x bell - from absolute to local mathematics

7/28/2019 X Bell - From Absolute to Local Mathematics

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J . L. BELL

FROM ABSOLUTE TO LOCAL MATHEMATICS

In this paper (a sequel to [1.]) I pu t forward a "local" interpretation ofmathematical concepts based on notions derived from category theory.The fundamental idea is to abandon the unique absolute universe ofsets central to the orthodox set-theoretic account of the foundations ofmathematics, replacing it by a plurality of local mathematical frameworks - elementary toposes - defined in category-theoretic terms. Suchframeworks will serve as local surrogates for the classical universe ofsets. In particular theywill possess sufficiently rich internal structure toenable mathematical concepts and assertions to be interpreted withinthem. With the relinquishment of the absolute universe of sets,mathematical concepts will in general no longer possess absolutemeaning, nor mathematical assertions absolute truth values, but willinstead possess such meanings or truth values only locally, i.e., relativeto local frameworks. There is an evident parallel between this approach to the interpretation of mathematical concepts and the interpretation of physical concepts within the theory of relativity: this wediscuss in section 2. Section 3 examines the procedure of passing fromone local framework to another, observing that it is an instance of thedialectical process of negating constancy. In particular, we show(following F. W. Lawvere) how the construction of models of Robinson's nonstandard analysis and the proofs of Cohen's independenceresults in se t theory may be construed as instances of this procedure.

1. CATEGORY THEORY AND LOCAL MATHEMATICALFRAMEWORKS

Category theory (d. [2] or [14]) provides a general apparatus fordealing with mathematica l s tructures and their mutual relations andtransformations. Invented by Eilenberg and MacLane in the 1940's, itarose as a branch of algebra by way of topology, but quickly transcended its origins. Category theory may be said to bear the samerelation to abstract algebra as the lat ter does to elementary algebra.For elementary algebra results from the replacement of constantSynthese 69 (1986) 409-426. 1986 by D. Reidel Publishing Company


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410 J. L. BELLquantities (i.e., numbers) by variables, keeping the operations on thesequantities fixed. Abstract algebra, in its turn, carries this a stagefur ther by allowing the operations to vary while ensuring that theresulting mathematical structures retain a certain prescribed form(groups, rings, or what have you). Finally, category theory allows eventhe form of the structures to vary, giving rise to a general theory ofmathematical structure or form. Thus the genesis of category theory isan instance of the dialect ical process of replacing the constant by thevariable, a theme which will play an important role in what I have tosay here.In ca tegory theory the transformations (called arrows) betweenstructures (called objects) play an autonomous role which is in no waysubordinate to that played by the structures themselves. (Thus cate-gory theory is like a language in which the verbs are on an equalfooting with the nouns.) In this respect category theory differs crucially from set theory where the corresponding notion of function isreduced to the concept of set (of points). As a consequence, the notionof transformation in category theory is vastly more general than theset-theoretic notion of function. In particular, for example, theconcept of category-theoretic transformation will admit interpretations in which one variable quantity depends functionally on anotherbut where the corresponding "function" is not describable as a set of(ordered pairs of) "points" (for instance when the functional dependence arises as the phenomenological description of the motion of abody).The generality of category theory has enabled it to play an increasingly important role in the foundations of mathematics. Its emergencehas had the effect of subtly undermining the prevailing doctrine thatall mathematical concepts are to be referred to a fixed absoluteuniverse of sets. Category theory, in contrast, suggests that mathematical concepts and assertions should be regarded as possessing meaning only in relation to a variety of more or less local frameworks. Toindicate what I mean, let us follow MacLane [14] in considering thecategory-theoretic interpretation of the concept "group". From theset-theoret ical point of view, the term "group" signifies a set (equipped with a couple of operations) satisfying certain elementary axiomsexpressed in terms of the elements of the set. Thus the set-theoreticalinterpretation of this concept is always referred to the same frame-


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FROM A BSOL UT E TO L OC AL MA TH EMA T IC S 411work, the universe of sets. Now consider the category-theoret ic account of the group concept. Here the reference to the "elements" ofthe group has been replaced by an "arrows only" formulation, therebyenabling the concept to become interpretable not merely in theuniverse (category) of sets, but in essentially any category. Thepossibility of varying the framework of interpretation offered bycategory theory confers on the group concept a truly protean generality. Indeed, the interpretation of the term "group" within the categoryof topological spaces is topological group, within the category ofdifferentiable manifolds it is Lie group and within the category ofsheaves over a topological space it is sheaf of groups.We see that the category-theoretic meaning of a mathematicalconcept such as group is now determined only in relation to anambient category, and this ambient category can vary. Thus the effectof casting a mathematical concept in categorical language is to confera further degree of ambiguity of reference on the concept.To some extent this ambiguity of reference of mathematicalconcepts is already present within classical set theory, since its axiomsare formulated in first-order terms and therefore admit many essentially different interpretations. Indeed, as far back as 1922, Skolem[15] remarked that for this reason set-theoretical notions - in particular, infinite cardinalities - are relative. He concluded that axiomatized set theory "was not a satisfactory ultimate foundation formathematics". Skolem's structures were largely ignored by mathema

ticians, but a new chal lenge to the absoluteness of the set-theoret icalframework arose in 1963 when Paul Cohen constructed models of settheory (i.e., Zennelo-Fraenkel set theory ZF) in which importantmathematical propositions such as the continuum hypothesis and theaxiom of choice are falsified (Godel having already in the late 1930'sproduced models in which the propositions are validated). The resulting ambiguity in the truth values of mathematical propositions wasregarded by many set-theorists (and even by more "orthodox"mathematicians) as a much more serious matter than the "mere"ambiguity of reference of mathematical concepts already pointed outby Skolem. In fact, the techniques of Cohen and his successors haveled to an enormous proliferation of models of set theory with essentially different mathematical properties, which in turn has engendereda disquieting uncertainty in the minds of set-theorists as to the identity


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412 J. L. BELLof the "real" universe of sets, or at least as to precisely what mathematical properties it should possess. The upshot is that the set concept- .in so far as it is capturable by first order axioms - has turned out tobe radically underdetermined."What I suggest is that we accept the radically underdeterminednature of the set concept and abandon the quest for the absoluteuniverse of sets in the form proposed by classical set theory. I t shouldbe emphasized that this does not necessarily require that we espousethe extreme formalist/finitist view (or gospel of despair) that settheoretical concepts (or those involving the infinite, at any rate) aremeaningless. No, I think the answer lies in recognizing that themeaning (or reference) of these concepts is determined only relative tomodels of ZF, or, more generally, to the local frameworks of interpretation to be introduced in a moment. In this event, an assertion likethe continuum hypothesis will no longer be regarded as the possessorof an absolute but unknown truth value,for the unique universe of setswhich was presumed to furnish the said truth value will no longerexist. Note, however, that although the concept of absolute truth ofset-theoretical assertions will have vanished from the scene, there willappear in its place the subtler concept of invariance, that is,validity in all local frameworks. Thus, e.g., whereas the theorems ofconstructive arithmetic will turn out to possess the property of invariance, set-theoretical assertions such as the axiom of choice, or thecontinuum hypothesis, will not, because they will hold true in somelocal frameworks but not in others.Having reached the point where models of set theory are treated aslocal frameworks of interpretation for mathematical concepts, or inother words, having replaced the concept of the unique universe ofsets by the concept of a varying framework of interpretation, itbecomes natural to a ttempt to formulate these ideas within the conceptual language best equipped to handle variation: the language ofcategory theory. Thus the first thing we require is a category-theoreticformulation of the notion of model of set theory. This we find inLawvere and Tierney's concept of an (elementary) topos (see, e.g., [3],[9], [10]).To arrive at the concept of a topos, we start with the familiarcategory S of sets whose objects are all sets (in a given model M of settheory) and whose arrows are all mappings (in M) between sets in M.We observe that S has the following properties.

...


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F R O M ABSOLUT E TO LOCAL MATHEMATIC S 413(i) There is a ' terminal ' object 1 such that, for any object X,there is a unique arrow X ~ 1 (for 1 we may take anyone-element set, in particular {O}).(ii) Any pair of objects A, B has a Cartesian product A x B.(iii) Fo r any pair of objects A, B on e can form the 'exponential'

object B A of all mappings A ~ B.(iv) There is an 'object of truth values' n such that for each

object X there is a natural correspondence betweensubobjects (subsets) of X an d arrows X ~ n. (For 0 wemay take the set {O, I}; arrows X ~ 0 ar e then characteristicfunctions on X, an d the exponential object OX correspondsto the power set of X.)

All four of th e above conditions ca n be formulated in purely categorytheoretic (arrows only) language: a (small) category satisfying them iscalled a topos.Th e concept of a topos is in fact much more general than that of(the category of sets in) a model of set theory in th e original sense.This is revealed by the fact that, in addition to S, all of the followingare toposes: (1) the category VB) of Boolean-valued sets an d mappings within a Boolean extension (d . [1]) of a model of set theory; (2)the category of sheaves (or presheaves) of sets on a topological space;(3) th e category of all diagrams of mappings of sets

X O ~ X l ~ X 2 ~ " '.Evidently the objects of each of these categories may be regarded assets which ar e varying in some manner: in case (1) over a Booleanalgebra, in case (2) over a topological space an d in case (3) overdiscrete time. (So, in this parlance, the category S itself is the categoryof sets "varying" over the on e point set 1.) These examples suggestthat a topos may be conceived of as a category of variable sets: thefamiliar category S we started with is the "limiting" case in which thevariation of the objects has been reduced to zero. Fo r this reason, S iscalled a topos of constant sets. Thus, the category-theoretic formulation of the s et concept - the notion of topos - turns ou t to be, asdoes th e notion of category itself, another consequence of the dialectical procedure of replacing th e constant by the variable.In any model of set theory on e has natural "logical operations" 1\


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414 J. L BELL(conjunction), v (disjunction), 'I (negation), ~ (implication), definedon the object of truth values 2 = {a, I} corresponding in the usual wayto the set theoretical operations n, U, - , ::} of intersect ion, union,complementation and residuation. The richness of a topos' internalstructure enables this correspondence to be carried through there aswell. Thus, in any topos E we ge t natural arrows defined on its objectnE of truth values which may be thought of as internally defined"logical operations" in K Since these logical operations are definedentirely in terms of the internal mathematical structure of E, a toposmay be regarded as an apparatus for synthesizing logic from mathe-matics. The remarkable thing is that the logic so obtained is, ingeneral, intuitionistic; in other words, the logical algebra fiE =(OE, II, v, I , : :}) is a Heyting algebra (d. [9]). For example, when E isthe topos Shv(X) of sheaves on a topological space X, 0E is theHeyting algebra of open sets in X. On the other hand, when E is aBoolean extension y(B), OE is the Boolean algebra B, so in this casethe internal logic of E is classical.The fact that proposi tional logic is interpretable in any topos E isreally only the beginning. For by employing the "internal completeness" of the truth value object n one can provide interpretations ofthe quantifiers V and 3 and so enable statements of first-order logic to

become interpretable in K Moreover, by exploit ing the presence, forany object A of E, of the exponentials nA , ni lA, etc., which may beregarded as representing the collect ions of propert ies, propert ies ofproperties, etc., defined over A, we find that statements of higherorder logic become interpretable in E, jus t as in an ordinary model ofset theory. (And then the truth value object n of E represents thedomain of possible "truth values" of such statements in K) In fact, itcan be shown (d. [7]) that toposes are precisely the models fortheories formulated within a natural typed higher order languagebased on intuitionistic logic. Each topos E is associated with such alanguage whose types match the objects of E and whose functionsymbols match the arrows of K A theory in such a language is a set ofsentences closed under intuitionistically valid deductions. Given sucha theory T, a topos E T can be constructed which is a model of T, andconversely, given a topos E, we can form a theory TE (the set ofsentences "true" in E) whose associated topos E T E is categoricallyequivalent to KThus higher-order typed intuitionistic theories (which we shall call


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FROM A BS OL UT E T O LOCAL MATHEMATICS 415simply "theories") and toposes are essentially equivalent. Each toposE may be regarded as being formally described by its associated theoryT E, and each theory T as being concretely realized or embodied by itsassociated topos E T A theory T may be regarded as a generalized settheory and a topos which is a model of T as a local universe ofdiscourse within which the mathematical assertions made by Taretrue and the constructions sanctioned by T can be carried out.

Two constructions of paramount importance in mathematics arethose of the natural numbers and real numbers. In set theory theconstruction of the set of natural numbers is sanctioned by the axiomof infinity, and the set of real numbers then essentially obtained as thepower set of the set of natural numbers. The procedure in topos theoryis similar, except that the axiom of infinity is replaced by the so-calledPeano-Lawvere axiom (d. [9]) which asserts the existence of an objectof natural numbers characterized by the universal possibility ofdefining functions on it by recursion. Given such an object N (whichcan be shown to be unique up to isomorphism) in a topos E, the objectof real numbers (the counterpart within E of the set of reals) may thenbe defined in terms of the exponential object nN by imitating the usualclassical procedures (i.e., Cauchy sequences or Dedekind cuts: butnote that , in contrast with the classical case, in a genera l topos thesetechniques may lead to non-isomorphic results!).

Henceforth I shall use the term local (mathematical) framework fortopos with an object of natural numbers. These local frameworks nowbecome the generalized models of set theory or local universes ofdiscourse to which mathematical concepts are to be referred: thusarises the local interpretation of mathematical concepts. Analogously,the theories associated with these local frameworks are the generalizedset theories in which mathematical constructions and assertions are tobe codified.

Return for a moment to "the case of the topos (now a local framework) Shv(X) of sheaves over a topological space X. Thinking of theHeyting algebra O(X) of open sets as a domain of truth values, it isnatural to regard Shv(X) as the generalized model of set theory"generated" by this domain of truth values. (When X is the one-pointspace 1, O(X) is essentially the two element set {O, I} so, since Shv(l)is a topos S of constant sets, the latter is, as one would confidentlyexpect, the model of set theory generated by the classical truth valuedomain {O, I}.) It is also suggestive (and consonant with the origins of


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416 J. L. BELLthe concept of topos) to regard Shv(X) as the generalized topologicalspace obtained from X by building a set-theoretical st ructure on theopen sets of X. With this idea in mind, any topos (local framework!)may also be conceived as a generalized space as well as a generalizedmodel of set theory. (This is the viewpoint suggested by algebraicgeometry: the original source of the concept of topos.) The resultinginterplay of topological and set-theoretical concepts is a key feature oftopos theory.I have emphasized that it is in the spirit of category theory to regardno framework as absolute. This tendency is realized by the possibilityof moving from one category to another via the concept of functor. Afunctor may be regarded as a transformat ion between categories thatpreserves their basic categorical structure. Now when the categoriesconcerned are local frameworks (toposes!), there is a st ronger notionof transformation available, which we shall call admissible or con-tinuous transformation. Formally, an admissible transformation f: E ~F between a local framework E and a local framework F is a pair offunctors f*: E ~ F, f*:F ~ E where f* is right adjoint to f*, which inturn is left exact. (Topos theorists will note that this is the opposite ofa geometric morphism.) In the "geometric" case where E and Farethe toposes of sheaves over topological spaces X and Y and we thinkof E and F as generalized spaces, the admissible transformationsbetween E and F correspond exactly to the continuous maps from Y toX: this is the source of the term "continuous". The functors r, f* arecalled the components of f. I f there is an admissible transformationbetween a framework E and a framework F, then F is said to bedefined over E. This terminology is suggested by the fact that, if S is aframework of constant sets and f: S ~ F is any admissible transformation to a framework F, then the components of f are given by(for I, X objects of S, F respectively):

rU) = I-fold copower (disjoint union) of 1 in F; f*(X) = setin S of "elements" of X, i.e., arrows 1 ~ X in F.That is, we may think of rU) as the "representative" in F of theconstant set I and f*(X) as the "extent" or "projection in S" of the"variable" set X.

The possibility of shifting via admissible transformations from onelocal framework to another is central to the interpretation of mathematical concepts proposed here, and points up its essentially kinetic


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FROM ABSOLUTE TO LOCAL MATHEMATICS 417and relational character. In this connection one is struck by theevident analogy with the physical geometric notion of change ofcoordinate system. And indeed, just as in astronomy one effects achange of coordinate system to simplify the description of the motionof a planet, so also it proves possible to simplify the formulation of amathematical concept by effecting a shift of local mathematicalframework. Consider , for example, the concept "real-valued continuous function on a topological space X" (interpreted in a topos S ofconstant sets). Any such function may be regarded as a real number(or quantity) varying continuously over X. Now consider the toposShv(X) of sheaves over X. Here everything is varying (continuously)over X, so shifting to Shv(X) from S essentially amounts to placingoneself in a framework which is, so to speak, itself "moving along"with the variation over X of the given variable real numbers. Thiscauses the variation of any variable real number not to be "noticed" inShv(X) ; it is accordingly there regarded as being a constant realnumber. In this way the concept "real-valued continuous function onX" is transformed into the concept "real number" when interpreted inShv(X). (To be strictly precise, the objects in Shv(X) satisfying thecondition of being (Dedekind) real numbers correspond, via theadmissible transformation S ~ Shv(X), to the real-valued continuousfunctions on X.) Putting it the other way around, the concept "realnumber" interpreted in Shv(X) corresponds to the concept "realvalued continuous function on X" interpreted in S. This observationprovides the basis for various proofs of independence in intuitionisticanalysis (analysis interpreted in Shv(X): d . [8]).We give two other examples of this procedure. Let B be a comple teBoolean algebra of commuting projections on a Hilbert space H.Then the real numbers in VB), the Boolean extension of V by VB),correspond to those self-adjoint operators on H whose spectral components lie in B (d. [16]). This provides the basis for Takeuti andDavis's approach ([6]) to the foundations of quantum mechanics inwhich they suggest that "quantizing" a statement of classical physicsamounts to interpreting it in VB). Finally, if B is the reduced measurealgebra of a measure space Z, then the real numbers in VB) corresponds to measurable functions on Z. This yields Takeuti's Booleanvalued analysis according to which real analysis interpreted in y


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418 J. L. BELLtransforms the concept "real number" into the concepts "continuousfunction", "measurable function" or even "self-adjoint operator".This provides striking testimony that under the local interpretation amathematical concept possessing a fixed sense now inevitably has avariable reference. Indeed, we may think of the sense of the concept of"domain of all real numbers" (i.e., the continuum) as being fixed by itsdefinition within an appropriate theory, while, as we have seen, itsreference varies with the local framework of interpretation. Thisresolves, or rather dissolves, the dilemma of classical set theory inwhich a concept such as "domain of all real numbers", although surelyintended to possess a unique reference in fact cannot because of thefirst-order formulation of its definition within the language of settheory. The local interpretation not only accepts this variability ofreference but welcomes it and assigns it a central position.

2. SOME ANA LOGIE S W IT H THE THEORY OF RELAT IV ITYThe local interpretation of mathematical concepts, based as it is oncategory theory, has an essentially relational character. According tothe local interpretation, the. reference of a mathematical concept,insofar as it can be const rued as an entity, is no longer to be regardedas being a thing in itself, whose nature is independent of other things,and whose characteristic properties are entirely intrinsic to it . On thecontrary, the propert ies of a mathematical ent ity are now determinedby, and indeed only have meaning in terms of, the totality of itsrelationships with other entities.

The recognition that properties originally held to be intrinsic mustinstead be treated as relat ional has arisen frequently in the history ofthought. For example, Leibniz recognized that a state of rest ormotion of a material body is not an intrinsic s ta te of the body but onlyhas meaning in relation to other bodies. One of the profoundestins tances of this phenomenon arose in the transition from classical(Newtonian) to relativistic physics, when physical concepts such assimultaneity of events and mass of a body formerly ascribed anabsolute meaning were seen to possess meaning only in relation tolocal coordinate systems.

There is an evident analogy between local mathematical frameworks and the local coordinate systems of relativity theory: each serveas the appropriate reference frames for fixing the meaning of mathe-


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FROM ABSOLUTE TO LOCAL MATHEMATIC S 419matical, or physical concepts respectively. (And we have alreadymentioned the analogy between admissible transformations of frameworks and changes of coordinate system.) Pursuing this analogy suggests certain further parallels.Thus, for example, consider the concept of invariance. In relativisticphysics invariant physical laws are statements of mathematical physics (e.g., Maxwell's equations) which, suitably formulated, hold uni-versally, i.e. , in every local coordinate system. Analogously, invariantmathematical laws are mathematical assertions which again hold universally, i.e., in every local mathematical framework. These in fact

turn out to be the theorems of higher order intuitionistic logic with anatural number system, which include, for instance, the theorems ofintuitionistic arithmetic but not the axiom of choice or the law ofexcluded middle. Thus the invariant mathematical laws are thosewhich are demonstrable constructively: this points up the significanceof constructive reasoning for the local interpretat ion. Notice in thisconnection that a theorem of classical logic which is not constructivelyprovable (e.g., the law of excluded middle) will not in general holduniversally until it has been transformed into its intuitionistic correlate(which, e.g., in the case of the excluded middle A v -,A is -,-,(A v-, A. The procedure of translating classical into intuitionistic logic(see, e.g., [5]) is thus the counterpart of casting physical laws intoinvariant form.The physical concept of inertial coordinate system also has itscounterpart in the local interpretation of mathematics. An inertial

coordinate system is one in which undisturbed bodies undergo noaccelerations, i.e., in which Newton's first law of motion holds. Thusinertial coordinate systems act as surrogates for Newtonian absolutespace. Analogously, a classical local mathematical framework is onein which objects undergo no variations (i.e., are "constant"), in otherwords, one which resembles the classical universe of constant sets asclosely as possible. This resemblance can be ensured by the satisfaction of the axiom of choice suitably formulated in a strong categorical form (see, e.g., [3]). For the truth of the (strong) axiom ofchoice in a framework E implies not only that the internal logic of E isclassical and bivalent (i.e., there are only two truth values "true" and"false") but also that the arrows of E resemble set-theoretic mappingsin that they are determined by thei r act ion on "points" of theirdomains. These features may be taken as distinguishing frameworks of


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420 J. L. BELLconstant sets among arbitrary frameworks. Since the axiom of choiceensures the presence of these features, we may define a classicalframework to be one in which this axiom is satisfied while at the sametime observing that it is an accepted principle of classical set theory.Accordingly, classical local frameworks correspond to inertial coordinatesystems and the axiom of choice to Newton's first law of motion.

These observations suggest the idea that the local interpretation ofmathematical concepts bears the same relation to classical set theoryas relativity theory does to classical physics.3. THE NEGATION OF CONSTANCY

We have remarked that the transition from the notion of model of settheory to the notion of topos (local framework) is an instance of thedialectical process of replacing the constant by the variable. A par-ticularly striking form of this process arises when the transition ismade by an admissible transformation. Suppose that we are given aclassical local framework S (i.e., one satisfying the axiom of choice:the objects of S may then be regarded as being "constant"), and alocal framework E defined over S, i.e., for which there is an admissibletransformation S ~ E. We may regard E as a framework which resultswhen the objects of S are allowed to vary in some manner. (Forexample, when E is Shv(X), the objects of E are those varyingsmoothly over the open sets of X.) Thus, in passing from S to E wedialectically negate the "constancy" of the objects in S, and introduce"variation" or "change" into the new objects of E. In passing from aclassical framework to a framework defined over it we are, in short,negating constancy.Now in certain important cases we can proceed in turn to dialectically negate the "variation" in E to obtain a new classical frameworkS* in which constancy again prevails. S* may be regarded as arisingfrom S through the dialectical process of negating negation. Ingeneral, S* is not equivalent to S and is therefore, according to awell-known result of topos theory (ct. [4] or [10]) not defined over S,so that the second "negation", i.e., the passage from E to S*, cannotbe an admissible transformation (but it is a functor, in fact a "logical"functor). Thus the action of negating negation in this sense transcendsadmissibility. This is the price exacted for reinstating constancy inpassing to S*.


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FROM ABSOLUTE TO LOCAL MATHEMAT ICS 421To establish the importance of this process of negating negation we

show that it underlies two key developments in the foundationsof mathematics: Robinson's nonstandard analysis and Cohen 's in -dependence proofs in set theory. (The discussion here owes much toLawvere's seminal article [12].)Fix a classical framework S, which we shall think of as consisting ofconstant sets: we shall reserve the term "set" for "object of S".Given a set I, each element i E I may be identified with theprincipal ultrafilter U; = {A c I: i E A} over I. This identificationsuggests that we think of arbitrary ultrafilters over I as "generalizedpoints" of I. The collection of generalized points of I forms a new setf3I (the Stone-tech compactification of I) . Elements of I (nowidentified as a subset of f3I) are called standard points of I, andelements of f3I - I ideal points of I. I f I is infinite, ideal points alwaysexist.Now consider the local framework E = SI of sets varying over I.Objects of SI (which we shall call variable sets) are I-indexed familiesof sets X = (Xi: i E I) . An element of an object X is an I -indexedfamily (Xi: i E I) such that Xi E Xi for i E I, i.e., a "choice function" onX. Thus the Cartesian product fliEI Xi is the set of "elements" of thevariable set X.Each (constant) set A is associated with the variable set A given bythe constant function with value A. The set of 'elements' of thevariable set A is then AI.The framework SI is defined over S via the admissible transformation S ~ SI given by j*(A) = A, f*(X) = lliEIXi .Given an element io E I, we can arrest the variation of any variableset X by evaluating X at io, i.e., by considering Xi". I f we apply thisin particular to the set A l of "elements" of the variable set A, thatis, if we evaluate each such 'element' at io, we just retrieve A. So inthis case, if we negate the constancy of (the elements of) A by passingto the set A l of (variable) "elements" of A, and then negate thevariation of these "elements" by evaluating at a standard point of I,we come full circle. This instance of "negation of negation" is,accordingly, trivial. The situation is decidedly different, however,

when the evaluation is made at an ideal point of I.Given an ideal point U of I, i.e., a non-principal ultrafilter over I,how shall we "evaluate" functions in A l at U? To this end, observethat the result of evaluating at a standard point lo of I is essentially the


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422 J. L. BELLsame as identifying functions in AI provided their values at io coincide, i.e., stipulating that for f, g E A I ,f - g ~ f (io) = g( io)

~ {i E I: f (i ) = g( i)} E U ~ I 'We use this last equivalence as the basis for evaluating functions in AIat an ideal point U of I. That is, we define

f "


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FROM ABSOLUTE TO LOCAL MATHEMATICS 423steadily over P. I f we think of X(p) as the collection of elements ofthe variable set X secured at state p, then the "steadiness" conditionmeans that no secured elements are ever lost. For pEP and sets X,Y varying steadily over P we write

p U- Xs: Yfor

V q ~ p V x E X ( q ) 3 r ~ q [ X E Y(r)],that is, given state p, X is eventually contained in Y. We write

pl l -X= Yfor

p l l - X ~ Yand pll- Yc X,that is, given state p, X eventually coincides with Y.Two elements p, qE P are mutually consistent if 3rE P [ p ~ r &q ~ r], A set of mutually consistent elements of P is called a body ofknowledge. A body of knowledge K is said to be complete if whenever

pEP is mutually consistent with every member of K, then p belongsto K.Given a complete body of knowledge K, define the equivalencerelation - K on the collection of sets varying steadily over P byX- K Y ~ 3 p E K[pll-X= Y].

Thus, X - K Y means that our body of knowledge K yields theassertion that X and Y eventually coincide. The collection S* ofequivalence classes modulo -T of steadily varying sets forms a newclassical framework, in general internally distinguishable from S in thesense that it does not possess all the elementary proper ties of S: S* isin fact a (possibly nonstandard) Cohen extension of S.To recapitulate: the framework E was obtained by negating constancy in allowing variation ("growth") over states of knowledge, andthe Cohen extension S* obtained from (the steadily varying objects in)E by using a complete body of knowledge to determine the "eventual"

identities between the variable sets, in other words, to negate theirvariation.In the "Cohen extension" case passage from S to S* (negation ofnegation) preserves some of the principles associated with constancyof sets (e.g., axiom of choice, classical logic) but, as Cohen showed, P


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424 J. L. BELLmay be chosen in such a way - now familiar to every set theorist - soas to ensure that other such principles (e.g., axiom of constructibility,continuum hypothesis) are violated in this passage. In passing from Sto E (negation of constancy) the classical bivalent logic of S isreplaced by the intuitionistic logic of E. And passage from E to S*(negation of negation) restores classical logic and constancy but, as wehave remarked, not all principles associated with constancy.Now, we could have refrained from performing the return passageto constancy (i.e., the second "negation") and instead remained in theframework E of variable sets. The set-theoretic independence proofscan be obtained by scrutinizing the internal properties of E (moreprecisely, by employing the Scott-Solovay method of replacing E byits associated Boolean-valued framework). I f we agree more generallyto abstain from returning to constancy then some startling possibilitiesbegin to emerge. For example in certain more' radical choices of theframework E of variable sets (where the "sets" vary over a category ofrings in a natural way), the "real line" in E will conta in non-trivialsquare zero infinitesimals, i.e., real numbers =f 0 such that 2 = O. Insuch frameworks (d. [11] or [13]) every function defined on the realline is infinitesimally linear, hence smooth, and therefore correspondsto the motion of a classical body. In these circumstances one can thenproceed to develop the calculus along the lines of Fermat and Newton,with no mention of infinite processes or limits. But for this to bepossible we must remain within a framework of variable sets, resolutelyadopting a local viewpoint in which constancy and classical logic nolonger prevail.

4. SUMMARY AND CONCLUSIONMy proposal is that the absolute universe of sets be relinquished infavour of a plurality of local mathematical frameworks. Mathematicsinterpreted in any such framework is appropriately called localmathematics; an admissible transformation between frameworksamounts to a (definable) change of. local mathematics. The reference ofany mathematical concept is accordingly not fixed, but changes withthe choice of local mathematics.Constructive provability of a mathematical statement now means thatit is invariant or valid in every local mathematics. So, on this account,the use of constructive proof procedures, far from hobbling mathema-


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FROM ABSOLUTE TO LOCAL MATHEMATICS 425tical activity as (classical) mathematicians are wont to claim, hasinstead the opposite effect of extending the validity of mathematicalreasoning to the widest possible spectrum of contexts, including thosein which "variation" is taking place. The role of the axiom of choice isto eliminate as much of this "variation" as possible, ensuring that anyframework in which it is satisfied is sufficiently similar to the classicaluniverse of "constant" sets to allow classical mathematical (i.e.,set-theoretical) reasoning to become valid there.The replacement of absolute by local mathematics results, in myview, in a considerable gain in flexibility of application of mathematicalideas, and so offers the possibility of providing an explanation of their"unreasonable effectiveness" (cf. [17]). For now, instead of being

obliged to force an intuitively given concept onto the Procrustean bedof absolute mathematics, with the attendant distortion of meaning, weare at liberty to choose the local mathematics naturally fitted toexpress and develop the concept. To the extent that the given conceptembodies aspects of (our experience of) the objective world, so alsowill the associated local mathematics; the "effectiveness" of the latter,i.e., its conformability with the objective world, thus loses its"unreasonableness" and instead is shown to be a product of design.So the local interpretation of mathematics implicit in categorytheory accords closely with the unspoken belief of many mathematicians that their science is ultimately concerned, not with abstract sets,but with the structure of the real world.

REFERENCES

[1] Bell, J. L.: 1977, Boolean-Valued Models and Independence Proofs in Set Theory,Oxford University Press.[2] Bell, J. L. : 1981, 'Category Theory and the Foundations of Mathematics', BritishJournal for the Philosophy of Science 32, 349-358.

[3] Bell, J. L.: 1982, 'Categories, Toposes and Sets', Synthese 51, 293-337.[4] Bell, J. L.: 1982, 'Some Aspects of the Category of Subobjects of ConstantObjects in a Topos ', Journal of Pure and Applied Algebra 24, 245-259.[5] Bell, J. L. and M. Machover: 1977, A Course in Mathemat ica l Logic, NorthHolland, Amsterdam.[6] Davis, M.: 1977, 'A Relativity Principle in Quantum Mechanics', InternationalJournal of Theoretical Physics 16,867-874.[7] Fourman, M. P.: 1977, 'The Logic of Topoi'; in Barwise, J. (ed.), Handbook of

Mathematical Logic, North-Holland, Amsterdam.


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x bell - from absolute to local mathematics

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