpoints. For aclass ofstructures thathe callsgeneralizedconcatenation structures whichhaveamonotonic13 n-ary operation,he gavea fairly complete description ofthe possibilities when the structure isboth finitelyuniqueandtranslationhomogeneous between adjacent singular points. What makes matterssimple is that such structures can have at most three singular points:a maximum, a minimum, and an interior one. An example of such astructure is the multiplicative, positive real numbers augmentedby 0and oo with translations x— ► xr,wherer>0. The singular pointsarethe two extremeones,0 and 00,andone interior one,1.

If,in addition, the group of translations commute,14 thenan interiorsingular point, call it c, acts like a generalized zero15 in the follow-ing sense: if F denotes the operation, thenF(e,...,c,Xi,c,...,c) ==6i(xi), whereoneither sideofc the function 6i agrees witha translationof the structure. Moreover,ifany singularpoint is a generalized zero,then any other singularpoint, c', acts like an infinity in the sense thatif c' is an argument of the function,then the value ofFis c!. Finally,for structures on a continuum,Alper's results can be used to derive anumerical representation in which translations on each side of the inte-rior singularity are multiplication byaconstant,the two constantsbeingsimply relatedbyapowerrelation. As we shallseein Section 6.3, theseandrelatedresults havebeen appliedeffectively in devising a theoryofcertainty equivalents for gambles (Luce, 1992b).

PROBLEM 11. What structures are there that are usefulfor appliedmeasurement, beyond those based ona generaloperation, for whichhomogeneityfails at selected points? And what structures, althoughfailing homogeneity more globally, still have a fairly rich automor-phismgroup, suchas x— > knx, where k>o is afixedconstant andnrangesover the integers?


Important non-homogeneous structures lie outside this framework.The mostnotable examples are qualitative probability structures. Theyare non-homogeneous not only because of their extremepoints - thenull anduniversal events-butbecause two events that are qualitativelyequally probable need not exhibit the same relationalpatterns to otherevents. Also, the only automorphisms of sucha structure are ones,likethe identity, that take an event intoan equallyprobable one. Obviously,the previous tacks we have taken in structures that are more or lesshomogeneous are completelyuseless in such contexts. Yet,clearly theprobability case exhibitsa great deal ofregularity.

PROBLEM 12. What kind of useful classification can be given forstructures that exhibit a great deal of regularity, such as is seen inprobability structures, but thatonly have automorphisms afor whicha(x) is equallyprobable to x?

6. APPLICATIONS OFRESULTS ABOUTSCALE TYPES

6.1. Product-of-PowersCompatibility inBoundedCases

One feature of physical measurementis the existence of two distinctkinds of attributes: those having an internal structure, like mass, time,length,and charge,and those havinga trade-off structure between com-ponents, like energy,momentum, density, etc. A major feature of clas-sical physicalmeasures is that theconjoint ones can be representedasproductsofpowersof theextensiveones. Forexample, kinetic energyisgiven by mv2 and densitybym/V. This aspectof the representationis reflected inthe fact that theunits of the conjointmeasures are alwaysproducts of powersof theunits of extensivemeasures. How this arisesfrom measurementconsiderations is discussed in Ch. 10 ofFMI andagain,much more generally,in parts ofChs. 20and 22of FM3.

The most generalresult to date involves two major properties: aqualitative concept of how astructure on one component of a solvableconjoint structure distributes16 in the latter structure (Definition 20.6,FM3),and the assumption that the component structure is such that itstranslations form a homogeneous Archimedean ordered group. Thesetwo properties force theconjoint structure to have amultiplicative rep-resentation involving a power of the representation of the componentattribute (Theorem 20.7,FM3).

238 R. DUNCANLUCEAND LOUIS NARENS

This model of interrelated measures covers much of classical mea-surement and makes clear exactly which generalizationof extensivestructures - basically any ratio scale structure - can be added to thephysical structure without disrupting the product-of-powers feature.Unfortunately, it fails to cover everything of importance. The mostnotable exceptionis relativistic velocity, which is bounded from aboveby the speedof light.

As is wellknown,physicists have elected to keep the multiplicativeconjoint relation s = vt among distance,velocity,and time,and to usea non-additive and bounded representationof the associative concate-nation of velocities,namely, u(&v= (v + v)/{\ +uv/c2), where cdenotes the maximum velocity, that of light. (In principle, they couldhave adopted an additive representationof ©, but at the very consider-able expenseof foregoing the simple relation s = vt). Itis important torealize that © is not reallyan operationonthe velocity componentof thedefiningconjoint structure. The reasonis that v and v © v are velocitiesin a single frame of reference whereas v is in a different frame, one inwhich the distances and times are modified relative to the first frame.Thus,from theperspective ofrelativistic measurement,itis inappropri-ate to think of © as an operation on the conjoint structure. But evenif one does, it fails tobe distributive in the conjoint structure. Withinthe context of a binary operation, distribution comes to: if ut =u't'and vt =v't', then (v © v)t = (u1 © v')t'. It is easy to verify that thisfails. More deeply, when distribution holds, the automorphism struc-tures of the conjoint and component structures agree in the sense thatevery translation of the component structure is that component's con-tribution to an automorphism of theconjoint structure, andconversely.This is not true in the case of the velocity component. Yet, at the sametime multiplication by a positive constantr,whichis an automorphismof the conjoint structure, simply maps one velocity representation intoanother, with c becoming re. This lack of a connection between theautomorphism groups of the two structures and yet their tight productof powers representation is not understood in terms of measurementtheory.

This problem is important not only to complete ourunderstandingof how relativistic velocity ties into theclassical measures,but also toallow the introduction of other bounded measures. The most obviousof these is probability which, in computingexpectations, has aproductrelation with other measures, as in (subjective) expectedutility theory

REPRESENTATIONALTHEORY OFMEASUREMENT 239

(Savage,1954; yonNeumann andMorgenstern,1947). Aside from theboundedness, the two cases are very different. Other examples mayarise in the behavioral sciences wheremeasures of subjective intensityare almost surelybestmodelled as bounded from above.

PROBLEM 13. What links the bounded component structures ofaconjoint structureand the conjoint structure itselfsoas, again, to leadto aproduct-of-powers representation? Possibly one should initiallyassume the components are homogeneousbetween bounds, as in thevelocity case, but ultimately that restriction must be removed to dealwith theprobability case.

6.2. Compatibility ofPsychologicalandPhysical Theory

Matching is a psychological procedure in which a subject 'matches' astimulus in one intensitydomain,suchas brightness, to agivenstimulusin another intensity domain, such as loudness. Formally, there are twophysicalstructures XandS, withdomains X andS, thateachhaveratioscale representations; classically, they are simply extensive structuresbut the theory applies to any with aratio scale representation. The psy-chologicaldatamaybe summarizedas follows: for eachannX, there issome s in S such that, according to the subject, s matches x,which wesymbolize as xMs. In this situation,there are two physical relationalstructures and a purely psychological connecting relation Mbetweenthem. Empirically, to a first approximation at least, the matching rela-tion can be described as a power relation between the two physicalratio-scale measures. The question is to what does thiscorrespond.

Luce (1990) suggesteda principle of compatibility between the twodomains thatmay be described as follows. Toeach translation r of X,thereis a corresponding translation aT of S such that for all x 6 X andall s £ S:

xMs if and only if r(x)MaT(s).This is easily shown to be equivalent to a power relation between theratio scale representations, and a number of other relationships areexplored. The multiplicative constant of the relation has, of course,dimensions thatdependupon theexponent involved.

Clearly, the application just described is highly special to a particu-lar situation,but the general idea of asking which psychological laws

240 R.DUNCANLUCEAND LOUISNARENS

relatingphysical variables are in fact compatible with the translationstructure of these variables is a far more general principle. The fail-ure of such compatibility means non-physical variables are required todescribe the phenomenon.

PROBLEM14. Is there any deepscientificorphilosophicalgroundingfor the supposition thatpsychological laws should be compatible withthe automorphism structuresof thephysicalvariables that they relate?Are there other examples in which application of this principle canbeillustrated? Andare thereexamples whereit clearly is violated?

6.3. Applications ofResults about Structures with SingularPointsTodate just onenew application(beyondrelativistic velocity) has beenmade in the measurementliterature of the results about structures withgeneral operations that are finitelyunique, have singular points, andaretranslation homogeneous between adjacent singular points (see Sec-tion 5.3). It concerns certainty equivalents to uncertain monetaryalternatives.17 For a fixed event partition, a certainty equivalent canbe viewed as a monotonic function of money arguments- the pay-offsassociated with the severalsubevents- intoan amountof money that isindifferent totheuncertain alternative. A sharpdistinctionismaintainedbetweengains and losses,making 0an interior singular point. Assum-ing the structure is homogeneousoneither sideof 0 and finitely unique,which has been typical ofutility theories,and definingutility to be theisomorphism that represents the translations as multiplication by pos-itive constants yields a linear weighted averageutility representation.Thefact that0must beageneralizedzerois veryimportant inconstruct-ing the weighting functions ofrank- andsign-dependentcharacter. Wedonot gointo anyof the details,for the only point inmentioningithereis as evidence that such generalresults do have applications.

PROBLEM 15. Given that we gain someresults about structures withvariousforms ofnon-homogeneity(seeProblems11and12) includingoneswithinterior singularpoints, are thereapplications thatup to nowhave beenoverlookedbecausepreviously we didnot know how to dealwith suchsituations?


7. CONCLUSIONS

The representational theory ofmeasurement,despite itspositivecontri-butions,has been subjectedto attack on anumber of fronts. Onebasicissue is how to formulate clearly what one means by measurement insuch a way that the representing structure is derived from the axiomsof the qualitative structure. Of considerable interest are structures thatdo not map into the real number system, but rather into families ofrandom variables or into structures basedonmulti-valued logics(Prob-lems 1-5). Further,even in the case of numerical representations, onecan wonder why anything as idealized as the continuum is relevant toscience (Problem 6). The remaining problems are all considerably lessphilosophical and involve somewhat complex issues in the theory ofscale types. We know a lotmore about scale type than wedid 12 yearsago,but much about structures remains shrouded;hard work and newideas are probably needed to gain a deeper understanding. Some ofthese problems arelisted as7-15.

ACKNOWLEDGMENT

This work has been supported in partby National Science Foundationgrant SES-8921494 to theUniversity of California,Irvine. We wish tothankJean-Claude Falmagne,DieterHeyer,ReinhardtNiederee,andA.A.J.Marley for detailed and helpfulcomments onearlier drafts of thismaterial.

InstituteforMathematicalBehavioralSciences,University ofCalifornia,Irvine, CA 92717, U.S.A.

NOTES

1Krantz,Luce,Suppes, andTversky (1971) willbereferredtoasFMI;Suppes,Krantz,Luce, andTversky (1989) as FM2; andLuce,Krantz, Suppes, andTversky (1990) asFM3.2 Some of the subtletiesinvolved in formulatinguniqueness results are dealt withbyRoberts andFranke (1976).3 A conjoint structure is a weakorderingof aCartesianproduct;it is additiveif itadmitsan additiverepresentationover its components.

242 R.DUNCANLUCE AND LOUISNARENS

4 Archimedeaness simply means that any bounded standard sequence is finite. Putanother way, no positive element is infinitesimal relative to another element of thestructure. Inpractice,the impact of Archimedeanessis to permit homomorphisms intothe realnumbers rather than orderedextensions of the realnumbers such as the non-standard reals.5 Among thepositiveuses weincludethe testingof mathematically formulatedtheoriesrelatingseveralvariables,although werecognizethat othersmay notwant to takesuchaconsiderationinto account in finding asolutionto this problem. Subjective expectedutility,mentionedlater,is oneexample;threemore aregiven inSection6.6 The primary exceptions are purely ordinal cases including variants such as intervalordersand semiorders.7 See Heyer and Mausfeld (1987) for adiscussion of some conceptualproblems con-nected withBoolean-valuedapproaches.8 A structure is Dedekind completeif every subset of elements that is bounded fromabove has a least upperboundin the domain. A structure thatcan be embeddedin aDedekindcompleteoneof thesamealgebraic formis said tohaveaDedekindcomple-tion.9For automorphismsa,(3,a <' (3ifandonlyifforeveryelementxofX,a(x) < P(x).10 Forany two translationsoneof whichis greater than the identity, finitely many appli-cationsof thepositiveonewillexceedthe other.11Ifx < y,there exists z in X such thatx < z < y.12 Homogeneityimplies, for allelements x,either weak positivity,x o x > x, idem-potence, x o x ~ x,or weak negativity,x o x < x. The first andthird are formallyidenticalif > is replacedby<.13 The definitionofmonotonicis theusualoneexcept thatsomecareisneededindealingwithextreme points,if such exist.14 This is true if theycanberepresentedin themultiplicativerealnumbers. The assump-tionmay beredundant,butLuce failed toderiveit fromthe otherassumptions.15 The term 'zero' is appropriatewhenone thinks of homomorphisms to the realnum-bers where0is the interiorsingularpointand thetranslations arex

—» rx.

16 Let C = (A x X,>) be the conjoint structure withA a relationalstructure on Awhoseorder is that induced fromC. For fixedx,y € X, let a be the functiondefinedby allsolutions to (a,x) ~ (a(a),y). Then,A distributes inC if every such ais anautomorphismof A.17 This is closelyrelatedto theprospect theoryofKahnemanandTversky (1979) and tosuch extensionsof it asLuce andFishburn (1991) andTversky andKahneman(1992).

REFERENCES

Adams,E.: 1966, 'On theNatureandPurposeofMeasurement',Synthese,16, 125-169.Alper,T.M.: 1987, 'AClassificationof AllOrder-PreservingHomeomorphismGroups

ofthe RealsThat SatisfyFiniteUniqueness',JournalofMathematicalPsychology,31, 135-154.

Bridgman,P.: 1922, 1931, Dimensional Analysis,New Haven, CT: Yale UniversityPress.


Cameron,P.J.: 1989, 'Groups ofOrder-Automorphismsoftheßationals withPrescribedScale Type',JournalofMathematicalPsychology, 33, 163-171.

Campbell,N.R.: 1920/1957, Physics: The Elements,Cambridge: CambridgeUniver-sity Press. Reprinted as FoundationsofScience: The PhilosophyofTheory andExperiment,New York: Dover, 1957.

Campbell,N.R.: 1928,AnAccount of thePrinciplesofMeasurementandCalculation,London: Longmans, Green.

Cohen,M.andFalmagne,J.-C: 1990, 'RandomUtilityRepresentationofBinaryChoiceProbabilities: A NewClass of Conditions', JournalofMathematicalPsychology,34, 88-94.

Cohen, M. and Narens,L.: 1979, 'FundamentalUnit Structures: A Theory of RatioScalability',JournalofMathematicalPsychology, 20, 193-232.

Cohen, M.R. andNagel,E.: 1934, An Introduction to Logic andScientific Method,New York: Harcourt,Brace.

Droste, M.: 1987, 'Ordinal Scales in the Theoryof Measurement',JournalofMathe-maticalPsychology, 31, 60-81.

Ellis,B.: 1966, Basic Concepts ofMeasurements, London: Cambridge UniversityPress.

Falmagne,J.-C: 1979, 'On a Class of ProbabilisticConjoint Measurement Models:Some DiagnosticProperties',JournalofMathematicalPsychology, 19, 73-88.

Falmagne,J.-C: 1980, 'AProbabilisticTheoryofExtensiveMeasurement', PhilosophyofScience, 47, 277-296.

Falmagne,J.-C.andIverson,G: 1979, 'ConjointWeberLawsandAdditivity',JournalofMathematicalPsychology,86, 25^43.

Helmholtz,H. yon: 1868, 'Über dieThatsachen,diederGeometriczu Grunde liegen',GottingenNachrichten, 9, 193-221.

Helmholtz, H. yon: 1887, 'Zahlen und Messen erkenntnis-theoretisch betrachtet',Philosophische Aufsatze EdwardSeller gewidmet, Leipzig.Reprinted inGesam-melteAbhandl, Vol. 3, 1895, pp.356-391. English translationby C.L. Bryan:1930, Counting andMeasuring, Princeton,NJ: VanNostrand.

Heyer, D. andMausfeld,R.: 1987, 'On Errors,ProbabilisticMeasurement andBooleanValuedLogic',Methodika,1, 113-138.

Heyer, D. and Niederee, R.: 1989, 'Elements of a Model-TheoreticFramework forProbabilisticMeasurement', in: E.E.Roskam (Ed.),MathematicalPsychology inProgress,Berlin: Springer,pp. 99-112.

Heyer, D. and Niederee,R.: 1992, 'Generalizingthe Concept of Binary Choice Sys-temsInduced byRankings:One Way ofProbabilizingDeterministicMeasurementStructures', MathematicalSocialSciences, 23, 31-44.

Hilbert,D.: 1899, GrundlagenderGeometric(Bth cd.,withrevisions andsupplementsby P. Bernays, 1956), Stuttgart: Teubner.

Holder,O.: 1901, 'DieAxiomederQuantitatund dieLehre vomMass',Berichte überdie VerhandlungenderKoniglich SachsischenGesellschaft der Wissenschaften zuLeipzig,Mathematisch-Physische Klasse,53, 1-64.

Holman,E.: 1971, 'ANoteonAdditiveConjointMeasurement',JournalofMathemat-icalPsychology,8, 489-494.

Iverson,G. and Falmagne,J.-C: 1985, 'StatisticalIssues inMeasurement',Mathemat-icalSocialSciences, 10, 131-153.

244 R.DUNCAN LUCEAND LOUISNARENS

Kahneman,D. and Tversky, A.: 1979, 'Prospect Theory: An Analysis of DecisionunderRisk',Econometrica, 47, 263-291.

Klein,F: 1872 (this was theoralErlangen address;transcribedintoEnglish, 1893), 'AComparativeReviewofRecentResearchesinGeometry',Bulletinof theNew YorkMathematicalSociety, 2,215-249.

Krantz, D. H.: 1964, 'ConjointMeasurement: The Luce-Tukey AxiomatizationandSomeExtensions',JournalofMathematicalPsychology, 1,248-277.

Krantz, D. H.,Luce, R. D., Suppes, P., andTversky, A.: 1971, FoundationsofMea-surement, Vol. I,NewYork: AcademicPress.

Kyburg,H.E.,Jr.: 1984, TheoryandMeasurement,Cambridge:CambridgeUniversityPress.

Lie,S.: 1886, 'BemerkungenzuHelmholtz'ArbeitiiberdieThatsachen, diederGeome-tric zuGrunde liegen',Berichteüber dieVerhandlungenderKoniglich SachsischenGesellschaft der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse,38,337-342.

Luce,R. D.: 1971, 'SimilarSystems andDimensionallyInvariantLaws', PhilosophyofScience, 38, 157-169.

Luce,R.D.: 1978, 'DimensionallyInvariantNumericalLaws Correspond toMeaning-ful QualitativeRelations',Philosophy ofScience, 45, 1-16.

Luce,R.D.: 1986, 'Uniqueness andHomogeneity of OrderedRelationalStructures',JournalofMathematicalPsychology, 30,391-415.

Luce,R. D.: 1987, 'Measurement Structures with ArchimedeanOrdered TranslationGroups', Order,4,165-189.

Luce, R. D.: 1990, 'On the Possible Psychophysical Laws Revisited: Remarks onCross-ModalMatching',PsychologicalReview,91,66-77.

Luce, R. D.: 1992a, 'Singular Points in GeneralizedConcatenation Structures ThatOtherwise AreHomogeneous',MathematicalSocialScience 24,79-103.

Luce, R. D.: 1992b, 'A Theory of Certainty Equivalents forUncertain Alternatives',JournalofBehavioralDecision Making 5, 201-216.

Luce, R. D. and Fishburn, PC: 1991, 'Rank- and Sign-Dependent Linear UtilityModels forFiniteFirst-Order Gambles',JournalofRisk andUncertainty, 4, 29--59.

Luce,R. D.,Krantz,D. H., Suppes, P., and Tversky, A.: 1990, FoundationsofMea-surement, Vol. 111, New York: AcademicPress.

Luce, R. D. and Narens, L.: 1985, 'Classification of Concatenation MeasurementStructures According to Scale Type', Journalof Mathematical Psychology, 29,1-72.

Luce,R. D. andTukey, J. W.: 1964, 'SimultaneousConjoint Measurement: A NewTypeofFundamentalMeasurement',JournalofMathematicalPsychology, 1, 1-27.

Michell,J.: 1986, 'MeasurementScalesandStatistics: A ClashofParadigms',Psycho-logicalBulletin,100, 398^107.

Michell,J.: 1990, AnIntroduction to theLogicofPsychologicalMeasurement,Hills-dale,NJ: ErlbaumAssociates.

Narens, L.: 1981a, 'A General Theory of Ratio Scalability withRemarks about theMeasurement-Theoretic Concept ofMeaningfulness', Theory andDecisions, 13,1-70.


Narens, L.: 1981b, 'On theScalesofMeasurement',JournalofMathematicalPsychol-ogy,24, 249-275.

Narens,L.: 1985, AbstractMeasurement Theory,Cambridge,MA:MITPress.Narens, L.: 1988, 'Meaningfulness and the Erlanger Program of Felix Klein',

MathematiquesInformatiqueetSciencesHumaines, 101,61-72.Narens, L.: 1994, 'The Measurement Theory of Continuous Threshold Structures',

JournalofMathematicalPsychology, inpress.Narens,L.andLuce,R. D.: 1976, 'The AlgebraofMeasurement',JournalofPure and

AppliedAlgebra, 8,197-233.Niederee,R.: 1987, 'OntheReference toReal NumbersinFundamentalMeasurement:

A Model-TheoreticApproach',in: E.E.Roskam andR. Suck (Eds.),Progress inMathematicalPsychology, Vol.1, pp.3-23, New York,Amsterdam:Elsevier.

Niederde,R.: 1992, 'WhatDo NumbersMeasure? ANew Approach to FundamentalMeasurement', MathematicalSocialSciences, 24,237-276.

Roberts,F. S. andFranke,C.H.: 1976, 'OntheTheoryofUniquenessinMeasurement',JournalofMathematicalPsychology, 14,211-218.

Roberts,F. S. andRosenbaum, Z.: 1985, 'SomeResultsonAutomorphismsof OrderedRelational SystemsandtheTheoryof ScaleType inMeasurement',in: V Alavietal.(Eds.), GraphTheoryandItsApplicationstoAlgorithmsandComputer Sciences,New York: Wiley, pp. 659-669.

Savage,C. W. andEhrlich, P. (Eds.): 1992, PhilosophicalandFoundationalIssues inMeasurementTheory, Hillsdale,NJ:ErlbaumAssociates.

Savage,L.J.: 1954, TheFoundationsofStatistics,New York: Wiley.Scott,D. and Suppes,P.: 1958, 'FoundationalAspects of Theories ofMeasurement',

JournalofSymbolic Logic, 23, 113-128.Stevens,S.S.: 1946, 'OntheTheory ofScalesofMeasurement',Science,103,677-680.Stevens,S.S.: 1951, 'Mathematics,MeasurementandPsychophysics', inS. S. Stevens

(Ed.),Handbook ofExperimentalPsychology, New York: Wiley, pp. \-A9.Suppes,P.: 1951, 'A Set of Independent Axioms forExtensive Quantities',Portugal.

Math., 10, 163-172.Suppes, P., Krantz,D. H.,Luce,R.D., and Tversky, A.: 1989, Foundationsof Mea-

surement, Vol. 11, New York: AcademicPress.Suppes, P. and Zanotti,M.: 1992, 'Qualitative Axioms for Random-VariableRepre-

sentationsofExtensiveQuantities',in: C. W. SavageandP. Ehrlich (Eds.),Philo-sophicalandFoundationalIssues inMeasurement Theory,Hillsdale,NJ: ErlbaumAssociates,pp. 39-52.

Suppes,P. andZinnes,J.L.: 1963, 'BasicMeasurementTheory',in:R.D.Luce, R.R.Bush,andE.Galanter (Eds.),HandbookofMathematicalPsychology, Vol.1,NewYork: Wiley, pp. 1-76.

Tversky, A. and Kahneman, D.: 1992, 'Advances in Prospect Theory: CumulativeRepresentationof Uncertainty', JournalofRiskand Uncertainty 5,297-323.

yonNeumann,J.andMorgenstern,O: 1947, TheoryofGamesandEconomicBehavior,Princeton: Princeton University Press,secondedition.

246 R. DUNCANLUCE ANDLOUISNARENS


LuceandNarens givea substantial listof problemsthat arecentral to therepresentational theoryof measurement,a topicon whichDuncan andIhave worked together with other colleagues for more than aquarterof acentury. DuncanandLouis bring outnicely the range of philosophicaland scientific problems still to be faced in the theory of measurement.Unfortunately, most of these problems as well as the ones that havebeen solved in the past have not attracted the interest of philosophersof science in the way onemight have thought they would. Ithas turnedout that in spite of the philosophical rootsof much of the work in thetheoryof measurement,the current developmentshavemainly beendueto scientists and mathematicians.

Theproblems of measurementin thebehavioral andsocial sciencespresent foundational and conceptual issues of considerable subtletywhich now have a large literature, especially those surrounding themeasurementofsubjective probability andutility. Without question theproblems formulatedbyLuceandNarensalldeserveattention,althoughof course some areof more generalinterest than others. It isareflectionof the generalnature of the theory of measurement that my own list ofproblems would overlapbut still be rather different from thatpresentedby them. Sayingsomethingaboutmy ownlist is notmeant to be acrit-icism of theirs,but is a way of emphasizing therange of philosophicalissues stillopen in the theory ofmeasurement. The two large topics thatIwouldorganize problems around and that are not directly mentionedby Luce and Narens are geometricalproblems of measurement-whatarealsocalled multidimensional scalingproblems-andsecondly,com-putationproblems.

Geometrical Problems. As Luce and Narens point out in variousplaces briefly and casually, there is a long history of representationand uniqueness problems in geometry. There is a substantial reviewof classical work inFoundations ofMeasurement, Vol.IIbut there areways in whichthe modern representationaltheory ofmeasurementcallsfor new developments in geometrical representation theory. The mostimportant direction,in my own judgment, is to developrepresentationtheories like those ofmeasurementthatareembeddingsandnot isomor-phisms to standard analytic representations. Typical examples wouldbe sufficient,and where possible,necessary andsufficient conditions to

REPRESENTATIONALTHEORY OF MEASUREMENT 247

embed abounded fragmentof Euclidean geometryinaEuclidean spaceof the same dimension and with the usual results on uniqueness forthatembedding. Anotherkind ofexample of interest in currentphysicsis a qualitative axiomatization of a discrete lattice of points for spe-cial relativity to provide aqualitative geometrical framework for latticecomputations andother discrete geometricalreasoningcharacteristic ofmuch modern physics.

Inasimilar vein,but withundoubtedlyasomewhat different concep-tual apparatus, wecouldmuch benefit from deeperqualitative analysisof the geometrical structures thatarise inmultidimensional scaling.

The closerelation between topology andmeasurementinmany con-textsisperhaps thepartof geometrymostneglectedin theFoundationsof Measurement, but deliberately so for reasons given in Volume I.Topologyprovides a general set ofconcepts for formulating axioms onqualitative structuresof adifferent sort thanhavebeenmainly exploitedin the theory of measurementthus far, although there has been a cer-tainbody of work ineconomics using topologicalconditions rather thanalgebraicones in the theoryofutility. Ontheotherhand,problemsof therelationbetween topologyand measurementin the theoryofperceptionhaveas yetbeen littleexplored.

Some of the most interesting recent work of Luce and Narens hasbeen on characterizing the automorphism groupof a suitablemeasure-ment structure as an Archimedean ordered group. There is a similarbut deeper and more far-ranging set of problems in geometry on theconnection between various transformation groups for geometry andthe characterization of these transformation groups as themselves beinginstances ofmanifolds. Forexample,the groupofrotations ofEuclideanthree-space is a nonsingular surface with a system of local coordinatesprovidedby familiar Euler angles. Exploitation of thiskind ofrelation-ship has scarcely begunas yet in the theoryof measurement.

ComputationandtheContinuum. Another important aspectofrecentworkbyDuncanandLouis inthe theory ofmeasurementis thatof scaletype. Here the work has depended almost exclusively on assumingthe measurementstructure is isomorphic to the continuum ofreal num-bers. Theseresults naturally raise philosophical questions thatcall fora deeper analysis of why the continuum is important in the theory ofmeasurement. Initially we think of measurement as a very finitisticconstructiveprocedure used in almost all domains of science to assign


numerical quantities to various empirical properties. It would be sur-prising if highly nonconstructive aspects of the entire continuum ofreal numbers really did play some essential philosophical role in ourconceptionof measurement. Insaying this,of course,Iam expressingphilosophicaldisagreement withLuce andNarens, adisagreement thatwehave discussed on various occasions.

AsIhave stated in commentsonother articles,Ilook uponclassicalanalysis and continuum mathematics as beingmainly computationallyimportant. The differential and integral calculus is just that,a calculus,not a foundational view of how the universe is really put together.Iasmuch as LuceandNarens,perhapseven more so, cherish infinitesimalsand what they can do for providing efficient methods of computation,especiallyinphysicsandengineering.Idonot for amomentnecessarilybelieve that infinitesimals are really out there in the real world.Iwouldbe quite prepared to accept the fact that space is ultimately discreteand we cannotgobelow a certain smallest measurementof length orof other quantities. This would not for a momentshake my confidencein the importance of infinitesimal methods in science which have beenused so successfully for over two centuries, but it is rather to insistthat they provide wonderfully efficient methods of computation, nota fundamental view of the world. Ilike very much the derivationof standard diffusion equations from taking the limit of very discreterandom walks.Iam quite happy to look upon the limit operation as anideal one abstracted from thereal detail of particles and the spaces inwhich they operate.

It is this kind of philosophical view of mine thathas led me to be amuch strongersupporterof finite structures andfinitistic or constructivemeasurementprocedures thanare Duncan andLouis. Thatdebate willcontinue andprobablyno endis in sight. That theyremainunconstruct-ed continuum advocates is clear from their statementof problems atthe beginning of their paper. ButIfound puzzling their statement inSection 3.4 that we couldhave "doubts about the heavy useof infinitestructures and continuum mathematics when, after all, the universe is,according to contemporary physics, composed of only finitely manyparticles." This is not the real problem of using the continuum. Wecould very wellrequire thecontinuum if there were only three particlesbut they were moving alongcontinuous paths. The realcommitment isto therebeingonly discretespace anddiscreteproperties or,ifyou wish,only a finite number of spatial points, at least in any bounded region,


anda finite numberof values of anypropertyof anyparticle. Thiskindof constructivism is clearly very far removed from a large number oftheir Problems.

In raising and pursuing once again this dialogue with them,Iamnot suggesting that Ihave any strong commitment that the wayIamsuggesting is the only way to proceed. In fact, it may well turn outthat the line of attack they have taken will be more fruitful than amore constructive finitistic approach. Actually, from a philosophicalstandpointInow favor a view thathas as yetnot been developedveryfar technically. Thisis theview that in the framework ofcurrentphysicaltheory we cannotempirically determine whether spaceis continuous ordiscrete,and possibly a decision that was empirically supported couldonly befor discreteness. In any case, given currentphysics, the choiceis transcendental,i.e.,beyondexperience. Consequentlyit isof interestto develop theories whose fundamental concepts are invariant underappropriatemappings from the infinite to the finite.

As a simple example, let (X,d) and (Y,d') be two metric spaceswith X beinga bounded infinite set and V a finite set. Then (X,d) ise-homomorphic to (V,d 1) if and only if there is amapping / from XtoV such that Vx,y in X

More to thepoint,rather than thissimple abstract example,are themeth-ods currentlyused to approximatecontinuous processesby the discreteones used in computer simulations,or the classical approximation ofthe discrete by the continuous,as in approximating the binomial by thenormal distribution. Studies of invariance andmeaningfulness in thesecontexts would in all likelihood be conceptually enlighteningnot onlyin terms of theories of measurement.

\x-y\<€ iff \f(x)-f(y)\<e.

FRED S.ROBERTS ANDZANGWILL SAMUEL ROSENBAUM

THEMEANINGFULNESS OF ORDINAL COMPARISONSFORGENERAL ORDERRELATIONAL SYSTEMS

ABSTRACT. This paperstudies themeaningfulness (invariance)of the ordinalcom-parisonf(a)> f(b),madeusingascaleof measurement/. Specifically,it investigatesscales/ whicharise fromhomomorphismsintogeneralorderrelationalsystems, whichare theunions of m-ary relations induced on the set of real numbers by rankings ofthe integers in {1,2,...,m}. The results generalize earlier work in the literature,whichsets out to make a systematic analysisof conditionsunder which the assertion/(a)> f(b) is meaningful.

1. INTRODUCTION

Intherepresentationaltheoryofmeasurement,aspioneeredin thepapersby Scott and Suppes (1958) and Suppes and Zinnes (1963) and in thebookbyKrantz,Luce,Suppes,andTversky(1971),oneconsiders scalesof measurementas arising from homomorphisms of relational systems.We will be specifically concerned with homomorphisms into a verygeneralkindofrelational system which wecallageneralorder relationalsystem. Inmeasurementtheory,one studies admissible transformationstaking one homomorphism into another and calls a statement usingscales of measurement meaningful if it is invariant under admissibletransformations. In this paper we study the meaningfulness of theordinalcomparisonf(a) > f(b). Asystematic studyof thiscomparisonfor homomorphisms intogeneralorder relational systems wasbegunbyRoberts (1984) and continued in Harvey and Roberts (1989). Thepurpose of this paper is to generalize the results in those two papers tocover a much wider variety of situations.

Inthis section we present some of thebackgroundmaterialonmea-surement theory needed for the rest of this paper. In Section 2 weintroduce the general order relations that will be a primary topic ofinterest here. We thenstudy the meaningfulness ofordinalcomparisonswith homomorphisms into general order relational systems. Section 3starts with meaningfulnessof equality comparisons f(a) = f(b). Sec-tion4 begins the study of f(a) > f(b). Section 5 showshow to reduce

251

P. Humphreys (cd.),Patrick Suppes: Scientific Philosopher, Vol. 2, 251-274.© 1994 KluwerAcademic Publishers. Printedin the Netherlands.

252 FRED S. ROBERTS ANDZANGWILL SAMUEL ROSENBAUM

the problemin many situations tohomomorphisms intospecial generalorder relations called strict and Section 6handles ordinal comparisonsfor homomorphisms into strictgeneralorder relational systems.

We shall follow the notation and terminology of Roberts (1979).Suppose A andBare sets andRand Sare ra-ary relations on AandB,respectively. We shall be interested in the systems (A,R) and (B,S),and sometimes we shall just refer to these systems as m-ary relations.A homomorphism from (A,R) to (B,S) is afunction / from A into Bso that for all a\,...,am from A (notnecessarily distinct),

We shall sometimes interchange the terms scale and homomorphism.We shall also speak of the representation (A,R) —

► (B,S) and theproblem of finding a homomorphism from (A,R) into (B,S) as therepresentationproblem. Anadmissible transformationof/ isafunction<p : f(A) —

► B so that cp o f is again a homomorphism. A statementinvolving scales of measurement is called meaningful if its truth orfalsity is unchanged if admissible transformations are applied to allof the scales or homomorphisms in the statement. This concept goesback to Suppes (1959) and to Suppes and Zinnes (1963). In somecases of relations (A,R) and (B,S), not every homomorphism from(A,R)to (jB,S)canbeobtainedfromeveryother homomorphismbyanadmissible transformation. In this case,RobertsandFranke (1976)callthe representation problem (A,R) into (B,S) irregular and introducethe following more general concept: a statement involving scales ofmeasurement is called meaningful if its truth or falsity is unchangedif all of the scales or homomorphisms in the statement are replacedby other scales or homomorphisms. This definition is reasonably wellaccepted,at leastas anecessarycondition for 'meaningfulness',and weshall adoptit here. However, thereare special situations whereit mightnot be appropriate. See Falmagne and Narens (1983) for a discussionand see Roberts (1985, 1989, 1994) and Luce, Krantz, Suppes, andTversky (1990) for surveys and general results about the concept ofmeaningfulness.

Suppose / is a homomorphism from (A,R) into (E, S), where Eis the set of real numbers. Many assertions can be made using thescale /. We shall be specifically interested in the simple assertion thatf(a) > f(b). When is it meaningful? We say that / is of ordinal scaletype if the representation(A,R) into (E,S) is regular (not irregular)

(1) (ai,...,am) G R<r+ (/(ai),...,/(am)) G5.

THEMEANINGFULNESS OFORDINAL COMPARISONS 253

and the collection of admissible transformations of / is exactly thecollection of order preservingmaps from f(A) into E, transformations(p so that

If/ isofordinal scaletype,then theassertionf(a) > f(b)ismeaningfulfor all a,b in A.

2. GENERAL ORDER RELATIONS

In this paper, the primary object of investigation will be a homomor-phism from an m-ary relation (^4,R) into a generalorder relation onthe reals. To define generalorder relations and explain why they areimportant, we first introduce some preliminary notions.

Letmbe afixed integergreater than1andletM — {1,...,m}.Bya ranking ofMwe willmean alinear ordering ofMwith ties allowed.(Roberts (1979)calls thisa weak order.) For example,ifm =4, we willwrite 3, 2-A,1 to represent that rankingofM which ranks 3 highest,2and 4 tied for next, and 1last. If aparticular ranking ofMhappens tohaveno ties, we will refer to it as astrict ranking ofM. The height ofaranking is thenumber of different levels it contains. For example theheightof theranking, 3, 2-4, 1is equal to 3, which we will frequentlydenote as ht(3, 2-4, 1) = 3. Note that if n is a ranking of Mthen1 < ht(7r) < m and thatht(7r) = mif and only if ir is a strict ranking.At theother extreme,the ranking1-2-...-m,whose heightisequal to1, will be called the tieranking ofMand designatedas c.

If 7r is a ranking ofM, wecan use it to define an m-ary relation onthe reals,Tn,as follows:

For example,if tt is theranking3,2-4,1,then we have that (14,23,87,23) G Tx while (94, 18,37,18) g Tv.

We now define an m-ary relation Sonthe reals to be ageneralorderrelation or GOR if

X\ > X2<-► ip(xi) > <p(x2).

(xi,...,x m) GTT iff(xi > Xj <->■ iis ranked ahead ofjin 7r).

s=7vlur,r2u...ur7rp


for rankings tt\,7r2,...,irp ofM,for somep > 1. We will let 11(5) =n= {7ri,...,7TP}. If all 7r G nhappen to be strict rankings, we willsay thatS is astrictgeneralorderrelation or SGOR.For example,withm= 2, some common binary relations on the reals qualify as GORs.For instance,Tj)2 is >,T2)i is<,Ti_ 2 is =,Th2UTi_2 is >,Th2UT2)lis Ti_ 2 UT2)i is<. Of these,>,<, and are SGORs.With'm =3,

UT3)2 i yields theusual 'betweenness' relationonthereals. Thus,many of therelations of interest in measurementtheory arise as GORs.

The GORs arose from the concept of m-point homogeneity whichplays a central role in the theory of scale type and meaningfulness. Arelation SonE is called m-point homogeneousifwhene-ver

x\ >x2> ... >xm and y\> y2 > ...> ym

for real numbers X{ and yi, there is an automorphism <p of (E, S) (one-to-one homomorphism from (E, S) onto (E, S)) so that <p(xi) = yi,i= 1, 2,...,m. The literature of m-point homogeneity is summarizedinLuce,Krantz,Suppesand Tversky(1990),Roberts (1989),andLuceand Narens (1986, 1987). Roberts (1984) showed that if S is m-ary,then (E, S) is m-point homogeneous ifandonly ifS is0 orSisa GOR.He then pointed out that in the systematic study of the meaningfulnessof ordinal comparisons f(a) > f(b) for homomorphisms /, it wasnaturalto studythecase where thosehomomorphisms were intom-pointhomogeneousm-ary relations or GORs,and he initiated that study forthecase where S = T*. Harvey andRoberts (1989) studied the sameproblem for generalGORs and in particular resolved all cases wherep= 2andm *= 2,3. Themainpurposeof thispaperis topresentresultsabout this problem in a much more generalsetting. Homomorphismsinto general order relational systems, systems (E,5i,... ,5&) whereeach Si is a GOR,are also ofinterest. However, their study will be leftto a later time.

Inthispaper,we shallbeinterestedinhomomorphisms fromrelations(A,R)whichare m-ary andfor whichR 0andR Am,whereAl7l istheCartesianproductofA with itselfmtimes; weshallcall such (A,R)nontrivial. A homomorphism from anontrivial m-ary relation (A,R)intoan m-ary GOR(E,S) willbecalledan m-aryGORhomomorphism.If (E, S) is an SGOR, / will becalled an m-ary SGORhomomorphism.

THEMEANINGFULNESSOFORDINAL COMPARISONS 255

3. MEANINGFULNESS OFEQUALITYCOMPARISON

Before studying the meaningfulness of the ordinalcomparison f(a) >f(b), weconsider thatof the equalitycomparison /(a) = /(&). We firstprove a lemma which we willuse throughout.

LEMMA 1. Suppose ir is a ranking of {1,2,...,m], fisa functionfrom A into E, |/(A)| > ht(7r) > 2, and a,b are elements of A withf(a) f(b). Then there are a\%...,am in A with as =a,at — bforsome 5,t € {1,...,m} such that

and

(/(a1 ),...,/(am))GT7r.Proof Assume without loss of generality that f(a) > f(b). There

are bu ..., bht{n) so that (*) /(&i) > ... > f(bht{7r)). If /(a) isany f(bi), replace bi by a. Ifnot, let ibe the largest index such thatf{bi) > f(a) or, if there is no such index,let ibe 1. Replace bi by a.We still have (*). Similarly, we canreplace some bjby b and preserve(*). A complication arises if bj is a. Then replace bj+\ by b, unlessj= ht(ir). In that case, drop b\, lower every index by one and letb = bht^y Again, (*)holds. Lastly, permute the bi appropriately and,at components where 7r declares a tie, throw in a repeat of the corre-sponding f(bj). This defines a\,...,am with the desired properties. ■

THEOREM1. Suppose f : (A,R) -► (E,S) is anm-ary GORhomo-morphism. Then the statement f(a) = f(b) is meaningful for alla,b G A.

Proof. Assume there are a,b G A for which /(a) = f(b) is notmeaningful. Then there are homomorphisms gand h from (A,R) into(E, S) so that g(a) = g(b) and h(a) h(b). Assume without lossof generality that h(a) > h(b). The analysis now breaks down intotwo distinct cases, depending on whether or not the tie ranking c =1-2-...-mappears inn.

Case 1. c G" 11. If we let k = min{ht(7r) : it G n}, then wehave k > 2. It is important to note that \h(A)\ > k. Otherwise,for any ai,...,am Giwe would have that (h(a\),...,h(am)) £

\{au...,am}\ =ht(7r)


S, which would in turn imply (because h is a homomorphism) that(0i,...,am) G" R. This contradicts our assumption thatR 0 (since(A,R) is nontrivial). Hence, \h(A)\ > k.ByLemma1,since \h(A)\ >k> 2, there must be a\,...,am in A with as =a,at=bso that

and (h(a\),...,h(am)) G5. Since /iis a homomorphism, this impliesthat (ai,...,am) G i?, which, because gis a homomorphism, inturn implies that (g(a\),...,g(am)) GS. By definition of k, G ={g(ai),...,g(am)} has cardinality at least k. However, by (2),it hascardinality at most k. Hence, it has cardinality k and, in particular,g{a) 7^ g{b). This is a contradiction.

Case 2. c G n. The proof proceeds much as in Case 1,but nowwe let k =min{ht(7r) : it G" n}. Note that k exists because IIdoesnot contain allpossible rankings ofM. Otherwise, for anya\,...,amin A, we would have (h(ai),...,h(am)) G S, which would in turnimply that (ai,...,am) GR. This would contradict our assump-tion thatR/ Am (since (^4,R) is nontrivial). Since we are assum-ing that c G n, we know that k > 2. It is important to note that\h(A)\ > k. Otherwise, for any ai,...,am GA, we would have(/i(ai),...,/i(am)) G5,so (ai,...,am) G#,and we wouldbe forcedto conclude that R = Am. Because \h(A)\ >k> 2, Lemma 1implies that there are a\,...,am in A with as

= a, at = bso that(2) holds and (h(ai),...,h(am)) 0 5. Since h is a homomorphism,this implies that [a\,...,am) G" J? and so, since pisa homomorphism,(p(ai),... ,g(am)) &S. Sinceeverym-tuple onEis insomeT^onM,by definition of k, G = {<?(ai),...,j(am)} has cardinality at least k.However,by (2),it has cardinality at most k. Hence, it has cardinalityk and, inparticular, g(a) g(b). This is againa contradiction. ■

The meaningfulness of the equality statement /(a) = f(b) plays afundamental role in the analysis of the measurement process. Specif-ically, Roberts and Franke (1976) prove that a homomorphism from arelation (A,R) to a relation (B,S) is regular if and only if the state-ment f(a) == f(b) is meaningful for alla,b G A. Hence, we have thefollowing Corollary.

(2) \{a1,...,am}\ = k


COROLLARY 1.1. Iff : (A,R) -> (E, S) is anm-ary GORhomo-morphism, then(a) / is regular;(b) f isoftheordinalscale typeifandonlyifthe statementf(a)> f(b)

is meaningfulforalla, 6 G A.

Proof Part (a) follows since by Theorem 1, f(a) = f(b) is mean-ingful for alla,b G A. Toprove part (b),suppose that /(a) > f(b) ismeaningful for all a,b € A. Thenfor anyhomomorphismg :(A,R) — >

(E,S) and anya,b G A,

Let (pbe an admissible transformation of / andletg = <p o /. Thenby(3), for any a,b G A,

It follows that (/? is order-preservingon f(A). Conversely, suppose </?is any order-preserving map from f(A) into E Then clearly ipo / isagaina homomorphism from (A,i?) into the GOR (E,S). Hence, <p isadmissible. We conclude that / has ordinal scale type. Next, supposethat /hasordinal scale typeandconsideranyhomomorphismg =p°f.Then (4) holds and hence (3) follows so f(a) > f(b) is meaningful. ■

4. THEMEANINGFULNESSOF ORDINAL COMPARISONSFORGORs

In the previous section, we showed that measurementinto a GORisof the ordinal scale type if and only if f(a) > f(b) is meaningful forall a,b G A. Ourmain result in this section,Lemma 2, shows exactlywhen meaningfulness fails, and thereby enables us to state conditionsthat insure measurementinto aGORbe of the ordinal scale type.

There is one specialcase which is not covered by the generalchar-acterization that is to appearinLemma 2. We deal with the specialcasefirst. Let ussay thata homomorphism / :(A,R) —

► (E, S) is singularif

The following theoremexhausts thetheoryofsingularhomomorphisms.Note thatpart (c) generalizesTheorem4of Roberts (1984).

(3) f(a) > f{b)~ g{a) > g(b).

(4) f(a)>f(b)~<pof(a)>tpof(b).

(ai,...,am) eR-> f{a\) =... = f(am).

258 FRED S.ROBERTS AND ZANGWILL SAMUEL ROSENBAUM

THEOREM 2. Iff : (A,R) -+ (E, S) is a singular m-ary GORhomomorphism, then(a) Everyhomomorphismfrom (A,R) into (E,S) is singular.(b) The tie ranking c = 1-2-...-m is in11.(c) IfUcontains any ranking other than c, then

(5) \f(A)\ < min{ht(7v) :TV eUandTT^e}

(d) |g(A)| > 1forany homomorphism g :(A,R) —► (E, S).

(c) f(a) > f(b) ismeaningless for some a,b G A.

Proof (a) Let qbeany other homomorphism. If (a\,...,am) G.R, we have f(a\) =...= f(am), which implies (by Theorem 1) thatg{a\)=... =g(am).

(b) Sincei? 7^ 0, there are ai,...,am GA.such that (ai,...,am) G#. Then(/(ai ),..., /(am)) G S. Since /(ai) =... = /(am), we areforced toconclude that c Gn.

(c) Suppose that 7r has minimum heightofallrankings inIIdifferentfrom c. If the inequality (5) is false, then \f(A)\ > ht(7r) > 2and sobyLemma 1 thereareai,...,am G A such that(f(a\),...,f(am)) G T^.But then (ai,...,am) Gi?even thoughit isnot true that f(a\) =...=f(am). This contradicts the fact that /is singular.

(d) If \g{A)\ =1, then, by part (b), R = Am,which is contrary toassumption.

(c) By part (d), we know that \f(A)\ > 1. Pick c,d G A such that/(c) > f{d). We now define a mapping g :A — » Eas follows: for alla G A, let

Note that <j(c) < g(d). Thus,it suffices to show thatg is a homomor-phism. Suppose that (a\,...,am) G i?. Then /(ai) = ...= f(am).By the definition of g, g{a\) = . .. = g(am). Hence, by part (b),(g{ai),...,g(am)) G5. Conversely,suppose that(#(ai),...,g(am)) GS. Suppose one of the /(a*) is /(c). Define 6j, j= 1,.. .,m, so that/(M=gfaj)- Thus,(/(6i),...,/(6m)) G 5andhence/(61) =...=/(6m). Then for all j,g{a3) = f(bj) = f{bz ) =g(ai) = f(d). Thus,for all j,/^)= /(c). Itfollows bypart (b) that(/(a1),..., /(am)) G

r /(c) if f(a) = /(d)g(a) = \ /(d) if /(a) = /(c)

( /(a) otherwise

THEMEANINGFULNESS OF ORDINAL COMPARISONS 259

S and hence that (ai,...,am) GR. The proof is similar if one of thef{ai) is /(d). Ifnone of the /(a*) is /(c)or /(d), then (/(a^) = /(%")for all j.Thus,(/(ai),...,/(am)) = (a(ai),... ,#(am)) G S and so(0'i,... ,am) G .R. ■

If we stay away from singular homomorphisms, we have the followingresult.

LEMMA 2. Supposef :(A,R) — > (E, 5) isanonsingularm-ary GORhomomorphism. Then thefollowing are equivalent:

(a) /(a) > /(&) w meaninglessfor some a, 6 G A.(b) 77z£re w a homomorphism g : (A,i?) — > (E, S) and there are

a\,...,am G A such that

for some ni, 7Tj G n, 7Tj 7^ ttj.

Proof. Fromthe definition ofsingularity, we know that theremust be6i,...,6m GAsuch that (6b...,6m) G i?and |{/(&i),...,/(&m)}| >1. Since /is a homomorphism, (f(b\),...,f{b'm)) G S and so(/(&i),..., /(6m)) € 2>. for some Gn.Note that

and inparticular \f{A)\ > ht(7Tj). Now assume (a), and inparticularassume that there is another homomorphism g : (A,R) —

► (E, S) sothat for some a,b G A,

fails. By Theorem 1, we know that f(a) / f{b). ByLemma 1, thereare a\,.. . ,am G A with as =a,at = b for some s, t G {1,...,m}and (/(ai),...,/(am)) € TT.. Hence, (/(ai),... ,/(am)) G 5 and(ai,...,am) G i?. Since gis also a homomorphism, this implies that(g(ai),..g(am)) G S and, inparticular, that (g(ai),..g{am)) GTtt for some -kj G n. Moreover, since (7) fails, 7r; 7^ ttj. Thisestablishes part (b).

Next,weassume (b). Since 7^ 7Tj, theremustbe r,s G {1,...,m}which are ranked differently by 7r; and ttj. Hence,(7) is violated with

(6) (/(ai),...,/(am))eTT. and (g(ai)t...,g(am)) G TV,

ht(7Tz ) = |{/(61),...,/(6m)}|>l,

(7) f(a) > fib)~ g(a) > gib)

260 FRED S. ROBERTS AND ZANGWILL SAMUELROSENBAUM

a=ar and b =as.Part (a) follows.

Iffor somepairofm-ary GORhomomorphisms /andgfrom (A,R)to (E, S) and ai,...,am in A, (6) holds, we will refer to 7r7; and -kj asco-rankings. Although the condition in part (b) is difficult to verifydirectly, it can be used in the following way: if it can be shown thatit is impossible to have a itiand -kj which are co-rankings, then weare guaranteed that f(a) > f(b) is meaningful for all a,b G A. As asimple example, consider the following, which is essentially the sameas Theorem3 ofRoberts (1984).

THEOREM3. Suppose f : (A,R) —► (E,S) is anonsingularm-ary

GORhomomorphismand \U\ =1. Thenf(a) > f(b) ismeaningfulforalla,b G A.

Proof. Since n contains only a single ranking, it cannot containco-rankings. ■

The next theoremuses the same approachand subsumes Theorems 4and 7 of Harvey and Roberts (1989). Suppose ~ is an equivalencerelation on {1,...,m}. We shall call this the signature of a ranking7r ifi~ jexactly when iand jare tied in tt. Thus, for example, therankings iti= 3-4,5, 1-2and -kj = 1-2,?>-A, 5 have the same signature,while therankings 7T; =3,2-1and -Kj =3,2,1do not.

LEMMA 3. Ifthereis anm-ary GORhomomorphismfrom (A,R) into(E, S), then co-rankings have the samesignature.

Proof Let 7r; and ttj be co-rankings. Since / andg are homomor-phisms,Theorem1tellsus that forallr,s G {!,».. ,m},/(ar) = f(as)if and only ifg(ar) =g(as). Hence,7^ and7Tj have the same signature.

THEOREM4. Suppose f : (A,R) — * (E,S) is anonsingularm-aryGORhomomorphism andno twodistinct iri and ttj inHhave the samesignature. Thenf(a) > f(b) is meaningfulforalla,b G A.

Proof ByLemma3,therearenoco-rankings. Hence,theconclusionfollows by Lemma 2. ■


Lemma 3 can actually be used to yieldan even strongerresult. Webegin by partitioningn according to signature. Letnbe partitionedinto disjoint, nonemptysets Hi,...,n„ where all 7Tj, itj G Ilk have thesame signature and whenever tt, and -Kj have the same signature, theyare in the same n^.For alliG {1,...,n}, let

Then there are three different signatures here,apartition ofnis

lii= {7Ti,7r4}, n2 = {7r2,7r3}, n3 = {7r5,7r6,7r7},and

The partition of n together with the homomorphism / induces acorrespondingpartition {R{,. ..,R^} ofR, where

(Note that R{ could be empty even if all the n^ are nonempty). By(12), /isa homomorphism from (A,RJ) into (E, Si ) for alli.We shallshow in thenext theorem that R( is the same for allhomomorphisms /from (A,R) into (E,S). Thenext theoremalso showshow thequestionof themeaningfulnessof /(a) > f(b) canbe decomposedaccording tosignature. It says, essentially, that f(a) > f(b) is meaningful for alla,b G A in the representation (A,R) — > (E, S) if it is meaningful forall a,b G A in some one of therepresentations (A,Rj) —> (E,Sj ).

THEOREM 5. Suppose f : (A,i?) -> (E,5) w on m-aryGORhomo-morphism. Then

Sz=U{2V : 7T G Ui}.

For example,suppose that(8) S — Ti,2-4,3-5 U25,2,1-3-4 U22,5,1-3-4 UT2_4,3_5,i

U 22_5,i,3,4 U T2_5,i,4,3 U22-5,3,4,1= T^UT„2 UT^U2V4 U2V5 U2V6 U2V7.

(9) S\ = 21,2-4,3-5 U22-4,3-5,1 >

(10) 52 =25,2,1-3-4U22,5,1-3-4 >

(11) 53 = 22_5,1,3,4 UT2-5,1,4,3 UT2_5,3,4,l .

(12) (0i,...,dm) €R{ ~ (/(ai),."" ,f(am)) G SZ.5Z.

262 FREDS. ROBERTS AND ZANGWILLSAMUEL ROSENBAUM

(a) Ifg : {A,R) — * (E,5) is another homomorphism, then g is alsoahomomorphism from (A,RJ) into (E, Si )for aliiG {1,...,n}and, moreover,R^ — Rf.

(b) Suppose there is j G {1,...,n} so that whenever h\, h2 arehomomorphisms from {A,R?) into (E, Sj), thenforalla,b G A,

Proof, (a) Suppose (0i,...,am) G R{,some i. Then (f(a\), ...,/(flm)) G sj,so(/(ai),...,/(am)) G2V for some tt G n,and(/(ai),"" ",f(a>m)) G5. It follows that (a\, ...,am) G -R, so {g{a\), ...,g{am)) G 5 and therefore, for some j, (g{a\),. ..,g{am)) G2V forsome 7r' Gnj and (<?(ai),... ,g(am)) G <Sj. But then, by Lemma 3,7r and 7r' must have the same signature. Thus,by the definition of thepartition,imustequaljand we conclude that (g(a\),...,g{am)) G Si.Conversely, suppose that (g(a\),...,g(am)) G S». Byreversing thedirection of the previous argument, one shows that (a\,... ,am) G

Rj for some j, and in particular that (#(ai),...,g(am)) G T^/ and(/(ai),...,/(am)) G Ttt for 7r G n^- and tt' G n^. Again, by usingLemma3,one shows that i— j.Hence,(ai,...,am) G i?f.Thus,wehave shown that g is a homomorphism from (A,R() into (E, Si). Bythis conclusion and by definition ofRf,wehave

(b) Let jbeasin the theorem, andletgbe anyhomomorphism from(A,R) into (E,5). By (12) and part (a), both / and g are homomor-phisms from (A,Rj) into (E,Sj ). Thus,by the hypothesisof part (b),(13) holds for all a,b G A with h\ = f and h2 =#. This implies that/(a)> f(b) is meaningful for all a,b c A. ■

Leti?7 denote the common R{ — Rg- of Theorem5.j j j

(13) hx(a) > hx {b) ~ h2(a) >h2(b).

Then f{a) > f(b) is meaningfulforalla,b G A.

£(ai,...,am) GR\ <-► (tf(ai),...,tf(am)) GSi<-► (ai,...,am) G i?f,

soR{ =Rf.


COROLLARY 5.1. Suppose f : (A,R) -> (E,5) is an m-ary GORhomomorphism. Then f(a) > f(b) is meaningfulfor all a,b E Aiffor some jandfor some homomorphism g : (A,Rj) — > (E,Sj),g(a) > gib) is meaningfulforall a,b G A.

It should be noted that the converse of Corollary 5.1 is false. LetA = {x,y,z}, R = {(x,x,y Jz),(x,y,y,z),(y,y,x,z),ix,z,z,y)}and 5=UJ=lTni where tti = 1-2, 3, 4, tt2

=3,1-2,4, tt3 = 1, 2-3,4,and 7T4 =1, 4, 2-3. ThenSi = 2V, U2V2

,52=2V3 U2V4

,f(x) = 100,f(y)=50,f(z)=0is ahomomorphism from (A,R) into (E,5),R\ ={(x,x,y,z),(y,y,x,z)}, and R2 = {(x,y,y,z),(x,z,z,y)}. Nowany otherhomomorphism g from (A,R) into (E, 5) has g{x) >g(y) >g(z), so the statement f(a) > f{b) is meaningful for alla, 6 G A Ifh(x)= 50, h(y) = 100, /i(z) =0 and k(x) = 100,k(y) =0, fc(z) =50,then / and hare homomorphisms from (A,R\ ) into (E,Si ) and / andk are homomorphisms from (A,R2 ) into (E,S2). Since f(x) > f(y)while h(x) < h{y), the statementg(x) > g(y) is meaningless for allhomomorphisms g from (A,R\) into (E,S\). Similarly, since f(y) >f(z) and k(y) < k(z), the statementg(y) > g(z) is meaningless for allhomomorphisms g from (A,R2) into (E, S2).

Next, wenote that Theorem4 is an immediate consequenceofThe-orems 3 and 5 because,if distinct -Xi and ttj in ncannothave the samesignature,then each consists of asingle ranking.

More importantly,Theorem5 anditsCorollary suggestthe followingapproach to themeaningfulnessquestion. When faced with anarbitrary(E,S), a good first step is topartition nby signature, and carry outthe analysis on each of the (E, Si) separately. If meaningfulness off{a) > f(b) for all a,6 G A can be demonstrated for any one ofthese for homomorphisms from {A,Ri) into (E,Si), then it is alsodemonstrated for theoriginal GOR. What is especially attractive aboutthis approach is that each Siuses only rankings of the same signature,and can essentially be treated as an SGOR. For instance, the GOR Siof (9) can be handledby studying

which is arrived at by merging the second and fourth positions and thethird and fifth positions in the original Sj. Similarly, the GOR S2 of(10) can be handledby studying

(14) S?=rwUT2>3,i,

(15) S2* =T3,2,iUT2,3,i

264 FREDS. ROBERTS AND ZANGWILL SAMUEL ROSENBAUM

and the GOR53 of (11) by studying

We discuss this idea in detail in the next section.

5. REDUCTION TO SGORs

Suppose ~ is an equivalence relation on {1,...,m} and there areq equivalence classes. Consider a set A. Given ai,...,am GA,define (ai,...,am)* by dropping all components but the first com-ponent from each equivalence class of subscripts. For instance, ifm = 9 and ~ has equivalence classes 1-4, 2-3-6, 5, 7-9, and 8,then (ai,.. .,ag)* is (ai,a2,a5,a-],aB). Given a\,...,aq GA, define(ai,...,aq)+ byinsertingcopies of theelements a\,...,aq in theprop-er places to obtain an ra-tuple with signature ~. For example, with~ as above, (b\,b2,hMM)+ is {hMMMMMMMM).Thenotations (ai,...,am)* and (ai,...,aq)+ are ambiguous since theydependupon the equivalencerelation ~. However, in what follows,~will always be clear.

If (C,S) is any m-ary relation,define S* on Cby

For example, if S3 is definedby (11) and ~is the common signatureofthe71-j's defining S3, then S3 of (17) is givenby (16). Ingeneral,as inthisexample,if(E, S) isaGORinwhich allitjhave the samesignature~, then (E,S* ) is an SGOR.(The exception to this statement is whenq= 1, i.e., thecommon signature ~of all the ttj declares all elementsofMtied. Inthis case,(E, S* ) is a1-ary relation,and wearenotcalling1-aryrelations SGORs or even GORs.) If(D,Q) is anyg-ary relation,let

(dv...,dq)eQ}.

Note that ifai =ajwheneveri~ j,and inparticular if [a\,...,am) G2V for 7r of signature ~,then

1Q"m)-

(16) S3* = 22,1,3,4U22,1,4,3 U22,3,4,1 .

(17) 5* = {(ci,...,cm)* :(ci,...,cm)GS}.

q+= {(<*!,...,<y+

(18) (ai,...,am)*+ = (ai


Thus,if(C,S) is anyra-ary GORinwhichall -Xihavethe same signature~, then S*+ =S. Also,we always have

THEOREM6. Suppose S is a GOR in which each -Xi has the samesignature ~ with q > 1 equivalence classes and f is an m-ary GORhomomorphism from (A,R) into (E, S). Then ifß* andS* aredefinedfrom~, thefollowing hold:(a) Any homomorphism from (A, 22) into (E, S) is also ahomomor-

phismfrom (A,R*) into (E, S*).(b) Any homomorphismfrom (A,R*) into (E, S* ) walso ahomomor-

phismfrom (A,22) into (E, S).(c) /(a) > f(b) is meaningfulfor alla,b G A in the representation

(A, 22)— > (E,S) ifandonly iff(a) > f{b) is meaningfulforall

a,b € Am the representation(A,R*) —► (E, S* ).

Proof, (a) Suppose that g is a homomorphism from (A, 2?) into(E, S). Let ai, ...,a 9 be elements of A. Suppose that (g{a\), ...,g{aq )) G S*. Then (g(ai), ..., g(aq))+ G S since S*+ =S. Sinceg is a homomorphism from (A,2?) into (E,S), we conclude that (ai,...,aq)+ G22 and hence (ai, ...,aq) G 22* by (19). Conversely,suppose that (a\,...,aq) G R*. We first show that (0i,...,aq)+ GR.Since (a\,...,aq) G R*, there must be (&i,...,bm) GR so that (&i,..., &m)* = (ai, ...,aq). Then (/(&i),..., f(bm)) G S since /isa homomorphism from (A,R) into (E,S), and so we must have that/(&i) = /(&j) whenever t ** jbecause Sisa GORin whicheach 7Ti hassignature ~. Hence, it must also be true that (/(ai),...,f(aq))+ G Sand then (ai,...,aq)+ G i?,as desired. Now,(g(a\),...,g(aq))+ G Sbecauseg is ahomomorphism from (A, i?) into (E, S). Hence, (#(ai),...,g(aq))+*

G S*. By (19), (^(ai), . ..,g(aq)) G S*. Hence, wehaveshown that g is a homomorphism from (A, 22*) into (E, S* ).

(b) Suppose thatg is a homomorphism from (A, jR*) into (E, S*).Suppose that (<xi, ...,am) G -R. Thenby definition of 22*, (ai,...,am)* G 22*, and sincegis ahomomorphism from (A, 22*) into(E, s* ),(0(ai),...,0(am))* G S*. Then(£(ai),...,£(a m))*+ G S*+

-S,

the equality by our earlier observation. To prove that ig(a\), ...,g(a>m)) GS, it suffices to show that (g(ai), ..., m))*+ = (g(a\),■ ■ "

i g(cbm))- This follows by (18) if we can show thatg{ai) = #(%")wheneveri~j.Note that since /isa homomorphism from (A, 22)into

(19) (au ...,ag)+* = (ai,...,a9).

266 FRED S. ROBERTS AND ZANGWILL SAMUEL ROSENBAUM

(E, S) and since {a\, ...,am) G 22, we have (/(ai),...,/(am)) G Sand so f{ai) — f{af) wheneveri~ j. By part (a), / is a homomor-phism from (A, 22*) into (E,S*). Applying TheoremIto / and g, weconclude that g(ai) = g{aj) whenever i~ /, as needed. (To applyTheorem 1, we need to show that R* / 0 and 22* Aq. The formerfollows since 22 / 0. Tosee the latter,note that since q > 1, there is aniso that 1~idoesnot hold. Pickias small aspossible. Then {a\,...,am) G22 implies a\ ai and so (ai,ai,...,a\) is not in 22*.)

Tocomplete theproofofpart (b),suppose that {g{a\),...,g(am)) GS. Then (g(cn),...,g(am))* GS* and so(a\,...,am)* G 22* since gis a homomorphism from (A,22*) into (E, S* ). Since {a\,...,am )* G22*, there is {bu...,6m) G22such that (&i,...,bm)* =(ai,...,am)*.Note that (/(&i),...,f(bm)) G S since / is a homomorphism from(A, 22) into (E,S). Wenext show that

To see why,note that since (bi,...,bm)* = (ai,...,am)*, we have

because (/(6i),..., /(6m)) G Sand

because (^(ai),... ,g(am)) GSby hypothesis. We can now applyTheorem1 to g and/. (By part (a) and thehypothesis ofpart (b), botharehomomorphisms from (A,22*) into (E,S*); andas above, 22* / 0and 22* / Aq.) By Theorem1, wecan conclude from (23) that

Now (21), (22), and (24) yield (20), as desired. Since (/(&i), ...,f(bm)) GS, we have (/(ai), "" ", f(o>m)) GS. Since /isa homo-morphism from (A, 22) into (E, S), weconclude that (a\,...,am) G 22,as required. We conclude that # is a homomorphism from (A, 22) into(E,S).

(c) This follows trivially from parts (a) and (b).

(20) (/(&i),...,/(6m)) = (/(ai),...,/(am)).

(21) (/(61);...,/(6m)r= (/(0,),...,/(om))*.Also,

(22) f(bi) = f(bj) whenever i~ j

(23) g(ca) =g(o>j) whenever i~j

(24) f(di) = f(aj) whenever i~ j.

THE MEANINGFULNESS OFORDINAL COMPARISONS 267

Let us apply Theorem 6 to the example (8). From (9) and (14),we note that we can reduce the question of the meaningfulness off{a) > f(b) for a homomorphism / from (A,22i) into (E, Si) =(E,Ti,2_4,3-5 U 22_4,3-5,i ) to the same question for / treated as ahomomorphism from (A,22*) into (E,S*) = (E,Ti,2,3 UTyj). Asimilar conclusion holds for S2 and S3.

6. THEMEANINGFULNESS OFORDINAL COMPARISONSFORHOMOMORPHISMSINTO SGORs

In this section we examine the meaningfulnessof the statementf{a) >f{b) in the specialcase where Sis an SGOR, i.e.,consists of only strictrankings.

A strict ranking isnothingmore than a permutation of the elementsof M. We can think of such a ranking as a mapping which assigns toeachpreference level aparticular position in the ra-tuple.For example,the strict ranking -k = 3, 1, 4, 2 can be thought of as the permutation7T :M -* Mfor which tt(1) =3,vr(2) = 1, tt(3) =4, 7r(4) = 2. Theinterrelationshipsbetween therankings will alsoplayacriticalrole. Foreach, i,j G {1,...,p}, wherep = \U\, we define a^ :M —

► Mby<*ij ° = 7Tj, i.e.,

For example,if 7Ti = 2, 4, 1, 3 and 7Tj =4, 3,1,2, then

andGij is givenby

Note that, in general,it is always true thata^ is the inverse of Oji andthat on is the identity map. For what is forthcoming, we will find ituseful to define Sj to be the following setof permutations:

Inparticular, notethat each £i contains an, the identitymap. Note alsothat

Oij(k) =7ij(7r. \k)).

7Ti(l)=2, 7T,(2)=4, 7Ti(3) =l, 7TZ(4)=3,

=4, 7^(2)=3, =!, =2,

o-ij(l)=1, aij(2) =4, <Tij(3) =2, o-y(4) = 3

Si = {(Tij :j= !,...,£>}.

268 FREDS. ROBERTS AND ZANGWILL SAMUELROSENBAUM

Tosee why (25) holds,let (yu...,ym)= {xa-i{l),..., (m)). Wewish to show the following: for allk,

(26) vis kthin 7Ti and xv is fcthhighest among x\,...,xm

holds ifand only if(27) vis kth in 7Tj and ?/w is A;th highestamong yi,...,ym

holds. Note that o~j = Oji. Suppose that (26) holds. Thus, 7Ti(A;) =v. Suppose v = 7Tj(k). Then cr~jl(v) = crji('u) = Oji(7Tj(k)) =7Ti(7r~ (7Tj(A;))) = 7Ti(A;) = v, yv is xv and is kth highest among2/b " "" > 2/mI (27) follows. Conversely,suppose (27) and suppose v =TTi(k). Then,againas above,cr~jliv) =v,andsoyv is xv and xv is kthhighest among x\,...,xm\ (26) follows. Hence,wehave (25).

Itisstraightforward to seethat ifaisapermutationof{1,...,m]andfor some (x v ...,xm) G T^,wehave(vi(i)v,V(m)i € 2V,-,then cr mustbe cty.

The next few results demonstrate the importance of the&ij and theSj in theanalysisof SGORs.

LEMMA 4.Iff:(A,R) —»" (E,S) isanm-ary SGORhomomorphism,

7Ti w m 11/or S, a«J (f{a\),...,f(am)) G T^, ?/ze«

Proof. If a G £j, then there isjG {1,...,p} such that 7Tj GIIforS and cr =c^-. But then,by (25),

Hence, (/(ao-i(i)), "" " ,/(^-'(m))) an^ since / is a homomor-phism, (aa-i(1),...,aa-ifm)) G 22.

Conversely,ifcr 0 Ei, thenfor alljG {1,...,p},by theobservationrightafter the verification of (25), wemusthave

(25) (xi,...,xm) 62V, <-> (^-"(i),.-.,^-!^))cTnj

a G Si <-► (aa-i(!),...,aCT-i(m)) G 22.

(/(aa-i(l))'--''/(a^-i(m)))

(/(a<r-'(l))> " " > /(V'(m))) 0 2Vr

269THEMEANINGFULNESS OFORDINAL COMPARISONS

The nexttwo results echoLemma 3 andTheorem 4.

LEMMA 5. IfS is an SGOR and 7Ti and ttj are co-rankings, thenZji = Zjj.

Proof. Let / and g be two m-ary SGOR homomorphisms from(A, 22)into (E,S) satisfying(6). Thenby twoapplications ofLemma4,we have

THEOREM 7. Suppose f : (A, 22) -> (E, S) is an m-ary SGORhomomorphism. Ifthere is an iso that Si Sj for all j i, thenf(a) > f(b) is meaningfulforalla,b G A.

Proo/ Letg be any other homomorphism from (A,22) into (E, S)and a, 6 be in A. By Theorem 1, we need only consider the situationwhere f(a) f(b) and g(a) g(b). Since 22 0, there is (&i, ...,bm) G 22. Hence, (/(&i), ..., f(bm)) G 2V fc for some k. Since 7rfcis a strict ranking, we conclude that |/(A)| > m ='ht(7rjt) = ht(7Ti).Since m> 1throughout this paper,Lemma 1 implies that there are a\,...,am in Aso that as =a,at = b and (f(a\), ..., f(am)) G TXi.Hence,(ai,...,am) G 22, so ig{ax ),...,g(am)) G S and (g(a\),...,g(a<m)) G 2V. for some j.Since Si Sj forz/ j,Lemma5 impliesthati — j. Thus,

as desired.

As an application of this result,consider thespecial caseof Theorem 6of Harvey andRoberts (1989) where S = 2V,U2V2 with tti = 1, 2, 3and 7T2 = 2, 3,1. ThenSj = {crn,cr^}, where a\\ is the identity mapand (Ti2 is givenby

and S2 — {(72i, 022},where 022 is the identity map and 021 is givenby

Hence, if(aa-l{l)), ..., /(aa-i(m))) G" S and (c^-i^,...,aa-wm) )

0 22. ■

oG Sz <-► (aa-i(1),...,(V-i(m)) G22 <-+ aG Sj

fia) > fib) «-> g{a) >gib),

<712(1)=2, (712(2) =3, (712(3)=1,

0-21(1) =3, (72l(2) =l, (72i(3)=2.

270 FRED S. ROBERTS AND ZANGWILL SAMUELROSENBAUM

Since Si S2,weconclude thatif /isanm-ary GORhomomorphismfrom(A, 22) into(E,S), then /(a)> /(&)ismeaningful forall a,b G A.

As a second application of Theorem 7, suppose that S is givenby(8) and / is an m-ary GORhomomorphism from (A, 22) into (E,S).If Si is given by (9) and ~ is the common signature for rankings forSi, then S* is givenby(14). By definition,/ isa homomorphism from(A,22i) = (A,22{) into (E, Sj"). It is an m-ary GORhomomorphism.To see why,note that since it is an m-ary GORhomomorphism from(A, 22) into (E,S), 22 Am, and so R{ C R Am. If 22i =0,then by (9), |/(A)| < 3. Hence, by (8), R = 0. Since all m inHi have q > 1 equivalence classes, Theorem 6 implies that / is ahomomorphism from (A, 22|) into (E, Sjf ). By (14) and the discussionin the previous paragraph, /(a) > fib) is meaningful for alla, b G Afor the representation (A,R*) — * (E, S* ) and thus,byTheorem 6, alsofor the representation(A, 22i ) — » (E, Si ). Note by way of contrast thatif S2 is givenby (10), then S| is givenby (15). For SJ, Si = S2 soTheorem7does notapply. Infact,byTheorem5 ofHarveyandRoberts(1989),forallm-ary GORhomomorphismsgfrom iA,R%)into(E,SJ),theconclusion gia) > gib) is meaningless for somea,b G A solong as\giA)\ > 2 and |#(A)| < 00. However, we still have meaningfulnessfor m-ary GOR homomorphisms from (A,22) into (E, S) for S of (8)sinceCorollary 5.1tells us that we need to find only one Si for whichwehave meaningfulnessand Si has thisproperty.

The binary relation SJ of Equation (15) gives us a counterexampleto the converse of Theorem 7. As noted above, Sj = S2. We caneasily find a set A and binary relation 22 on A such that 22 0, 22 7^Am,and there is a homomorphism / from (A, 22) into (E, SJ) with/(A) unbounded. (Thus, in particular, |/(A)| = 00 and Theorem 5of Harvey and Roberts (1989) does not apply.) Now given a, b in Awith /(a) > fib), there is a c in A such that /(c) > /(a) > fib).Hence, (/(&),/(a), /(c)) G r3)2)i g SJ, so (&,a,c) G 22. But then,for anyhomomorphismg from(A, 22) into (E, SJ), (#(£>),#(&), #(c)) GSJ =23,2,1U22,3,1, which implies that gia) > gib). We conclude thatfia) > f(b) is meaningful for alla, b in A.

7. CONCLUDINGREMARKS

In this paper, we have studied the meaningfulness of the ordinal com-parisonfia) > fib), for alla,b c A, for m-ary GORhomomorphisms


/ : (A, 22) —♥ (E,S). We have given necessary and sufficient condi-tions for meaningfulness inmany situations,though in afew situationswehave just given sufficient conditions.

Tosummarize the situation,Theorem2settlesall the situations where/ is a singular homomorphism. Theorem 4 settles all the situationswhere/ is nonsingular and all the 7Ti havedifferent signatures.

InTheorem 5 (Corollary 5.1), we consider the case of / with somesignatures allowed to be the same and partitionninto sets of rankingsof the same signature and,correspondingly, let Sj be the union of TlTifor the rankings 7Ti of the jth signature. We study meaningfulnessof ordinal comparisons for homomorphisms into (E,Sj) and note thatmeaningfulness in one of these cases implies meaningfulness for thewhole case. However, the converse of this statement is false, andexactly how to handle the situation when meaningfulness fails in all ofthehomomorphisms into (E, Sj ) is still anopenquestion. Thequestionof determiningmeaningfulness in eachsuchcase (E, Sj ), i.e.,where allthe rankings inS= Sjhave the same signature, is handledby reducingto SGORs.

Theorem 6 gives a reduction to the SGOR situation. Theorem 7gives a sufficient condition for meaningfulness if the homomorphismis into an SGOR. However, the converse of this Theorem is false. Weare left with theproblem ofhandling meaningfulnessif there isnoiforwhich Si Sj for allj i.

As we havenoted,the m-ary GORscorrespond to m-point homoge-neousm-ary relations. The theory of the meaningfulnessof the ordinalcomparisons /(a) > fib) remains to be systematically developed forhomomorphisms into other kinds of relations or relational systems.Some specialcases of interest with which to start wouldbe homomor-phisms into m-point homogeneousn-ary relations where m n;andhomomorphisms into relational systems (E,5i,...,S&) where each Siis an rrii-ary GOR.

DEDICATION

This paper is dedicated to Patrick Suppes, on the occasion of his 70thbirthday and hisretirement from Stanford University.

272 FREDS. ROBERTS AND ZANGWILL SAMUELROSENBAUM

ACKNOWLEDGEMENTS

FredRoberts acknowledges the support of the National ScienceFoun-dation under grant number IRI-89-02125 to RutgersUniversity. Bothauthors thank DeniseSakai,ShaojiXv,and Wenan Zangfor their help-ful comments.

Fred S. Roberts,DepartmentofMathematics andCenter forOperationsResearch,Rutgers University,NewBrunswick,NJ08903, U.S.A.ZangwillSamuelRosenbaum,DepartmentofMathematics and Computer Science,Wilkes University,Wilkes-Barre, PA 18766, U.S.A.

REFERENCES

Falmagne,J.-C. and Narens, L.: 1983, 'Scales and Meaningfulness of QuantitativeLaws',Synthese, 55, 287-325.

Harvey,L.H. and Roberts,F. S.: 1989, 'On theTheoryof Meaningfulness of OrdinalComparisons inMeasurement ll',AnnalsN.Y. Acad ofSci., 555, 220-229.

Krantz, D. H.,Luce,R.D., Suppes, P, andTversky, A.: 1971, FoundationsofMea-surement, Vol. I,New York: AcademicPress.

Luce,R.D.,Krantz,D. H., Suppes,P., andTversky, A.: 1990, FoundationsofMea-surement, Vol. 111,New York: AcademicPress.

Luce,R. D. and Narens,L.: 1986, 'Measurement: The Theory ofNumerical Assign-ments', Psychol.Bull.,99, 166-180.

Luce,R. D. and Narens,L.: 1987, 'MeasurementScales on the Continuum', Science236, 1527-1532.

Roberts,FS.: 1979, MeasurementTheory, withApplications toDecisionmaking, Utility,and theSocialSciences,Reading,MA: Addison-Wesley.

Roberts, F. S.: 1984, 'On the Theory of Meaningfulness of Ordinal Comparisons inMeasurement',Measurement,2, 35-38.

Roberts,FS.: 1985, 'ApplicationsoftheTheoryofMeaningfulness toPsychology',J.Math. Psychol., 29,311-332.

Roberts,F S.: 1989, 'MeaninglessStatements, MatchingExperiments,and ColoredDigraphs(Applicationsof GraphTheoryandCombinatoricsto theTheory ofMea-surement), in: F S. Roberts (Ed.), Applications of Combinatorics and GraphTheory in the BiologicalandSocial Sciences, IMA Volumes inMathematics andItsApplications,Vol. 17,New York: Springer-Verlag,pp. 277-294.

273THEMEANINGFULNESS OFORDINAL COMPARISONS

Roberts,F S.: 1994, 'Limitationson Conclusions Using Scales ofMeasurement', in:A. Barnett, S.M.Pollock, andM.H. Rothkopf (Eds.), Operations ResearchandPublicSystems, Amsterdam:Elsevier,pp.621-671, inpress.

Roberts,F.S. andFranke,C.H.: 1976, 'On theTheoryofUniquenessinMeasurement',J.Math. Psychol., 14, 211-218.

Scott,D. andSuppes,P.: 1958, 'FoundationalAspects ofTheoriesofMeasurement',J.SymbolicLogic,23, 113-128.

Suppes,P.: 1959, 'Measurement, EmpiricalMeaningfulness and Three-Valued Log-ic', in: C. W. Churchman and P. Ratoosh (Eds.), Measurement:Definitions andTheories,New York:Wiley, pp. 129-143.

Suppes,P. andZinnes, J.: 1963, 'Basic Measurement Theory', in: R.D.Luce, R. R.Bush,and E.Galanter(Eds.),HandbookofMathematicalPsychology, Vol.1,NewYork: Wiley, pp. 1-76.

COMMENTS BYPATRICK SUPPES

The study of themeaningfulness of astatementwhenvarious measure-mentsarereferred toin thestatementhasreceivedconsiderable attentionin various articles in these three volumes. Several of the outstandingproblems formulated by Luceand Narens, for example, are concernedwith meaningfulness. The article by Roberts andRosenbaumis a con-tinuationofaseriesofpapersbyFredandhiscolleaguesthatprovideoneof the most thorough studies ofmeaningfulness to be found anywhere,namely,of the meaningfulnessofordinalcomparisons asrepresentedbyanumerical function. This means thatthe papersareconcerned with themeaningfulness of the ordinal numerical comparison /(a) > fib). Inthe early days of meaningfulness theory,it wasassumed that the mean-ingfulness ofordinalcomparisons wasastraightforward andcompletelyelementary, indeed,almost trivial,exercise. Some thirty years later itis clear thatcomplexities liebeneath the surface evenin what appear tobe thesimplest casesof meaningfulness.

Theanalysis ofRobertsandRosenbaumon thesemattersis thoroughindeed. Ionly regret that they did not amplify their detailed discussionwithadditional details ofactual scientific examples wherecontroversiesaboutmeaningfulnesshavearisen,whichiscertainly thecase forordinalcomparisons subject to statistical analysis.

Itwouldalsobeinteresting to seehow their results work for theaffineorder relation of betweenness which they mention briefly at onepoint,and also for the projective order relation of separation which has itsrepresentationtheorem formulated ordinarily in terms of the projective

274 FRED S. ROBERTS AND ZANGWILL SAMUEL ROSENBAUM

conceptof the crossratio of thenumerical representationoffour points.The familiar condition being that distinctpoints a, b, c, d stand in theseparationrelation ab Scd if andonly if the cross ratioof a,b, c,dwithrespect to a numerical representation/ is negative. (The cross ratio isdefinedby

f(a) ~ /(c) fjb)- fjd)fia) -fid) fib) -fie)

where we have to extend thenumerical operations on the real numbersto include infinity -thedetails are given in Section13.2 ofFoundationsofMeasurement,Vol. II.)

Still another direction of interest mentioned in my comments onLuce and Narens, is the introduction of topological ideas where therequirement is made that the order relation be continuous, the kind ofassumptionusedcritically in someof Debreu's (1954) well-known workonordinalutility. Havinginmind application to theutility ofcommoditybundles in aspace ofann-dimensionalEuclidean space,Debreuprovesa theoremofamore generalcharacter, i.e.,for a topologicalspace that isconnected,separable and for which there isa completeorder relation ofpreference. Anassumptionof continuity is oftenused inessential waysineconomics. Itwould beuseful toenlarge the studyof meaningfulnessto include preservationof continuity.

Theselast remarks are in thespiritof remarksIhavemade to severalof the measurementpapers, to the effect that we needmore extensionsof concepts from the theoryof measurementto the rich array of exam-ples and applications of the same general ideas that may be found ingeometry.

REFERENCES

Debreu,G.: 1954, 'Representationof aPreferenceOrderingby aNumericalFunction',in: Thrall,Coombs, andDavis (Eds.), DecisionProcesses,New York: John Wiley&Sons.

Suppes,P., Krantz,D.,Luce,D.,andTversky,A.: 1989,FoundationsofMeasurement,Volume11, SanDiego: AcademicPress.

C.ULISES MOULINES AND JOSEA. DIEZ

THEORIES AS NETS:THE CASE OF COMBINATORIAL

MEASUREMENTTHEORY1

ABSTRACT. The firstpart ofthis essay deals with thegeneralproblem of identifyingempiricaltheories;an identitycriterionfor theoriesasnets ofmodelclassesisproposedwhichmaybeseenas anexpansionof Suppesianideas onthis issue. In thesecondpart,the criterionproposedis applied to the particularcaseof combinatorialmeasurementtheory. This theory is represented as a tree-like net of eighteen so-called model-elements. Some of them are only implicitly (or not at all) containedin the literatureon measurement, especially inFoundationsofMeasurement. The corresponding set-theoretical predicates are precisely defined for each case and some new results areproven. A particular innovationof thepresentreconstructionof measurement theory isthesystematic treatmentofso-callednullandlimitelements.

Philosophy of science owes Pat Suppes a number of far-reaching in-sights. One of them concerns the identity of empirical theories,anotherthe foundations of measurement. The two are interrelated (though notin an immediately obvious way) and havebeen veryinfluential on sub-sequent work. Suppes' views on the identity of theories have led toa family of approaches that have come tobe known as 'the semanticview of theories'; more specifically, it has inspired a program in theformal reconstruction of empirical theories usually called 'the struc-turalist view. Now, there may be some quarrel over the question ofwhether structuralism belongs to the semantic approach or not. Inourown view, this is a rather uninteresting quarrel about words. Thereare certainly some significant differences between the ideas about thenatureof empirical theories propoundedby,say, Bas vanFraassen andFrederick Suppe on the one hand, and those of Joe Sneed and Wolf-gang Stegmuller on the other. But, in our opinion, the differences areoverarched by some convergent views on the identity of a theory at avery fundamental level-one might say: at thelevel of 'ontology'. Thissharedmetatheoryofempiricaltheories maybestbe summarizedby thefollowing tenets:

275

P. Humphreys (cd.),Patrick Suppes: Scientific Philosopher, Vol. 2, 275-299.© 1994 Kluwer Academic Publishers. Printedin the Netherlands.

276 C.ULISESMOULINES AND JOSEA. DIEZ

(1) An empirical theoryis not alinguistic entity;more concretely,it isnot just aset of axiomatic sentencesoraconjunctionof them. Itisrather an abstract, mathematical sort of entity.

(2) The most basic component of a theory's identity is a class ofstructures, andmorespecifically aclassofmodels inTarski'ssense.

(3) Typically, thisclassofmodels isnotasingleton,notevenaclassofisomorphic structures. Categoricity is never to be found in devel-oped empirical theories,nor is it a desideratum. The desideratumfor empirical theories is rather to have a genuinelyheterogenousclass of empirical applications.

(4) Themostconvenient way to pick outtheclassofmodels essentiallycharacterizing a theory'sidentity is by means of a set-theoreticalpredicate,i.e. by defining a 'second-order' predicate in terms ofnaive set theory.

Now this set of tenets, shared by allphilosophers of science who callthemselves semanticists,or structuralists,or,perhaps, something else,is the common gift we allhave gotten from Pat Suppes. In a more orless explicit way we found them expoundedin his writings on generalphilosophyof scienceand,particularly, in the book he has (alas!) neverpublished,Set-TheoreticalStructures in Science (Suppes,1970, Chs.1and2).

In his characteristic way, Suppes has not gone on to philosophizemuch around his general view on theories but has preferred instead toimplementhisapproach in anumber ofreconstructions ofconcretecas-es,rangingfromrelativity theoryandclassical mechanics to psychologyandeconomics. Andhere we find the connection with the second greatgift we have received from him in philosophy of science, namely ahuge program in the construction and reconstruction of measurementtheories inmany different disciplines, but especially in the social sci-ences. This programhas culminated inthe three volumes of the Summametrica he has set up with Krantz,Luce,and Tversky (Krantz et al,1971-1990, in the following abbreviated as FM for 'Foundations ofMeasurement'). Put in a nutshell, the connection between the twostrandsof Suppes'philosophyof science consists in explicating differ-ent kinds of measurementas different classes of qualitative structuresas models characterized by set-theoretical predicates. The axiomaticconditions defining thesepredicateshave to be satisfiedby observation-al structures which consistof certain qualitative objects, relations,andoperations. Thisproposal to explicatemeasurementis whatSuppesand

MEASUREMENT THEORY AS A THEORY NET 277

his collaborators call the 'axiomatic-representational approach'; theycharacterize its essentials as follows:The mostpervasive abstractionin measurement theory consists in formalizingbasicobservations as arelational structure, that is, a set with someprimitive relationsandoperations. This abstractionarises fromconsideringthe natureof empirical,qualitativeoperations(FM, Vol. 3, p.201).

As in physical theories, we should not expectcategoricity for the classof models determining the identity of ameasurement theory:[The] theoryof measurement avoids categoricalsystems of axioms. The reason... isapparent: diverse empirical applications are intended, and it would be surprising ifdifferent applications led to isomorphic models of the theory. The same attitude isstrikinglyevidentinphysics...Unless the object of physics is to create theone truemodelof the entireuniverse, categoricity of aphysical theory is a defect, not a virtue(FM, Vol. 3,pp.247-248).

Consequently, according to the Suppesian view of science, there isno essential difference between the discipline of measurementand thediscipline of physics, or any other empirical science for that matter.Bothconsist of theories in the sense ofabstract entities thatmaybestbeidentified through a class of models characterized by a set-theoreticalpredicate; and since each theory has a plurality of genuinely differentintended applications, categoricity is not to be supposed to be amongthe characteristic features of such theories.

In the rest of this essay, we shall dwell upon these two strands ofSuppes'philosophy ofscience: whatexactly is the identitycriterion foran empirical theory and what consequencesdoes it have for therecon-struction of the discipline of measurement? We will do this from thestandpoint of the structuralist program which was startedby SneedandStegmuller in the early 'seventies. As pointed out above,structuralismhas at least some of its roots in the Suppesian approach. This does notmean that there is complete convergenceof views about the natureoftheoriesbetween Suppes and the structuralists. There are some signif-icant divergencesover the semantics of empirical theories. They werealreadydiscussedinanother Festschriftessaysomeyearsago(Moulinesand Sneed, 1979,as well as Suppes,1979). Itis not the purposeof thisessay togoback to this discussion. We shallrather concentrateon thoseaspectsof a theory's identity whichstructuralists think areessential andwhich,in our view,are already implicit in Suppes' approach, so that hemay agreeonthem. Moreconcretely,we try to enlargeSuppes' conceptof an empirical theory in a way which, we think,makes the structural

278 C. ULISES MOULINES ANDJOSE A. DIEZ

pictureof atheorymoreadequate. And weillustrate thisenrichednotionin thecase of a particular measurementtheory.

Suppes apparently identifies a given theory with a class of (model-theoretical) structures. This class is defined by a set-theoretical pred-icate. And the conditions or 'axioms' which, in turn, define this set-theoretical predicate allhave the same logicaland epistemological sta-tus.

Wenow wish tomodify thispicture in the following way. A theoryisnot tobe identified with aclass ofstructures but rather with ahierarchi-cally organizedarray of classes of structures. Each class of structureswe call a 'model-element' and the whole array we call a 'theory-net.That is,agiventheory(usually) is ahierarchicalnetofmodel-elements.

The reason to undertake thismodification is mainly this: allaxiomsof a given theory are equal, but some are more equal than others.That is, they are allaxiomatic in the sense of beingneither definitionsnor theorems, but they have a different status - they play differentmethodological andepistemological roles. We distinguish at least twocategories:

(a) basic axioms;(b) special laws.

Basicaxioms are those thatshould be satisfiedin anyconceivable appli-cation of the theory. They are common to all cases. This categoryshould, in turn, be subdivided into two subcategories: what we callcharacterizations and fundamental laws. Thefirstare purelyconceptualor mathematical innatureand determine the theory's conceptual frame.The second are empirical laws of very generalscope. But we need notgointo the details of this subdivision in the present context.Suffice it toknow thatboth characterizations and fundamental lawsare common toall applications of the theory, however heterogeneous they might be -but that, for that veryreason,theyaloneusuallyarescarcely informative.Togetmore concrete informationabout thesystems which aresupposedtobemodelsof the theory,youhaveto require,in addition,speciallaws.These special laws are not common toall cases of application of thetheory but areputup according to thekind of application we envisage.The more specialized they are, the more empirical information theyprovide,but their application in the theory presupposes the validity ofthe basic axioms (i.e. of the characterizations and fundamental laws).Ifwe callMq[T] theclass of structures satisfying the basic axioms,this


means that theclass of structures Mi[T] satisfyingany of these speciallaws fulfils the condition: Mi[T] C Mo[T]. The specializations maygo in different directions,i.e. they may be branched out. Therefore,atheory's identity is not just givenby the class of structures determinedby thebasic axiomsbutby anopenarray ofclasses andsubclasses. Thepicture of the structure of a theory T we get accordingly is that of aramified tree with a basic common element - for example, somethingof the sort shown in Figure 1 -where Mj C Mfj for all i, j,and thesuperindex indicates thelevel of specialization, so that, for anyi> 1,and for any j,there is a ft such that iMj C Ml

k~l).

Now the assertion thatany theory can be brought into this form is atheoretical claim about theoretical science,or if you wish,a metatheo-retical claim. Such a claim, as in anyother theoretical endeavour,hasto be not only made as precise as possible but also checked against theempirical data. Now, the data here is somewhat different from whatwe usually think of when we speak about the empirical data of a the-ory. It does not consist of natural items like particles,gases, genesormental states;itrather consists ofcultural products,namely the theoriesthemselves that deal with particles, gases, and so on. These prod-ucts are typically found in standard scientific literature like textbooks,encyclopedias,and so forth. Leaving this ontological difference aside,

o o oo o o o

o o o o

Fie8.1.

C. ULISESMOULINES AND JOSE A.DIEZ280

however, the methodological situation is completely analogous to theone weknow from the methodologyofcheckingany theorybymeansofdata. This means that wehave to findout whether the theory's intendedinstances really fit the theoretical model. The particular procedurebymeans of which the philosopher of science checkshis/her general the-ory of science against the data is the logical reconstruction of concretecases of scientific theories. This procedure only works if we are readyto acceptacertaindegreeof abstractionandidealization withrespect tothe standardpresentations of theoriesby workingscientists. But this isno specific problem of the methodology of philosophy of science: anytheoretization of data - be it in physics or metaphysics - only worksto some degreeif we accept abstractions and idealizations of the 'rawmaterial. Up tonow,the generalmetatheoretical claim to the effect thatany empirical theory may be represented as a tree-like theory-net hasbeen checked in about thirty case studies from different disciplines -fromphysics throughbiology andpsychology toeconomics. Thedegreeof abstraction and idealization that had to be assumed to consider theclaimpositively confirmed inall thesecases has varied but it hasneverbeen intolerablyhigh-at least,it has notbeensignificantlyhigher thanthedegreeof abstraction andidealization we encounter when checkingthebest theories of the social sciences. Inthis sense,itmaybe said thattheclaim that theories are tree-like theory-netshasbeenconfirmed for awide spectrumof scientific theories withoutmuch 'forcing' of the 'rawmaterial.

On the other hand, the claim in questionhas been confronted withanother sort of methodologicalproblem, which seems tobe specific tothe metatheory of science, or at least does not appear to be so acutein other theoretical sciences. This problem has to do with the extremevarianceofidentity criteria for theobjects ofstudyin ourcase. Whereasin usual empirical science there are intuitive criteria of identity forthe objects investigated which are reasonably precise and commonlyaccepted,it is justa fact that the presystematic intuitions about what anempirical theory has got to look like are extraordinarily vague. This,of course, poses a problem when trying to check our central claim.For example, if you think that the thing called 'classical mechanics'is just one theory (as a superficial glance at physics textbooks maysuggest), then you will find that thereis no reasonable way to representthe structure of this thing as a tree-like theory-net. So, the claim wouldappear to be falsified in an important case. However, the situation


changesradically if you do not consider that 'classical mechanics' isthe name of one theory, but rather an amorphous denomination for abunch of different theories like Newtonian particle mechanics, rigidbody mechanics,classical hydrodynamics, anda few more. Then, youcanfindquiteniceandnaturalrepresentationsofthese 'smaller'items astree-like theory-nets. A similar situation emerges whenanalyzing thatthing called 'phenomenological thermodynamics' and in many othercases. The things representable as tree-like theory-nets are just therebut it may not be a completely trivial task to detect and differentiatethem.

Therepresentationof given theories as tree-like theory-netshaspri-marily a metatheoretical value: it allows us to identify each theoryprecisely, to analyze its formal structure in detail, and also to investi-gate the natureof possible intertheoretical relationships of allkinds asformal links between two or more graphs of this sort. But in the caseof young,still not well-established disciplines, this representationmay,in addition to the epistemologicalrole,also playamethodological one.We mean by this that it may help the working scientist herself -andnot only the philosopherof science -in providing her with some cluesaboutpossible developments of her theory she might nothave thoughtof before. Since oneof thebasic ideas of the theory-netrepresentationconsists in viewinga theoryasasortof 'pyramid' withdifferentprecise-ly defined levels of generality, it follows that if we notice somelacunaein the hierarchy of levels of existingtheory-elements, we may wonderwhether we might not fill theselacunae with new elements havingsomepossible use in actual research. In other words, the completion of atheory-net in a natural way may have not only an aesthetic but alsoaquite practical value.

We now come to the promised illustration of the foregoing ideasby means of an example from the discipline of measurement. Thepurpose of this reconstruction is not only to provide an illustration ofour generalmetatheory,but also to make acase for twomore particularclaims suggestedabove. First, in the caseof 'measurement theory',asin so many other cases, we havea misnomer: there is no such thing asthe one and only theory of measurement, but rather there are several(though not many) different theories of measurement(if we accept theidentity criterion proposed above). We reconstruct the historically andmethodologically most conspicuous one: the theory of combinatorialmeasurement. The second thing to note is that, by representing this

282 C.ULISES MOULINES ANDJOSE A. DIEZ

theory asa tree-like theory-net, we can say exactlyat whatplace in thepyramid the usual expositions of this theory seem to have overlookedsome potentialities that might be empirically relevant. These lacunaemay be filled in a rather easy and natural way and the correspondingrepresentation theorems maybe obtained.

To abbreviate, let us call the theory of combinatorial measurementTCM'.We claim that TCMis a theory-net with abasicmodel-elementand seventeenspecializations so far detected. Some of them appear tohave beenoverlookedby the authors ofFM.

TCMneeds three basic concepts: a domain of objects, an orderingrelation, and an operation of combination. That is, themodels of thistheory, whichmake up the conceptualframework for all its specializa-tions,are structuresof the sort (D, £, o). Thisis oneof the reasons whyTCMis a theory different from other measurement theories (accordingtoour identity criterion): itsmodels have adifferent structurebyhavingacomponent (thecombination operation) the othermodels do nothave.Theother reason(which is related to the first one) is that this theory hasabasic axiom (a weakmonotonicity condition) whichis notcommon toall theories of measurementbut iscommon toallkinds of combinatorialmeasurement.

TCM's basic class of models, Mq [TCM], is givenby the followingbasic axioms.2

DEFINITION1. x GMO°[TCM] iff there are D, £, o such that:(i)Z>^o(2) £C D x D isstrongly connected and transitive(3) o is a function from D x Dinto D

Condition (3) of Definition 1is more restrictive than is actuallyneededfor a thoroughreconstruction of TCM.Thepresent formulation requiresthat the operation o be closed, i.e. that combinations be defined on allpairs of objects. However, we know that, in some cases, combinationsmight be empirically impossible (for example, because the resultingobject would be just too 'big', or would explode, or whatever). Wecould weaken Definition l-(3)so that the domain of obe onlya subsetofDxD,andthe restof thereconstruction would retainasimilar form.However, the formulation of definitions and theorems would become

(4) (a) (a ~b —► a oc ~ b o c)

(b) (a~ 6—

► coa ~ cob)3

MEASUREMENTTHEORY AS A THEORY NET 283

more cumbersome. For lack of space, we choose thesimplified (morerestrictive) formulation. Froma metatheoretical pointof view,nothingessentialis thereby lost.

For the sake of abbreviation, let uscall anymodel of TCMa 'com-binatorial structure' and write 'C(x)' insteadof x G Mq[TCM]; that is,'Cis the fundamental set-theoretical predicate ofour theory.

FromDefinition 1an obvious corollary follows:4

THEOREM1. a~b&c~d— >aoc~bod&coa~dob.Proof. From a ~ b and Definition l-(4b) we get a o c ~ bo c.

From c ~ dand Definition l-(4b) we getb oc ~ b o d. Therefore,byDefinition l-(2), ao c ~ b o d. Analogously for c oa~ d ob.

The first level of specializations of the basic element makes a fun-damental distinction. Itdistinguishes between the way the orderingofobjects behaves after combination. This differentiation is fundamentalin the sense thatall otherinterestingproperties ofspecifickindsof com-binatorial measurement, and in particular the form the representationtheorem adopts, will dependonit. When combining two objects aandb, it maybe that theresulting object iseither 'greater'or 'lesser' than aand b but that, in any case, it preserves the same ordering with respectto both; or else,on the contrary, the resulting compound ao b may besometimes greater thanabut lesser than b, or conversely.

For the first possible situation let us choose the label 'externalcom-binatorial metric' iEC).The second situation we callacase of 'internalcombinatorial metric' (IC).We shalldefine themjnamoment,butbeforethat, let us make a remark ona further propertyexternalcombinatorialmetrics mayhave. Ininternal measurement,combination is necessarilyidempotent. Notso in external measurement.But,here, there mightbesomespecialobjects for whichaparticular kind ofequivalenceis valid.A sort of object / - let us call them 'limits' -might appear, for whichit is the case that, for any other object aof the domain, a oI= I;whencombining velocities, the speedof light might come to our minds asan example of this situation. On the other hand, there might also be akind ofobjects n -call them 'null'- for which it is true that aon = afor any a; when combining chemical substancesand considering theirtherapeutic effects, a placebo might be a case in point here. The pos-sibility of havingboth limits and nulls is responsible for some formal

284 C.ULISESMOULINES ANDJOSE A. DIEZ

complications in the reconstruction of the theory. But since there aregoodempirical as well as formal reasons not toexclude these possibili-ties apriori, we willhave to consider them in our definition of externalcombinatorial metrics. In internal measurement, they have no role toplay. Letus introduce first the correspondingauxiliary definitions:5

(a) Nia) iffao6~6oa~6(b) L(a) iffao&~6oa~a

Our limits are reminiscent of the notion of an 'essential maximalelement' as introducedinFM (Vol. 1,Ch.3, Sec.7),though theydo notfully coincide.6 Atany rate,nulls andlimitshaveanumber ofpropertiesweintuitively expectfrom them. First,nulls areequivalentonly tonullswhile limits are equivalent only to limits, and all nulls are mutuallyequivalent while the samegoes for limits.

THEOREM2. (1) N(a) & a ~ b -> N(b)(2) L(a)& a~ b -> L(b)(3) Nia) &Nib) -* a ~ b(4) L(a) &Lib) -» a ~ bProof. (1) Assume Nia) and a ~ b. Then,by Definition l-(4), for

any c,ao c ~ b o c. Since Nia),ao c ~ c. Thus,for any c, c ~ 6 oc.Therefore, N(b). The proof of (2) is entirely analogous. (3) FromNia) it follows that a ob ~b and from Af(6) it follows that a ob ~ a.Therefore a ~b. Similarly for (4).

Secondly, nulls and limits have the mutual relationship we shouldexpect: if there are at least twonon-equivalent objects in the domain,then nonull may be equivalent toa limit.

Proof. Takesome a,b with a *> b. Suppose there were ac withN(c)and Lie). Then,byiV(c), aoc~a&6oc~6;and,byL(c),aoc ~ cand 6 oc ~ c. Therefore, a~aoc~6oc~6,i.e. a~ b (!).

DEFINITION2. Let x = (D, £, o) with C(x).

THEOREM3. 3a,b(a *b) -> ->(N(c) &L(c))

MEASUREMENTTHEORY ASA THEORY NET 285

For the rest of thisessay, we shall just assume that the premiseof The-orem 2 holds. Further,null objects allow for a stronger monotonicitycondition with respect to o:

Proof. AssumingNia), we get 6~ao6^aoc~c,i.e. byDefinition l-(2), 6 >3 c. Similarly for the rest of the theorem.

Wenow introduce the first specializationofTCM which is,in turn, thebasic model-element of one of the two main 'branches' of thecombi-natorial 'tree.

EXTERNALCOMBINATORIAL METRICS

DEFINITION 3. EC{x) iff there are D,fc, o such that x = {D, fc,o)and Cix) and(1) a o b £ a,b V a, 6 £ a ob(2) a o 6 ~ a -* £(a) V JV(6)

In external combinatorial measurement, nulls and limits have the ex-tremalproperties weintuitively expect from them: when combination isalwaysincreasing, i.e., when the combination of two objects is alwaysgreater thanboth of them,nulls are minimal andlimits maximal; whencombination isalwaysdecreasing,nulls aremaximalandlimitsminimal.

Wehave said that thecharacteristic feature of this kind of measure-ment is that the result of the combination of a and b is either greaterthan a andb or lesser than both. But this does not imply that, for allpairs (a,b), itwill alwaysbe greateror always.lesser thanboth. Itcouldbe thecase that it is sometimes greaterand sometimes lesser than botha and b. If the latter is the case, we speak of a cyclic metric. If theformer is the case, i.e. if ao b is greater than a and b for all (a,b) orless than aand b for all (a,b), we speak of a linear metric. Moreover,all linear metrics share a strict monotonicity condition. Linear metricsmay, in turn, be subdivided into thepositive case (where the result ofcombination is always greater than the components) and the negativecase (where the result is always lesser than thecomponents). The mostconspicuoussubcaseof thepositive linear metric is in turnthe extensive

THEOREM4. N(a) -* (ao& £ aoc *-* b y c)& (fcoa y coa <-► b y c).

286 C. ULISESMOULINESANDJOSE A. DIEZ

one, which historically was the first one to be studied and which hasfound very wide application inphysics.

LINEARCOMBINATORIALMETRICS

DEFINITION4. LECix) iff there are D,y,o such that x = {D,fc, o)and ECix) and(1) Va,b(a ob y a,b) V Va, 6(a,b fc a ob)(2) (a) iayb*-+aoC yboc)

(b) ->L(c) a ycob)

For thenextspecialization (positive metrics), besides positivity,a natu-ralcondition shouldbe imposed: the well-known conditionof solvabil-ity.

POSITIVELINEARCOMBINATORIAL METRICS

DEFINITION5:PLECix) iff thereare D,£, o such thatx = (D, £, o)andLECix) and(1) aob>za,b(2)ayb^ 3c(-^N{c) & a y b oc)

The most important specialization on this branch of the tree are theextensive metrics, i.e. combinatorial metrics with additive represen-tation. They are those metrics everybody thinks of when consideringpositive linear combinatorial measurement, though they are possiblynot the only specializationof this kind relevant for empirical science.Inorder to get this important specializationwith the correspondingrep-resentation theorem, weneed to add therequirements ofcommutativity,associativity,and archimedianity to the conditions already set forth.

EXTENSIVEPOSITIVELINEARCOMBINATORIAL METRICS

DEFINITION6. EPLECix) iff there are £>, £, ° such that x = {D, fc,o) andPLECix) and

MEASUREMENTTHEORY AS A THEORYNET 287

(1) a ob ~ b o a1(2) (a ob) o c ~ a o (6 oc)(3) Any sequence (a;);eN of combinable objects for which it is true

that: ->N(ai) & Va^i >2-*a{ = a<_i oaj) & 36(-iL(6) &Vai(6 >- Oi)), is finite.

Theseextensive metrics are essentially those representedinFM(Vol. 1,Ch. 3, Definition 3), but enriched with nulls and limits; the followingrepresentation theoremcan be proved:

(1) There is f :D —► [0, oo] such that

(2) Anyfunction f satisfying (1) is such thatforany a € D:

(3) //"/ arcd /'satisfy (1) then there is an r £ R+,such that,for any

Proof The proof is rather easy, though somewhat lengthy. We canonly sketch it here.

Toprove Theorem5-(l), the essentialideais to show that,if we takeallnulls and limits out ofD, then the resulting structure (£)', y' o') isan extensive metric in the usual sense and the representation theoremcan be provedfor itin the way it is shown,for example,inFM (Vol. 1).That is, this 'reduced' structurehasarepresentationfunction /* with therequired properties. If we then construct / = /* U {(a,0) :a G D &N(a)} U {(a,oo) :a € D& £(a)}, it is easily seen that / is afunctionsatisfying theorderingand the additivity conditions.

Toprove Theorem 5-(2), first assume /(a) =0; for any b e D, wethenhave /(a)+fib) = fib), i.e. /(ao6) = fib); therefore aob ~b,i.e. Nia); thus: /(a) = 0 -+ Nia). Now, assume /(a) = 00,for any b, we then have /(a)+ f(b) = fia), i.e. /(a o 6) = /(a);therefore ao b ~ a, i.e. L(a); thus: /(a) = oo — > L(a). On theother hand, assuming ./V(a) take any b with ->L(6) (such a b exists byTheorem3);wehave justseen thatthis implies/(6) oo; JV (a),in turn,

THEOREM5.Ifx = (D, £ o) withEPLECix) then:

(a) a y b <- fia) > fib)(b)/(ao6)= /(a)+ /(6)

(a) Nia) «- fia) =0(b) L(a) «-> /(a) =oo

aeD: fia) =r " /(a).

288 C. ULISESMOULINES ANDJOSE A.DIEZ

implies ao b ~ b and thus /(a ob) = fib); i.e. /(a)+ /(&) = f{b);since /(o) 7^ 00, it must be /(a) = 0; thus N(a) —

► /(a) = 0.AssumingLia), take any 6 with -»JV(6); we have seen that this implies/(&) 7^ 0; Lia), in turn, implies ao b ~a, i.e. /(a 06) = /(a),i.e. /(a) + /(&) = /(a); since f(b) =^ 0, it must be /(a) = 00; thus:Lia) -+ /(a) =0.

Finally, Theorem 5-(3) follows easily from the fact that,if / and/'are two functions satisfying Theorem 5-(l), then f'(a) = r " /(a) foranyr € M,ifa is anulloralimit (by Theorem5-(2)); and for the restofobjects whichare neither nulls nor limits,justapply the usualargumentas given inFM (Vol. 1).

NEGATIVEMETRICS

Thoughextensivepositive measurementwith itscorrespondingadditiverepresentation is the mostoutstanding case of combinatorial measure-ment, it isbut oneof themainbranches ofTCM's tree. Yougetanotherbranch - which seems to have been ignored in the literature -if youconsider negativemetrics within the linear case. They correspond tothoseproperties whichdecrease-insteadofincreasing-after combina-tion. Aphysicalexample wouldbe theparallelcombination ofelectricalresistances, where the total resistance of the system decreases as moreresistors arecombined.

As may be gathered from the proof of the representation theorembelow, negative metrics may be seen, from a purely formal point ofview, just as trivial 'rewriting' of the positive ones. However, what isnot so trivialis the fact that we really get the conversecaseof the posi-tivemetricsby justassumingthat combinationmakes thecorrespondingproperty decrease. Also,it is interesting tonote that (as the exampleofelectrical resistance shows) one and the same qualitative orderingmay'behave' in different ways depending on the kind of combination weuse. It would notbe natural to take one ordering for one combinationand another for the other one. Negative metrics are therefore concep-tually necessary toprovide a complete account of different empiricalconfigurations.

Ina waycompletely analogous to thepositivecase, wefirst introducenegative combinatorial measurementiNLEC) anditsextensive version{ENLEC) as a specialization.

289MEASUREMENT THEORY AS A THEORY NET

DEFINITION7. NLEC{x) iff thereare D,fc,o such that x = (D, fc,o)andLECix) and(1) a,by aob{2)ayb-+ 3c{^Nic)& a o c £ 6)

DEFINITION8. ENLECix) ifthereare D,y,o suchthat a;= (D, £, o)andNLEC{x) and(1) a o6 ~ 6 oa(2) (a o 6) oc ~ a o (6 oc)(3) Any sequence (a2)ieN of combinable objects for which it is true

that: &Vai{i > 2 -» a{- a;_i o ai) & 36(^L(6) &

Maiiai y b)),is finite.

Now,extensivity in thecaseofnegativemetrics doesnotentailadditivityin the numerical representation on the positive real numbers. Therepresentationtheorem we get in this case is rather different. To stateit,let us first introduce anumerical operation,call it '*', such that, foranyr,s G [0, oo]:

The representationtheorem thenruns as follows

THEOREM6.Ifx = {D,y,o) withENLEC{x), then:(1) There isf :D — > [0, oo] so //*atf

(2) Anyfunction fsatisfying (1) is such thatforanya G D:

(3) Same as Theorem5-(3).

(a) r *s = -£&, ifr,s G E+(b) r* 5 = r,if s =oo(c) r* s = s,ifr =oo(d) r * 5 =0,ifr=0 or 5 =0

(a) a y b «-» f{a) > f{b){b) f{aob) = f{a)*f{b)

(a) Nia) <-> f{a) = oo(b) Lia) <- f{a) =0

290 C.ULISESMOULINES ANDJOSE A.DIEZ

Proof(sketch). The first thing to notice is therather trivial fact that,if (D,y,o) isanegativemetric,then (D, o) (where is theconverseof y)is apositive one. This entails,in turn, that (by Theorem 5) therewillbea homomorphism from (£>, ;<,°) into ([O, oo],> +). Then, takethe isomorphism

as definedby the condition:

Itis easily seen that the functional composition / oFis itself ahomo-morphism from (D, o) into ([O, oo],<, *), which,in turn, entails thatitis also ahomomorphism from (D,y,o) into ([O, oo],>,*).

The composition / oFis thus ahomomorphism from {D,£,o) into([O, oo], <,*), whichis equivalent to saying that itis itselfa homomor-phism from (D,y,o) into ([O, oo],>,*).

CYCLICMETRICS

A third important branch of TCMcontains the measurement of cyclicmagnitudes {CEQ.Aportionof thisbranchhas already beenstudied inFM (Vol. 1,Ch. 3), though within a different setting. Limits make nosensehere but there maybe null objects. Monotonicity takes a specialform.

DEFINITION9. CEC{x) if there are D, y,o such that x = (D,y,o)andEC{x) and

Thereare twokinds ofcyclic metrics: inferior and superior ones. Theformer {ECEC) are characterizedby the fact thatnull objects are thosethat correspond to no cycle at all (minimal objects), whereas, in thelatter(SCEC), thenull objectscorrespondtoacompletecycle (maximalobjects). Correspondingly, we shall have two different set-theoreticalpredicates:

F:([o,oo],>,+)^<[o,oo],<,*)

Fir) =1/r.

(1) 3a, 6(a ob ya,6) & 3a, 6(a,b y aob)(2) a^^c^aocV&oc^c)

MEASUREMENTTHEORY ASA THEORY NET 291

DEFINITION10. ICECix) iff thereareD,y,osuch that z = (£>,£ o)andCECix) and

DEFINITION11.SCECix)iff thereareD,y,o suchthat x= (Z>, £, o)and CEC{x) and

a^&âocHoc^cVcâocHocV&oc^câoc.

Ininferior cyclic metrics,nulls areminimal, while they are maximal insuperior cyclic metrics.

Ifweadd therequirementsofcommutativity, associativity,andarchi-medianity to theconditions ofDefinition 10, respectively Definition 11,we get the extensive version of inferior cyclic metrics, respectivelysuperior cyclic metrics.

DEFINITION12. EICECix) iff thereare D,y,o such that x= (D,y,o) and ICECix) and(1) a ob ~ b o a(2) (a o 6) o c ~ ao (6 oc)(3) If for any integern,na = a when n — \,and na = in — \)a o a

DEFINITION13. ESCEC{x) iff there are D,>%o such that x= (D,y,

o)andSCECix) and(1) Same as Definition 11-(l).(2) Same as Definition 11-(2).(3) Under the same assumption as in Definition 11-(3), a y b — >

3n(a y na &nb y b)

Both kindsofcyclic metrics allow fora 'natural'representation theorem.The range of the representation function is an interval [0,r) forEICECand an interval (0,r] for ESCEC- in the first case it assigns 0, in thesecond r, to the null objects. There is a sort of additivity herebut it is'modulo r\ Toexpress it formally, weneed someauxiliary definitions:let rGR+ andh,k £R +

U {o}. Then

o^6+-+aoc^6octcVc)-aocHocV6oc^c>-aoc.r\j r\j r^*j rsj rsj

when n> 1, then: a y b —+ 3n(a y na& nb y b).

292 C. ULISESMOULINESAND JOSE A. DIEZ

(a) nr{h) is the greatestinteger such thatn "r < h(b) h+ ~ k =df (h+k)— r " nr{h+k) (where the index li* stands

for 'inferior').(c) h+fk=h+^k,ifh+ (whereV stands for 'superior');

otherwiseit is =r.

THEOREM7. Ifx = (D,y,o) withEICECix), then,forany rG K+,f/zere is <zunique f :D -^> [o,r) so that:

Proof. Theproofof Theorem7-(l)and (2) is completely analogousto the one to be found in FM (Vol. 1, Ch. 3, Theorem 2), except forthe fact that here, for any a, fia) r. (This is actually the markof inferior cyclic metrics.) Indeed,suppose it were /(a) = r. Then,for any b, fia ob) = fib) + j f{a) = f{b)+ j/(&) (by definition of'+;:'). This entails N(a). Sincenulls areminimalhere,b y a,thereforefib) > /(a)- Since, by construction of /, r > fib), the assumptionr = /(a) would imply /(a) > /(&), thus /(a) = f{b) for any 6, i.e.a ~ 6 for any 6, which violates the general assumptionof Theorem 3.As for Theorem7-(3), the — >-part follows from the fact that, for any b,a o b ~ b, thus fia) + l- fib) = /(&),but by definition of '+£', thisis true only if f{a) = 0or it is amultiple of r,but since /(a) G [0,r),it must be /(a) = 0. Now, take any a with /(a) = 0. This entailsf(b) + j f\a) = fib) for any b. Therefore f(b oa) = /(&) for any 6,i.e. a o 6 ~ 6 for any 6, thus N{a).

THEOREM8. Ifx = (£>,£, o) vWtf* ESCECix), thenforany r G M+,//zerg is auniquef :D — ■> (0,r] so //za/:

Proof (sketch). First, it has to be shown that ESCEC{{D,y,o))iffEICECi(D,3,o)). Assume ESCECi(D, £, o». It is well knownthat< is alsoa weak order, o is the same operation,andcommutativityandassociativity are still satisfied. As for monotonicity, a b iff by a

(X)a£b~f(a)>f(b)(2)/(ao6) = /(a)+ i/(6).(3) Nia) <- f{a) =0

(1) atb «-> fia) > fib)(2)/(ao6)= /(fl) + J/(6)(3) Nia) «-» fia) = r

293MEASUREMENT THEORY AS A THEORY NET

iff {bocyaocyc or cybocyaocor aocycyboc) ifficy aoc^boc or aoc^boc^cor boc-<c^,aoc). As forarchimedianity, assume a-< b, which means b y a. By Definition13-(3),3n{b ynb &na y a), which means: 3n(a -< na &nb ■< b),which is Definition 12-(3). By a completely analogous argumentwegetESCECi(D,y,o» from EICEC{{D,3,o)).

Now, assume ESCEC((D,y,o)). It follows EICEC{(D,£,o)).This implies, by Theorem 7, that there is a representation function/' from D into [0,r). We now define a function / as follows:f{a) = r — fia). Clearly, / is a function from D into (0,r). Fur-thermore, a >z b iff b * aiff /'(&) > f'(a) iff r -f'(a) > r- /'(&)iff /(a) > fib), thus we get Theorem 8-(l). Using the numericalfact that, for any s, t G [0,r), r - (s + M) = (r - s) + J (r - £),we obtain: /(aob) = r - /'(a o b) = r - if{a) + J /'(&)) =(r- /'(a))+J (r-- f'(b)) = /(a) +J /(&), thus weget Theorem 8-(2). Theorem 8-(3) is immediate.

INTERNALMETRICS

TCM'slast 'bigbranch' arises from what we have called 'internalcom-binatorial metrics (/C)'. In this case, if a and b are combinable, thena^3 ao^^3^' or^^ ao^~ a- Nulls and limits make no sense inthesemetrics. Indeed,in internal metrics,if two objectsare equivalent,then the property of internality obviously entails that they will also beequivalent to their combination, (i.e. idempotence), so that equivalentobjects would always play the role of nulls and limits simultaneouslywith respect to each other. ConditionDefinition 14-(2)does not followfrom internality but is the 'converse' of idempotence.

DEFINITION 14. IC{x) iff there are D,y,o such that x = (D,y,o)and C(x) and(1) a>zaobybVbyaobya(2) ao6~a^a~6

A particular kind of internal metrics has already been studied in theliterature,although under a different label. They are those requiring,

C.ULISESMQULINES AND JOSE A. DIEZ294

in addition to internality (actually: idempotence),monotonicity andso-called 'bisymmetry' (seeDefinition 15— (2)). We may call these internalmetrics 'proportional' {PIC), since their numerical representation issuch that wegetone and the same proportionality factor for allobjects,as weshall seebelow.

DEFINITION 15. PlC{x) iff there are D,y,o such that x = (D,£, o)andIC{x) and(1) ayb<r->aocyboc

(2) (a ob) o {c od) ~ (a oc) ob o d)

Proportional internal metrics,in turn,maybe subdividedinto twocasesto allow for a 'nice' numerical representation. Inone case, the com-bination operationis necessarily closed and the domain of objects maybe infinite {CPIC); in the other case, the operation is notclosed but thedomain is assumed to Definite in anyapplication {FPIC). We may fur-ther specializeboth kinds into the case where o is commutative, whichallows for a still 'nicer' numerical representation.

DEFINITION16. CPIC{x) iff thereareD,£, osuch thatx = {D,y,o)andPIC(x) and

(2) Anysequence (a^)i€N such that, first, thereare c,dwith cy dand{ai oc ~

ai+ \ odV coai~ doa^+i),and second,thereis ab such

that b y ai for any ai, is finite.

The next theorem is therepresentationtheorem for thiskindof metrics.A similar theorem hasbeenproveninFM. But the representation func-tion there is somewhat more complicated. It can be shown, however,thatour representation is derivable from theirs (seeProof'below).

THEOREM9. Ifx = (D, £, o) with CPIC{x) then:(1) Thereis f:D — >" Eand anr G (0,1) so that

(1) (a) c ob y ay d o b —► 3e(eo b ~a)

(b)6oc>-a>-6od— » 3e(6o c ~ a)

(a) a y b <- f{a) > f{b)

MEASUREMENTTHEORY AS A THEORY NET 295

(2) Iff andf satisfy (1) then thereare s GM+ andtG X such that

Proof. InFM (Vol.1,Ch.6,pp.295-296)itisproved that,under theassumptionof Theorem 9, without internality there are functions / and/'satisfying (la) and (2a), and there are real numbers k, I,m with k,I> 0such that,for anya, 6 G D: {A) f{aob) =k- f{a)+l-f{b)+m,and so that,for any other /', k'= k and /'= /.

Now, weonly need to prove that (lb) follows from(A)plus internal-ity (i.e. idempotence). Assume (A). We know thatany objecta has anequivalent b (atleast itself). So, takeanya,b witha ~ b. Itfollows fromDefinition 14-(l)thatao& ~ a ~ b. Therefore,/ {aob) = f{a) = f{b).By(A) and f{aob) = f{a), wegetk ■ f{a)+/"/(&)+m = f{a). Butsince f{a) = f{b), we havek " f{a) + / " f{a)+m= f{a). Since thisis true for anya,itmustbe m =0. Therefore {k+l— l)-f{a)=0, i.e.k +l-\ =o,i.e. k = \-l.By (A), /(ao6) =(1-/) -/(a)+ /"/(&).This is Theorem 9-(lb).Theorem 9-(2b) is immediate.

It is easily seen that, by adding commutativity to the conditions ofCPIC, we would geta very simple representation, which we might call'bisection' {BCPIQ:

Consider now the case of proportional internal metrics over a finitedomain.

DEFINITIONI7.FPIC{x) iff thereare D,£, o such thatx = (£>,£, °)andPlC{x) andDis finite.

Since D is finite, it is moreover well ordered by the linear orderinggeneratedby yon the~-equivalence classes. This allows for anaturaldefinition of 'distance' between any two objects a, b as the numberof 'intermediate' objects separating a and b in the ordering. Clearly,therecouldbe various waysof metricizing this sort of distance,therebygetting different metrical representations. We shall not get into an

(b)/(ao6)= (l-r)-/(a)+ r-/(6)

{a) f{a) =s-f{a)+ t(b) r'=r.

fin « M /(a) + /W/(a°o) = 5 '

296 C. ULISESMOULINES AND JOSE A. DIEZ

FPIC

BFPIC

Fig.2.

explorationof these possibilities here. Suffice it to notethat, if we addcommutativity to the conditions of FPIC, we obtain a bisection again{BFPFC),as in Theorem 9-(lb). A corresponding theorem, which hasalready been proven in FM (Vol. 1,p. 297), could easily be translatedinto ourown terms here.

CONCLUSION

If we use the abbreviations of the set-theoretical predicates defined sofar to indicate each model-element of TCM,we may represent the formandcontentof this theoryas a tree-like theory-netas showninFigure 2.

C. UlisesMoulines,InstitutfurPhilosophic,Logikund Wissenschaftstheorie,Munich,Germany

JoseA.Diez,Departamentode Filosofia,UniversidaddeBarcelona,Tarragona, Spain

MEASUREMENT THEORY AS A THEORYNET 297

NOTES

1 We owetoProfessorR. DuncanLuce somehelpful remarkson apreviousversion ofthis paper. They contributed toan improvedpresentation.2 To simplifythe formalization we agree thatopen formulae are to be interpreted asuniversallyquantifiedover thedomainofobjects. Wemakeexplicituse oftheuniversalquantifier only in somecases whereconfusion might arise.3 In (4a) (and correspondinglyin (4b)) we do not require that a oc ~ b o c impliesa ~ 6.The reason is that thepossible existenceof so-called 'limitobjects' (seebelow)wouldviolate this requirement.4 Fromnowon, we implicitlyassumeineach theorem that thedomainof objects is thedomainofamodelof TCM.5 For thesake of perspicuity, we have slightlysimplifiedthe followingdefinitions. Athorough treatment ofthesenotions wouldrequire some formalcomplicationsbut thiswouldaddnothing essentialto the argument.6 When thispaper wasalready finished,ProfessorLucekindlysent usacopyofa recentresearch report of his, where he introduces the notions of 'infinite point' and 'zeropoint', which are quite similar to our 'limits' and 'nulls' (see Luce, 1990, especiallyp. 8).7 Actually,it can be provedthat commutativity follows from therest of conditions ofEPLEC, under the assumption that o is closed. However, for the sakeof perspicuityand toallowfor animmediategeneralizationto thecase whereo isnot closed,westatecommutativity explicitlyas a conditionforEPLEC.

REFERENCES

Krantz, D., Luce, R. D., Suppes, P., and Tversky, A.: 1971-1990, Foundations ofMeasurement, Vol. 1(1971), Vol. 2 (1989), Vol. 3 (1990), San Diego: AcademicPress.

Luce,R. D.: 1990, 'GeneralizedConcatenationStructuresThatAre TranslationHomo-geneous betweenSingularPoints', MathematicalBehavioralSciences, TechnicalReportSeries, Irvine: University of California.

Moulines, C. U. and Sneed, J.D.: 1979, 'Patrick Suppes' PhilosophyofPhysics', in:R.J. Bogdan(Ed.),Patrick Suppes, Dordrecht: D.Reidel,pp.59-91.

Suppes,P.: 1970,Set-TheoreticalStructures inScience, Typescript,Stanford: StanfordUniversity.

Suppes,P.: 1979, 'Replies to Moulines and Sneed', in: R. J. Bogdan (Ed.), PatrickSuppes, Dordrecht: D.Reidel,pp.207-211.


Ican hardly disagree with the general things that Moulines and Diezsay at thebeginningof their essayaboutmy approach to thephilosophy

298 C. ULISESMOULINES AND JOSE A. DIEZ

of science. Consideringother parts of the structuralist program withwhich Ulises has been associatedclosely with Stegmuller andSneed,Iwas surprised to find no discussions of a specific kind about empiricalinterpretations, butrather a quite detailed characterization of differentstructures of measurement.Ifind the latter enterprise,the one engagedin here, very much easier to understand and to appraise. Ihave likedtheattempt by the structuralist to characterize the subset of models thatare empirical butIhave not as yetbeen overwhelmed by the detailedarguments. Ontheotherhand,the kindof fine-grainedanalysisgivenbyMoulines andDiez of special axioms in the general theoryof extensivemeasurementleads to a veryrefined classification anda tree-like graphfor classesofstructures orderedbyset-theoretical inclusion. Atthemostbasic level the structures satisfy the axioms of the theory of combina-torial measurement.Characterization of this class of structures and theresulting many different specializations remind me of the Bourbakianenterprise of identifying mother-structures that are basic inmathemat-ics. In alimited way this is howIwould understand the Moulines andDiezenterprise.

On the other hand,Ido have one skeptical query for them. UnlikeBourbaki,they claim a strong empirical component for their theorizingand remark, in a vein that Imuch approve of, that the identificationof their hierarchical net of model elements, to use their terminology,is an empirical enterprise concerned with the data in just the way anyempirical theory is concerned with data. In their case the data aregeneratedby theories of measurementthat have been produced. WhatIdo not find is the thorough canvass that one might expect to backup the empirical side of the claim. Essentially everything that is saidis sensible and all the structures identified make sense,but there is noreal support for the claim that theanalysis hasbeen exhaustive or thatit corresponds in any detailed way to what is present in the very largeliterature of measurement.

Interesting examples that lie outside their tree structure are the mid-point algebras introduced by Wanda Szmielew (1983) in affine geom-etry. These midpoint algebras are closely related to the bisymmetricstructures introduced in Definition 16 byMoulines and Diez. But it isnatural in geometry to first introduce them without order, and this isthe case for Szmielew. In fact, more generally there are various rea-sons for introducing structures without order which may still lead tomeasurementresults. Iam not claiming that there are notgoodreasons


to exclude midpoint algebras without ordering in the analysis givenbyMoulines andDiez. WhatIfind missing is theempirical pursuitof themany different alternative structures lying on theedgesof the standardtopics they address. How are we to make the argumentone way or theother for exclusion or inclusion of a givenclass of somewhat aberrantstructures? From a philosophical standpoint this is the aspectof theirmethodology that is notclear to me, if weare toregard theenterprise asempirical in character.

REFERENCE

Szmielew, W.: 1983, From Affine to Euclidean Geometry: AnAxiomatic Approach,Dordrecht: Reidel.

NAMEINDEX

Abrahao, S.M. 190Adams,E.W. 129-131passim, 138, 145,

224,242Aharanov,Y. 21Albert,D.19,20,21,25,28Alper,T.M. 232,233, 235, 242Amaral, do, A.F.Furtado 189,190Archimedes 59Aristotle 55,59Arnold, V.I. 152, 161, 181, 182, 190Aspect,I.12Atiyah,M.F. 189

Chuaqui, R. 154, 189, 189, 192, 193Church, A. 119

Corrada, M.189Costa, da,N.C.A. 151-193Cuche, V 54

Daneri, A.21Davis,M. 170,190Debreu,G. 274Dell, J. 162, 190

Descartes, R.46, 51,53,60Detles, R.21Dewey 98Diez, J. 275-299Div,B. 54Doria, F.A. 151-193Dorling79Droste, M.235, 243Duhem 37Durr,D. 21

Einstein, A.11, 21, 29, 31, 36, 38, 45,77,83,93

Ellis, B.220,243EudoxusofKnidos 59Euler,L. 60,120

Falmagne,J.-C. 228, 241, 243, 252,272Feld 79

Fishburn, PC. 244Forge,J. 212Fourier43,47, 48 f.,54,56

Francia, di, G. Toraldo 189Franke,CH.241,245, 252, 256, 273Fresnel46,47,51Friedman,M.77, 83, 95

Galileo 59,60, 61Gleason, A.M.12, 19, 21Godel,K. 191Goldstein, S. 21Gridgeman,N. 113, 117Griinbaum, A. 80,145, 145

Harvey, L.H. 251, 254, 260, 270, 272Heerman,D.W. 117

301

Balzer, W. 144, 145, 212Barros,de,J. Acacio 26, 28, 189, 190Bell, J. 12, 19,20,21,32,35,38Bertrand 112Beth,E.144, 145Bethe,H. 41Binder,K. 117Blaquiere,A. 54Bogdan,R.J. 19,21Bohm,D. 11, 12,21,32Bourbaki,N. 154, 189, 298Bridgman, P. 220, 229, 242Broglie, de,L.45Burgess, J. 66, 95

Cameron, PJ. 230, 243Campbell, N.R. 220, 243Carnap,R. 125, 143, 145Catton,P. 77, 95Chiara, Dalla, M.L. 189Cho, Y. 162, 189

Cohen,M. 220,228, 231,243Cohen-Tannoudji, C. 54Comte, A.47

Earman, J.74, 94, 95, 212Eddington, A.S.79, 95Ehlers, J. 91,95Ehrlich, P. 224

Fermat60Fermi 52Feynman,R.40, 42, 43, 54, 55Field,H. 66, 95Fine, A. 29^13

Fraassen, van,B.20, 22, 115,118, 126, 144146, 213, 275

302 NAMEINDEX

212

32.

243

Heisenberg, W. 52, 56Helmholtz, yon,H. 102,219, 220, 222, 243Hempel, C.G.198, 205,212

Hirsch, M. 189Holder, O.102, 219, 222,223, 232, 243, 223Holman, E.243Horwich, P. 72,95Howson, C. 213Hughes, R.I.G.20, 21Hume, D.47Humphreys,P. 103

Inagaki, V 190Iverson, G. 243

Jones, J.P 190

Keller, E. 117Kelvin,Lord60Kepler,J. 60

Kobayashi, S. 162, 190Kochen, S. 19,21,32,38Kolmogorov,A.N. 119

Lagrange 45,60Lakatos, I.143, 145Laloe,F. 54

Lewis, A.A.151, 190Lie,S. 220, 244Lifshitz, E.M.41,43Loewer,B.3-28Longer,A. 21

Machtey,M.172, 188, 190Malebranche53Manders,K.74, 95Margenau,H.54Marley,A.A.J.241Maudlin,T. 19, 21Mausfeld,R.243Maxwell,J.C. 42, 48McClelland, J. 118McKinsey, J.C. 213Metropolis,N. 111,117Michell, J. 226, 244Minkowski 77Moivre, de, A. 102, 121Moler,N.70, 95Morgenstern, O.219, 239, 245Moulines,CU.63, 65, 96, 144, 145,

275-299Mundici,D. 189Mundy,B. 59-102Murphy, G.M.54

Nagel,E.220Narens,L.190, 219-249, 252, 272Naylor,T.H. 105, 117

Neumann, yon,J., 7, 8, 12, 19, 22, 3139,118,219,239,245

Newton, I.41,46,60, 120Niederee, R. 225,226, 227, 229, 241,

245Nomizu, K.190Norton, J. 94, 95

Pearce,D.145Pearle,P. 6, 21Peirce,C.S. 98, 131Peres, A.29, 32,35, 37,38, 39Pierce, M.212Pirani,FA.91,95Planck, M.53Podolsky,B. 11,21,38Poincare, de,H. 191Popper,K.30, 39Posidonius, Stoic 46Prosperi, G.M.21Pythagoras 49

Heyer,D.227, 229, 241,243Hilbert, D.243Hiley,D.21

Huygens, C.46,51,53,55, 60

Kahneman,D. 242, 244, 245Kallen, G. 41,43Kant,I. 27,47, 51,52Kanter,J.J. 54

Klein,F. 68,244

Krantz, D. 59,95, 97, 190, 223, 244, 251,252, 254, 272, 274,276, 297

Krause, D. 189Kreisel, G. 152, 190Kuhn,T.S. 79, 95,143, 145Kyburg, H.E., Jr. 224, 244

Landau, L.D. 41,43Laplace60, 61, 112, 121Laudan,L. 143, 145Leibniz, yon,G.F.W. 53, 60Leighton,R.H. 54,55Leverrier138

Luce,R.D. 59, 95, 97, 190, 219-249,251252, 254, 272, 274,276,297

Nelson,E. 19,21,25,28

Parijs, van,P. 213Pasch, M.102Pauli, W. 41

NAMEINDEX 303

Rayleigh 120Redhead,M.20,22Reichenbach, H. 79, 80, 96Rice 188Richardson, D. 152, 187, 190, 192Rimini 20

Rosenbaum, Z.S. 235, 245,251-274Rosenbluth, M.117Rosenbluth, A.117Rosenkrantz,R.D. 213Rubin, H.213Rumelhart, D. 118Russell, B. 153

Sakai, D.272

Savage,C.W. 224, 228, 239, 245Scarpellini, B. 152, 191Schild, A.91,95Schrodinger, E.29—43 passimSchwinger, J. 41Scott,D.220, 223, 245, 251, 272Sintonen,M.198, 200, 213Smolin, J. 190Sneed, J. 63, 65, 96, 129-131passim, 138,

144, 145, 195-216, 275, 277, 297Solomon, W. 77

Stegmuller, W. 131, 191,275Sternberg, S. 191Stevens,S.S.68. 96,231Stewart,L.188, 190

comments onFine 39comments onHumphreys 118comments onLoewer 22comments onLuce andNarens 246comments onMoulines and Diez 297comments onMundy 97comments onRoberts andRosenbaum

273commentson Sneed213comments onVuillemin 55comments onWojcicki 146

Suszko,R. 144, 145Szmielew,W. 298, 299

Tarski, A. 119, 126, 129,144, 146, 276Taubes, CH. 191

Tukey, J.W. 222, 223,244Turing, A. 119

Ulam, S. 118Urbach, P. 213

Vivian,R.G. 213Voirol, V. 54Vuillemin, J. 45-57

Weber, P. 20Whitehead, A.N.74Wigner, E.P. 19, 22Winnie, J.A. 80, 97Wittgenstein, yon,L. 48Wojcicki, R. 125-149Woodhouse, N.91,97Woodward, J. 144, 146

Xv,Shaoji 272

Robb, A.A.78,79, 80, 96Roberts,F.S. 235, 241, 245,251-274,251Rogers,H., Jr. 154, 162, 165, 167,169, 170,

172, 173, 178, 179, 188, 190Rohrlich, F. 105, 117Rosen,N. 11,21,38

Sands, M.H. 54,55

Sparrow,C. 119 ff.Specker,E.P. 19,21,32,38Spinoza53

Teller, P. 32, 38,77, 81, 82, 97,117Tsuji,M.189,190

Tversky, A.59, 95, 97, 190, 242, 244, 245,251,252,254,272,274,276,297

Sugar, A.C.213 Young, P. 46, 51,190Suppe, F. 213, 275Suppes, P. 12, 19, 28, 29, 37, 95, 96, 145, Zangi N 2i

151, 154, 187, 188, 189, 197, 220, Zanotti, M.102, 228, 245223, 228, 245,251, 252, 254, 272, Zeemail)EC 78> 80> 92> 97274, 275, 276 Zinnes, JL. 220, 245, 251,252, 272

comments onda Costa andDoria 191

SUBJECT INDEX

affine manifold93algebra, causality productof47

mid-point 298annealing, simulated 116application, firm, successful 197

intended 138Archimedean group 223argument,explanationas 203-207Arrow-Debreu equilibrium 151atom,Bohr model 140

collision 46representationofnature 45

average,quantumprobability as 23axiom,basic 278

necessary 81structural 77

axiomatic-representationalapproach277axiomatics, empirical theories 157

Bohmtheory,dfficulties of 23Boolean logic 8Born rule 3 f., 5-8,10, 14, 18,41

contradiction with other quantumlaws 7Bourbaki species ofstructures 154 ff.Brownian motion 25Buffon needle112Buffon-Laplace method113

calculus, differential andintegral 60cat,quantum state of11category 86Cauchy condition 160causal structure 80causality, product ofalgebra 47causation 79chaos 191chaos theory 119 f.,187, 188Churchthesis 114claim, empirical 131Co-ranking 260collapsepostulate7

empirical testing of 10collision, atomic 46combinatorial measurement theory 282communication theory 50comparison,ordinal, meaningfulness of

251-274

complexity, computationand119computability 114

insufficiency of189computation, complexity and119

continuum and 247connectionism 116conservationrule 46consistency, arithmetical 162, 172contextuality, quantum17continuum, computation and 247

representation230f.correspondence, failure of30

rules 125counting argument75,77

data, Suppes model of137decision procedure, Tarski 191Dedekindcompletion230definition, coordinating 125degree,comlete 169

in arithmetical hierarchy 165determinism, transcendental 26f.diagonalization 178differentiable manifold93

structure 91, 92, 93Diophantineequation151, 164, 179Dirac set 159dynamical system,polynomial,

undecidability of181 f.dynamics,Kantian 51

eigenfunction 48 f.eigenstate-eigenvalue rule 5, 6

contradiction with other quantumlaws 7eigenvalue 48 f.embedding 71empricism105epistemic interest 196EPRparadox 11, 12

Schrodinger version 29-43equality comparison, meaningfulness 255 ff.

undecidability of165equivalence, demonstrable 171essential maximal element 284ether, luminiferous 47existence theorem 221experiment,hierarchy of models for 148

305

306 SUBJECT INDEX

experimentation, numerical 103-121theoretical model105

explanationas argument 203-207evaluationof 207-212functional 206partial 205pragmatic aspects 201scientific 198-203sketch 199structural 195-216

expressibility,arithmetical 171

factual interpretation 142Fermatconjecture 151, 173ferromagnetism 106ff.Feynmanpropagator 24field magnitude 48

classical 45-57field theory,classical, Suppespredicates for

157theory, XIX century 51

finite uniqueness,homogeneity and,structures 235 ff.

structures without 235formalization, semantic 65

set theoretical 64syntactic 65

functor, forgetful 197

game, finite, incompleteness of theory 152gauge field 159general order relation 253

strict 254system257-264

geometry,analytical 60axiomatic 67-71differential, representationtheory 81-90Euclidean 134representation, quantity and 59-102representation theorems in 98space-time,axiomatic 77-80, 90-94, 101

Gleason theorem 4Gbdel diagonalization 170

number 168gravitation,Newtonian 134group, totally ordered Archimedean 223

halting function 165, 166,187Hamiltonian mechanics 161heat, diffusion of 47Heisenbergmodel, ferromagnetism 106 ff.hidden variable, algebraic constraint 32-35

quantumtheory, Bohm 12-19

theory 25hierarchy, arithmetical, rank in 173Hirsch problem187holeargument 94homogeneity 254

finite uniqueness,scale type and 231 f.homomorphism252

into SGOR, meaningfulness of ordinalcomparison267-270

Hopfieldnet 116Huygens equation 26hypothesis,nearly true 132

idealization71immaterialism 72impact, laws of 46incommensurability 79incompleteness, central to mathematics 188

finite Nash Game 186f.nastiestkind of181non-arithmetical 175polynomial dynamical systems 181 ff.theorem 169-174undecidability and 162-181

infinite regress137instrument, scientific 105integration, MonteCarlo methodand

108-112intended application doctrine 129-131interpretation, factual 142into representation71intrinsic axiom92

statement 90invariance 65Ising model110, 111

ferromagnetism 106 ff.

jump 167

Kaluza-Kleinunified field theory 162Kochen-Specker theorem 4, 12, 17, 38

language, formal, undecidability in 153informal, unlimitability of149representational 87

large numbers, strong law of109law,Newton's second 131

special 278Lie algebra 160light,mechanical theory 47

wave,Huygens51logic,propositional,quantum,classical 9Lorentztransformation 78

SUBJECT INDEX 307

Lorentz-Poincare action 160group 159

Lorenz equation119 f.Lyapunov theorem 181

Mach principle 93magnitude, physical, quantum definition of

45manifold, topological 84map, Richardson first 163

second 164materialism 71-77mathematics, applied vs. pure 128

pure, empirical theory vs. 130theoretical science vs. 147

reality and113Maxwellequations 26mean value theorem 108 ff.meaningfulness 65, 229 f.

condition 62generalized theory 87-90, 100group63, 82ordinalcomparison 251-274syntactic criterion 88theory of68

measurement, fundamental 64homomorphismdefinition 224problem,quantum 18Bohm solution 14orthodox resolution 7, 11quantum6representationaltheory 59,61-64,

219-249,251ancestry 98history 219 f.principles 220f.problems 224 ff.structure, finite 230theory,combinatorial, net275-299

error in 227 ff.mechanics, analytical 60

classical 66Hamiltonian 161Newtonian134non-Markovian stochastic 37, 40

Mercury,movement of,theory 133 ff.method, quantitative physical 60methodology, physical, XVIIIcentury

consensus 60, 98metric, cyclic 290-293

external combinatorial 285 f.internal 293-296linear cobinatorial 286

linearcombinatorial, positive286extensive 286 ff.

negative 288-290Metropolisalgorithm 105, 109, 111, 112,

113,116Minkowski space 159, 160model class, theory as 195-216

element, hierarchical net 278finite 76hierarchy of, for experiments 148mathematical 103optimization 206semantic 127semantical vs. theoretical 139-142,146set theoretical, Suppes approach

126-129theoretical, of empiricalphenomenon

133-139theory and 125-149theorybased 140

MonteCarlo estimator 113method 104

integrationand 108-112

Nash equilibrium 152Nash game 181

finite, incompleteness 186f.net,hierarchical 278

theory as275-299no-go theorem 35, 36ff.numerical unease 119

observable 105, 114contextual 17quantum,Bohm theory 16

onto representation 71operator,noncommuting 50

theory 56oracle Turingmachine 167order relation, general 253

temporal202ordinal comparison, meaningfulness

257-264for homomorphisms267-270

scale 252

pairing function 170paradigm 128,143particle interaction, causal thinkingon41phenomenon, empirical, theoretical model of

133-139photonpropagator 41

properties 26

308 SUBJECT INDEX

scattering42physical law, intrinsic formulation 66

numerical 67quantity 64

formal theory 94theory, intrinsic formalization 64-67

physics, abstract vs. concrete 53finitism in 99psychological theory and 239 f.

pi, value of, empiricalestimation 112 f.point, singular, structure with 240

stationary, stability problem 181possible world, Bohmian14

orthodox quantum 8predicate,nontrivial 171

Suppes 155variable degree 76

probability, classical, inBohm quantumtheory 15

law, quantum13quantum,average23

interpretationof 4theory and 3-28

problem, analytically intractable 115unsolvable, construction of151-193

product ofpowers 237rule 32,35

proof,algorithmic 188proposition,Bohmian 14

representaton 87psychology, physical theory and239 f.

quantity, physical 64formal theory 59 ff.,94representation, geometry and 59-102

quantumelectrodynamics 41entanglement 30logic,nonstandard 8 f.state 5

collapse 8theory 60Bohm 12-19

probability in 15locality of 12nonrelativistic 5orthodox, inexplicitnessof10 ff.probability and 3-28

random walk 112quantum,25 f.

ranking 253rationality, Bayesian 207realism, classical 5

reality, mathematics and113problem6

quantum,Bohm solution14reasoning, scientific, Bayesian 208reduction 264relation, intrinsic 61

meaningful221relational structure, primitives of221

system,general order, ordinalcomparison in 251-274

homomorphismof251relationism 71-77, 94

embedding 75,77materialist 75second order 73 f.

relationist theory 73relativity 60,78,79

general,classification schemes in 187principle 83

representationcondition 62, 63numerical 61

construction 232problem252quantity, geometry and 59-102theorem 62, 68Klein-Stevens approach 62

representationalcondition 70, 90proposition87

research program143, 196tradition 143

Rice theorem 188Richardson functor 164Riemann hypothesis 173Riemannian geometry 92

sampling,simple 110scale type,applications237-240

combination with structural conditions234 f.

homogeneity, finite uniqueness and231f.

scaling 64,67scattering, photon-photon42

theory 41Schrodinger's cat 5, 6

equation 5, 6, 8, 13, 20, 23, 24contradiction with otherquantum

laws 7science, axiomatic 151-193

computational 103theoretical, pure mathematics vs. 147

semantics, modern126sequence, standard 222

309SUBJECT INDEX

set, finite, structure 230theory,Suppes approach to 126-129

ZF 151-193 passimsignificance, empirical222situation, qualitative 220space-time geometry, axiomatic 77-80spatialism 71-77spectrum rule 35square rule 31state, initial 18statement, meaningful 252structuralist view 275structure, classification by scale type

231-234deduced and derived 156ground 156mathematical 155partial 156set theoretic, partof 201

sumrule 31Suppes predicates 151-193symmetrygroup 63, 82syntactic transform 88system, economic, incompletenessof

Hamiltonian model151systems theory,dynamical 151

tensoranalysis 89testability,emprical 221theory applicationvs. formation 140

asnet 275-299better thanordering 211competitionbetween 210definition127empirical,axiomatics of157

identity 276pure math vs. 130

formal core of130formation 199Maxwellelectromagnetic 158-160modelbased 140model class 195-198natureof 120

net, treelike 278-282passimphysical, intrinsic formalization 99psychological andphysical 239 f.semantic view 275theoretical model and 125-149

thought experiment,Bohm11topological manifold 91transcendental determinism 26 f.transformation, admissible 252translation, structural equivalent to 233 f.

structure 233trigonometry 70truth 126

approximate 131-132partial or relative132

Turing degree 169machine 165, 168, 170, 188

uncertainty relation 18undecidability, extra-arithmetical 174-181

incompletenessand 162-181nontrivialproperties 176polynomial dynamicalsystems 181 ff.Suppespredicatesand 151

uniqueness, finite,homogeneity,scale typeand 231f.

generalized theory 83group 63, 82, 84problem 85theorem 62, 68,77, 221

unsolvability, algorithmic 177

vector analysis 48 ff.field, planepolynomial, limit cycle 181quantity 69

velocity law, quantum13

wave, analysis ofconcept53equation,boundary constraint48

temporal form50wave-particle dualism 52world view 134

TABLEOF CONTENTSVolume 1:Probability andProbabilistic Causality

Paul Humphreys /Introduction xiii

PARTI:PROBABILITY

KarlPopperandDavidMiller /SomeContributions toFormal Theory ofProbability /Commentsby Patrick Suppes 3Peter J. Hammond /ElementaryNon-ArchimedeanRepresentationsofProbability for DecisionTheory andGames /Commentsby Patrick Suppes 25

RolandoChuaqui/RandomSequencesand HypothesesTests /Commentsby Patrick Suppes 63Isaac Levi/ ChangingProbability Judgements /Commentsby Patrick Suppes 87TerrenceL.Fine / UpperandLower Probability /CommentsbyPatrick Suppes 109PhilippeMongin / SomeConnections betweenEpistemic Logic and the Theory ofNonadditive Probability /Commentsby Patrick Suppes 135WolfgangSpohn / On the Properties of ConditionalIndependence/Commentsby Patrick Suppes 173Zoltan Domotor / Qualitative Probabilities Revisited/Comments byPatrick Suppes 197Jean-Claude Falmagne/ TheMonks' Vote:A Dialogue onUnidimensionalProbabilistic Geometry /Commentsby Patrick Suppes 239

311

312 tableofcontentsTO VOLUME 1

PARTII:PROBABILISTICCAUSALITY

Paul W Holland/Probabilistic Causation WithoutProbability /Comments by Patrick Suppes 257I.J.Good/Causal Tendency,Necessitivity andSufficientivity: AnUpdatedReview /Comments byPatrick Suppes 293Ernest W. Adams /Practical CausalGeneralizations /Commentsby Patrick Suppes 317ClarkGlymour,PeterSpirtes,andRichard Schemes /InPlace of Regression/Commentsby Patrick Suppes 339

D.Costantini / TestingProbabilistic Causality /Commentsby Patrick Suppes 367

PaoloLegrenziandMaria Sonino /PsychologistsAspectsof Suppes'sDefinition ofCausality /CommentsbyPatrick Suppes 381

Name Index 401SubjectIndex 407

Table of Contents toVolumes 2and 3 415

TABLE OFCONTENTSVolume 3: Language,Logic,andPsychology

PART VI:PHILOSOPHYOFLANGUAGE AND LOGIC

DagfinnFOLLESDAL / Patrick Suppes' Contribution to thePhilosophy ofLanguage /Comments by Patrick Suppes 3Michael Bottner /OpenProblems inRelationalGrammar /Commentsby Patrick Suppes 19William C.Purdy /A Variable-Free Logic for Anaphora/Commentsby Patrick Suppes 41J.Moravcsik /Is Snow White? /Commentsby PatrickSuppes 71PaulWeingartner /CanThereBe Reasons forPuttingLimitations on ClassicalLogic? /Commentsby Patrick Suppes 89Jaakko Hintikka and IlpoHalonen /QuantumLogicas aLogic onIdentification / Comments by Patrick Suppes 125Maria LuisaDalla Chiara andRobertoGiuntini /Logic andProbability in QuantumMechanics /Comments byPatrick Suppes 147

PARTVII:LEARNING THEORY,ACTION THEORY, ANDROBOTICS

W.K.Estes / From Stimulus-Sampling toArray-SimilarityTheory /Commentsby Patrick Suppes 171Raimo Tuomelaand GabrielSandu /Action asSeeing toItthat SomethingIs theCase /Commentsby Patrick Suppes 193ColleenCrangle /CommandSatisfaction and theAcquisition of Habits /Commentsby Patrick Suppes 223

313

314 TABLEOFCONTENTS TO VOLUME 3

PART VIII:GENERAL PHILOSOPHY OF SCIENCE

Maria Carla Galavotti /Some ObservationsonPatrickSuppes'Philosophyof Science /Comments by Patrick Suppes 245

EPILOGUE

Patrick Suppes / Postscript 273Bibliography of Patrick Suppes 275Name Index 325Subject Index 329Tableof Contents to Volumes 1and2 333

syntheselibrary

106. K. Kosik, Dialectics of the Concrete. A Study onProblems of Man and World.[BostonStudies in thePhilosophyof Science,Vol.LII] 1976

ISBN90-277-0761-8;Pb 90-277-0764-2107. N. Goodman, The Structure of Appearance. 3rd cd. with an Introduction by G.

Hellman.[BostonStudies in thePhilosophyof Science,Vol.LIII] 1977ISBN90-277-0773-1;Pb 90-277-0774-X

108. K. Ajdukiewicz, The Scientific World-Perspective and Other Essays, 1931-1963.TranslatedfromPolish.Edited and withan Introductionby J. Giedymin.1978

ISBN 90-277-0527-5109. R.L.Causey,Unity ofScience. 1977 ISBN 90-277-0779-0110. R.E.Grandy, AdvancedLogicfor Applications. 1977 ISBN 90-277-0781-2111. R.P.McArthur, TenseLogic. 1976 ISBN 90-277-0697-2112. L. Lindahl, Position and Change. A Study in Law and Logic. Translated from

Swedishby P.Needham. 1977 ISBN90-277-0787-1113. R. Tuomela,Dispositions. 1978 ISBN90-277-0810-X114. H. A. Simon, Models of Discovery and Other Topics in the Methods of Science.

[BostonStudies in thePhilosophyof Science,Vol.LIV] 1977ISBN90-277-0812-6; Pb90-277-0858-4

115. R. D. Rosenkrantz, Inference, Method and Decision. Towards a BayesianPhilosophy of Science. 1977 ISBN 90-277-0817-7; Pb 90-277-0818-5

116. R. Tuomela, Human Action and Its Explanation. A Study on the PhilosophicalFoundations ofPsychology. 1977 ISBN 90-277-0824-X

117. M. Lazerowitz, The Language of Philosophy. Freud and Wittgenstein. [BostonStudies in thePhilosophyof Science,Vol.LV] 1977

ISBN 90-277-0826-6;Pb 90-277-0862-2118. Notpublished119. J. Pelc (cd.),SemioticsinPoland, 1894-1969.TranslatedfromPolish. 1979

ISBN90-277-0811-8120. I.Porn,ActionTheory and SocialScience. SomeFormalModels.1977

ISBN90-277-0846-0121. J. Margolis,Persons andMind.TheProspects ofNonreductiveMaterialism.[Boston

Studies in thePhilosophyof Science,Vol.LVII] 1977ISBN 90-277-0854-1;Pb 90-277-0863-0

122. J. Hintikka, I. Niiniluoto, and E. Saarinen (eds.), Essays on MathematicalandPhilosophicalLogic. 1979 ISBN 90-277-0879-7

123. T.A.F.Kuipers,Studiesin InductiveProbabilityandRationalExpectation. 1978ISBN90-277-0882-7

124. E. Saarinen, R. Hilpinen,I.Niiniluoto andM. P. Hintikka (eds.), Essays inHonourofJaakkoHintikkaon theOccasionofHis 50thBirthday. 1979

ISBN90-277-0916-5125. G Radnitzky andG. Andersson (eds.), ProgressandRationalityinScience. [Boston

Studiesin the Philosophy of Science,Vol.LVIII] 1978ISBN 90-277-0921-1;Pb 90-277-0922-X

126. P. Mittelstaedt,QuantumLogic. 1978 ISBN 90-277-0925-4127. K. A. Bowen,Model Theoryfor ModalLogic. Kripke Models forModal Predicate

Calculi.1979 ISBN 90-277-0929-7128. H. A. Bursen, Dismantling the Memory Machine. A PhilosophicalInvestigationof

Machine Theories ofMemory.1978 ISBN 90-277-0933-5

SYNTHESELIBRARY

129. M. W. Wartofsky, Models. Representation and the Scientific Understanding.[BostonStudies in thePhilosophyofScience, Vol.XLVIII] 1979

ISBN90-277-0736-7;Pb 90-277-0947-5130. D. Ihde, Technics andPraxis. A Philosophy of Technology. [Boston Studies in the

Philosophy of Science, Vol.XXIV] 1979 ISBN90-277-0953-X;Pb 90-277-0954-8131. J. J. Wiatr (cd.), Polish Essays in the Methodology of theSocial Sciences. [Boston

Studiesin thePhilosophy of Science, Vol.XXIX] 1979ISBN90-277-0723-5;Pb 90-277-0956-4

132. W. C. Salmon (cd.),HansReichenbach:LogicalEmpiricist. 1979ISBN90-277-0958-0

133. P. Bieri, R.-P. Horstmann and L. Kriiger (eds.), Transcendental Arguments inScience. Essays inEpistemology.1979 ISBN90-277-0963-7;Pb 90-277-0964-5

134. M.Markovic andG.Petrovic(eds.), Praxis. YugoslavEssays in the PhilosophyandMethodology of the Social Sciences. [Boston Studies in the Philosophy of Science,Vol.XXXVI] 1979 ISBN 90-277-0727-8;Pb 90-277-0968-8

135. R. Wojcicki, Topics in the FormalMethodologyof Empirical Sciences. TranslatedfromPolish. 1979 ISBN9O-277-1004-X

136. G. Radnitzky andG. Andersson (eds.), TheStructure andDevelopmentof Science.[Boston Studiesin thePhilosophy of Science, Vol.LIX] 1979

ISBN90-277-0994-7; Pb90-277-0995-5137. J. C. Webb, Mechanism, Mentalism and Metamathematics. An Essay on Finitism.

1980 ISBN90-277-1046-5138. D.F.GustafsonandB.L.Tapscott (eds.),Body,MindandMethod.Essays inHonor

ofVirgil C.Aldrich. 1979 ISBN90-277-1013-9139. L.Nowak, The StructureofIdealization. Towards a SystematicInterpretationof the

MarxianIdeaof Science. 1980 ISBN90-277-1014-7140. C. Perelman, The New Rhetoric and the Humanities. Essays onRhetoric and Its

Applications. Translated from French and German. With an Introduction by H.Zyskind. 1979 ISBN 90-277-1018-X;Pb 90-277-1019-8

141. W. Rabinowicz,Universalizability. A Study inMorals andMetaphysics. 1979ISBN 90-277-1020-2

142. C. Perelman, Justice, Law andArgument. Essays on Moral and Legal Reasoning.TranslatedfromFrench andGerman.With an IntroductionbyH.J.Berman. 1980

ISBN90-277-1089-9;Pb 90-277-1090-2143. S. Kanger and S. Ohman (eds.), Philosophy and Grammar.Papers on the Occasion

of theQuincentennialofUppsalaUniversity. 1981 ISBN90-277-1091-0144. T.Pawlowski,Concept Formation in theHumanities and theSocialSciences. 1980

ISBN90-277-1096-1145. J. Hintikka, D. Gruender andE. Agazzi (eds.), Theory Change, Ancient Axiomatics

and Galileo's Methodology.Proceedings ofthe 1978 PisaConference on theHistoryandPhilosophy ofScience,Volume1.1981 ISBN90-277-1126-7

146. J. Hintikka, D. Gruender and E. Agazzi (eds.), Probabilistic Thinking, Ther-modynamics, and the Interaction of the History and Philosophy of Science.Proceedingsof the 1978 PisaConference on theHistory and Philosophy of Science,Volumell. 1981 ISBN90-277-1127-5

147. U. Mdnnich(cd.), AspectsofPhilosophicalLogic. SomeLogicalForays intoCentralNotionsof Linguistics andPhilosophy. 1981 ISBN90-277-1201-8

148. D.M.Gabbay,SemanticalInvestigations inHeyting's IntuitionisticLogic. 1981ISBN90-277-1202-6

SYNTHESELIBRARY

149. E. Agazzi (cd.), ModernLogic-A Survey. Historical,Philosophical,andMathemati-calAspectsofModernLogicand Its Applications.1981 ISBN 90-277-1137-2

150. A. F. Parker-Rhodes, The Theory of Indistinguishables. A Search for ExplanatoryPrinciplesbelow theLevelofPhysics. 1981 ISBN 90-277-1214-X

151. J.C. Pitt, Pictures,Images, and ConceptualChange. AnAnalysis ofWilfridSellars'Philosophy of Science.1981 ISBN 90-277-1276-X;Pb 90-277-1277-8

152. R.Hilpinen(cd.), New Studies inDeontic Logic. Norms, Actions, andthe Founda-tions ofEthics. 1981 ISBN 90-277-1278-6;Pb90-277-1346-4

153. C. Dilworth, Scientific Progress. A Study Concerning the Nature of the RelationbetweenSuccessive ScientificTheories. 2nd,rev. andaugmentedcd.,1986. 3rd rev.cd.,1994

ISBN0-7923-2487-0; Pallas Pb0-7923-2488-9154. D. Woodruff Smith andR. Mclntyre, Husserl andIntentionality. A Study of Mind,

Meaning, andLanguage. 1982 ISBN90-277-1392-8;Pb 90-277-1730-3155. R.J. Nelson, TheLogicofMind.2nd. cd.,1989

ISBN90-277-2819-4; Pb90-277-2822-4156. J.F. A.K. vanBenthem,The Logic ofTime. A Model-TheoreticInvestigationinto

the VarietiesofTemporalOntology, andTemporalDiscourse. 1983; 2ndcd.,1991ISBN 0-7923-1081-0

157. R.Swinburne(cd.),Space, TimeandCausality. 1983 ISBN 90-277-1437-1158. E. T. Jaynes, Papers on Probability, Statistics andStatisticalPhysics. Ed. by R. D.

Rozenkrantz. 1983 ISBN90-277-1448-7;Pb (1989) 0-7923-0213-3159. T.Chapman, Time:APhilosophicalAnalysis. 1982 ISBN 90-277-1465-7160. E.N. Zalta,AbstractObjects. AnIntroductionto AxiomaticMetaphysics. 1983

ISBN 90-277-1474-6161. S. HardingandM.B.Hintikka(eds.),DiscoveringReality.FeministPerspectives on

Epistemology,Metaphysics,Methodology,andPhilosophy of Science. 1983ISBN 90-277-1496-7;Pb 90-277-1538-6

162. M.A. Stewart(cd.), Law, Morality andRights. 1983 ISBN90-277-1519-X163. D.MayrandG. Sussmann (eds.),Space, Time, andMechanics. Basic Structuresof a

Physical Theory. 1983 ISBN90-277-1525-4164. D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic. Vol. I:

Elements of ClassicalLogic. 1983 ISBN90-277-1542-4165. D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic. Vol. II:

Extensionsof Classical Logic. 1984 ISBN90-277-1604-8166. D. Gabbay and F. Guenthner (eds.), Handbook of PhilosophicalLogic. Vol. Ill:

AlternativetoClassical Logic. 1986 ISBN 90-277-1605-6167. D. Gabbay and F. Guenthner (eds.), Handbook of PhilosophicalLogic. Vol. IV:

Topics in thePhilosophy of Language. 1989 ISBN 90-277-1606-4168. A. J. I.Jones, Communication and Meaning. An Essay in AppliedModal Logic.

1983 ISBN90-277-1543-2169. M.Fitting,ProofMethodsfor ModalandIntuitionisticLogics. 1983

ISBN90-277-1573-4170. J.Margolis,CultureandCulturalEntities. TowardaNewUnity ofScience. 1984

ISBN90-277-1574-2171. R.Tuomela,A Theory ofSocialAction. 1984 ISBN90-277-1703-6172. J. J. E. Gracia, E. Rabossi, E. Villanueva and M. Dascal (eds.), Philosophical

Analysis inLatinAmerica. 1984 ISBN90-277-1749-4173. P.Ziff,Epistemic Analysis. A Coherence Theory of Knowledge. 1984

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ISBN 90-277-1751-7174. P.Ziff, Antiaesthetics. AnAppreciationoftheCow with theSubtile Nose.1984

ISBN 90-277-1773-7175. W. Balzer, D. A. Pearce, andH.-J. Schmidt (eds.),Reduction in Science. Structure,

Examples,PhilosophicalProblems.1984 ISBN 90-277-1811-3176. A. Peczenik,L. Lindahl and B. vanRoermund (eds.), Theory of Legal Science.

Proceedings of the Conference on Legal Theory andPhilosophy of Science (Lund,Sweden,December1983). 1984 ISBN90-277-1834-2

177. I.Niiniluoto,IsScience Progressive? 1984 ISBN90-277-1835-0178. B. K. Matilal and J. L. Shaw (eds.), Analytical Philosophy in Comparative

Perspective.Exploratory Essays inCurrent Theories and Classical Indian Theoriesof Meaningand Reference.1985 ISBN90-277-1870-9

179. P.Kroes, Time:ItsStructureandRoleinPhysical Theories.1985

180. ].H.Fetzer,SociobiologyandEpistemology. 1985ISBN90-277-1894-6

ISBN90-277-2005-3; Pb 90-277-2006-1181. L.HaaparantaandJ. Hintikka(eds.), FregeSynthesized.Essays on thePhilosophical

andFoundationalWork ofGottlobFrege.1986 ISBN 90-277-2126-2182. M.Detlefsen,Hubert'sProgram. AnEssay onMathematicalInstrumentalism.1986

ISBN 90-277-2151-3183. J.L.Goldenand J. J.Pilotta (eds.), PracticalReasoning inHuman Affairs. Studies

inHonor of ChaimPerelman. 1986 ISBN 90-277-2255-2184. H. Zandvoort,ModelsofScientific Developmentand the CaseofNuclearMagnetic

Resonance. 1986 ISBN 90-277-2351-6185. I.Niiniluoto, Truthlikeness.1987 ISBN90-277-2354-0186. W. Balzer, C. U. Moulines and J. D. Sneed, An Architectonicfor Science. The

StructuralistProgram. 1987 ISBN90-277-2403-2187. D.Pearce,Roadsto Commensur-ability. 1987 ISBN90-277-2414-8188. L.M. Vaina(cd.),MattersofIntelligence.ConceptualStructures inCognitiveNeuro-

science. 1987 ISBN90-277-2460-1189. H. Siegel, Relativism Refuted. A Critique of Contemporary Epistemological

Relativism.1987 ISBN90-277-2469-5190. W. Callebaut and R. Pinxten, Evolutionary Epistemology. A Multiparadigm

Program,with aCompleteEvolutionaryEpistemologyBibliograph.1987ISBN 90-277-2582-9

191. J. Kmita,ProblemsinHistoricalEpistemology. 1988 ISBN90-277-2199-8192. J.H.Fetzer(cd.),Probabilityand Causality.Essays inHonor of Wesley C. Salmon,

withan Annotated Bibliography. 1988 ISBN90-277-2607-8; Pb1-5560-8052-2193. A. Donovan, L. Laudan and R. Laudan (eds.), Scrutinizing Science. Empirical

Studiesof ScientificChange. 1988 ISBN90-277-2608-6194. H.R. OttoandJ.A.Tuedio (eds.),PerspectivesonMind. 1988

ISBN90-277-2640-X195. D. Batens and J.P. vanBendegem (eds.), Theory and Experiment. Recent Insights

andNewPerspectivesonTheirRelation.1988 ISBN90-277-2645-0196. J.Osterberg, Selfand Others. A Study ofEthicalEgoism. 1988

ISBN 90-277-2648-5197. D.H. Helman (cd.), Analogical Reasoning. Perspectives of Artificial Intelligence,

Cognitive Science,andPhilosophy. 1988 ISBN 90-277-2711-2

SYNTHESE LIBRARY

198. J. Wolenski, Logic andPhilosophy in theLvov-WarsawSchool. 1989ISBN90-277-2749-X

199. R. Wojcicki, Theory ofLogicalCalculi.Basic Theory of ConsequenceOperations.1988 ISBN 90-277-2785-6

200. J. HintikkaandM.B.Hintikka,The Logic ofEpistemologyand theEpistemology ofLogic. SelectedEssays. 1989 ISBN 0-7923-0040-8; Pb 0-7923-0041-6

201. E.Agazzi (cd.),Probability in theSciences. 1988 ISBN 90-277-2808-9202. M.Meyer-(ed.), FromMetaphysics toRhetoric.1989 ISBN 90-277-2814-3203. R.L. Tieszen, Mathematical Intuition. Phenomenology and Mathematical

Knowledge. 1989 ISBN0-7923-0131-5204. A.Melnick, Space, Time, andThought inKant. 1989 ISBN0-7923-0135-8205. D.W. Smith, The CircleofAcquaintance. Perception,Consciousness, andEmpathy.

1989 ISBN0-7923-0252-4206. M.H. Salmon (cd.), The Philosophy of Logical Mechanism. Essays in Honor of

Arthur W. Burks. With his Responses, and with a Bibliography of Burk's Work.1990 ISBN 0-7923-0325-3

207. M. Kusch, Language as Calculus vs. Language as UniversalMedium. A Study inHusserl, Heidegger, andGadamer.1989 ISBN 0-7923-0333-4

208. T.C. Meyering,Historical Roots of Cognitive Science. The Rise of a CognitiveTheory ofPerceptionfromAntiquity to theNineteenthCentury. 1989

ISBN0-7923-0349-0209. P. Kosso,Observability andObservationinPhysicalScience. 1989

ISBN0-7923-0389-X210. J.Kmita,Essays on the Theory ofScientific Cognition. 1990 ISBN0-7923-0441-1211. W. Sieg (cd.),ActingandReflecting.The Interdisciplinary Turn inPhilosophy.1990

ISBN 0-7923-0512-4212. J. Karpiriski,Causality inSociologicalResearch. 1990 ISBN0-7923-0546-9213. H.A.Lewis(cd.), PeterGeach:PhilosophicalEncounters. 1991

ISBN 0-7923-0823-9214. M. Ter Hark, Beyond the Inner and the Outer. Wittgenstein's Philosophy of

Psychology. 1990 ISBN0-7923-0850-6215. M. Gosselin, Nominalism and Contemporary Nominalism. Ontological and

EpistemologicalImplications of the Work of W.V.O. Quine and of N. Goodman.1990 ISBN0-7923-0904-9

216. J.H. Fetzer, D. Shatz and G. Schlesinger (eds.), Definitions and Definability.PhilosophicalPerspectives.1991 ISBN 0-7923-1046-2

217. E. Agazzi and A. Cordero (eds.), Philosophy and the Origin and Evolution of theUniverse. 1991 ISBN 0-7923-1322-4

218. M. Kusch, Foucault's Strata and Fields. An Investigation into Archaeological andGenealogicalScienceStudies. 1991 ISBN0-7923-1462-X

219. C.J.Posy, Kant'sPhilosophy ofMathematics.Modern Essays. 1992ISBN0-7923-1495-6

220. G. Van de Vijver,New Perspectives onCybernetics. Self-Organization,AutonomyandConnectionism.1992 ISBN0-7923-1519-7

221. J.C.Nyiri, TraditionandIndividuality. Essays. 1992 ISBN0-7923-1566-9222. R.Howell,Kant's TranscendentalDeduction. An Analysis ofMain ThemesinHis

CriticalPhilosophy. 1992 ISBN 0-7923-1571-5

SYNTHESELIBRARY

223. A. Garciade la Sienra, The Logical Foundationsof the Marxian Theory of Value.1992 ISBN 0-7923-1778-5

224. D.S. Shwayder, Statement and Referent. An Inquiry into the Foundations of OurConceptualOrder.1992 ISBN 0-7923-1803-X

225. M.Rosen,Problemsof the HegelianDialectic.DialecticReconstructed as aLogicof HumanReality. 1993 ISBN0-7923-2047-6

226. P. Suppes,ModelsandMethods in thePhilosophyofScience:SelectedEssays. 1993ISBN0-7923-2211-8

227. R. M. Dancy (cd.),KantandCritique: New Essays inHonorof W. H. Werkmeister.1993 ISBN0-7923-2244-4

228. J. Woleriski(cd.),PhilosophicalLogic inPoland.1993 ISBN0-7923-2293-2229. M.DeRijke (cd.),DiamondsandDefaults. Studies inPure and AppliedIntensional

Logic. 1993 ISBN0-7923-2342-4230. B.K. Matilaland A. Chakrabarti (eds.), Knowing from Words. Western and Indian

PhilosophicalAnalysis of Understanding andTestimony.1994ISBN0-7923-2345-9

231. S.A. Kleiner, The Logic of Discovery. A Theory of the Rationality of ScientificResearch. 1993 ISBN0-7923-2371-8

232. R. Festa, Optimum InductiveMethods. A Study in InductiveProbability,BayesianStatistics, andVerisimilitude.1993 ISBN 0-7923-2460-9

233. P. Humphreys (cd.),Patrick Suppes: Scientific Philosopher.Vol. 1:Probability andProbabilisticCausality. 1994

ISBN 0-7923-2552-4; Set Vols.1-3 ISBN0-7923-2554-0234. P. Humphreys (cd.), Patrick Suppes: Scientific Philosopher.Vol. 2: Philosophy of

Physics, TheoryStructure, andMeasurementTheory. 1994ISBN 0-7923-2553-2; Set Vols.1-3 ISBN0-7923-2554-0

235. P. Humphreys (cd.), Patrick Suppes: Scientific Philosopher. Vol. 3: Language,Logic,andPsychology.1994

ISBN0-7923-2862-0; Set Vols. 1-3ISBN 0-7923-2554-0236. D.Prawitz andD. Westerstahl (eds.),Logic andPhilosophy ofScience in Uppsala.

Papers from the 9thInternationalCongress of Logic,Methodology,andPhilosophyof Science. 1994 ISBN 0-7923-2702-0

237. L. Haaparanta(cd.), Mind, Meaning andMathematics. Essays on the PhilosophicalViewsofHusserl andFrege.1994 ISBN0-7923-2703-9

238. J. Hintikka (cd.): AspectsofMetaphor.1994 ISBN0-7923-2786-1239. B. McGuinness and G. Oliveri (eds.), The Philosophyof MichaelDummett. With

Replies fromMichaelDummett.1994 ISBN0-7923-2804-3240. D. Jamieson (cd.),Language, Mind, andArt. Essays in Appreciationand Analysis,

InHonor ofPaulZiff. 1994 ISBN0-7923-2810-8241. G. Preyer,F. Siebelt and A. Ulfig (eds.), Language, Mindand Epistemology.On

DonaldDavidson's Philosophy. 1994 ISBN0-7923-2811-6242. P. Ehrlich (cd.), Real Numbers, Generalizationsof the Reals, and Theories of

Continua. 1994 ISBN0-7923-2689-X243. G Debrock andM.Hulswit (eds.): LivingDoubt. Essays concerning theepistemol-

ogy of Charles SandersPeirce. 1994 ISBN0-7923-2898-1

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