pragmatic holism (or pragmatic reductionism)

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BRUCE EDMONDS PRAGMATIC HOLISM (OR PRAGMATIC REDUCTIONISM) ABSTRACT. The reductionist/holist debate is highly polarised. I propose an intermediate position of pragmatic holism. It derives from two claims: firstly, that irrespective of whether all natural systems are theoretically reducible, for many systems it is utterly impractical to attempt such a reduction, and secondly, that regardless of whether irreducible ‘wholes’ exist, it is vain to try and prove this. This position illuminates the debate along new pragmatic lines by refocussing attention on the underlying heuristics of learning about the natural world. KEY WORDS: pragmatism, holism, reductionism, modelling, proof, heuristics, learning, reduction, philosophy, science In the face of complexity, an in-principle reductionist may be at the same time a pragmatic holist. (Simon, 1981) 1. INTRODUCTION This paper is written from a practical perspective, namely the perspective of someone attempting to understand and control the world by modelling it using limited resources. From this perspective, if an abstract ‘truth’ does not give its possessor any leverage over the world then it is irrelevant. Also, if the application of this abstract ‘truth’ consistently requires a specific set of other practical considerations, then a practically orientated person would tend to use these considerations in place of the original ‘truth’. The reductionist/holist debate is highly polarised. Many of the participants are not seeking useful accounts of modelling or under- standing phenomena but are more concerned with defending posi- tions. The two camps have adopted distinct terminology, styles, journals, conferences and criteria for success and thus are largely self-reinforcing and mutually exclusive. Foundations of Science 4: 57–82, 1999. © 1999 Kluwer Academic Publishers. Printed in the Netherlands.

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Page 1: Pragmatic Holism (or pragmatic reductionism)

BRUCE EDMONDS

PRAGMATIC HOLISM (OR PRAGMATIC REDUCTIONISM)

ABSTRACT. The reductionist/holist debate is highly polarised. I propose anintermediate position of pragmatic holism. It derives from two claims: firstly, thatirrespective of whether all natural systems are theoretically reducible, for manysystems it is utterly impractical to attempt such a reduction, and secondly, thatregardless of whether irreducible ‘wholes’ exist, it is vain to try and prove this.This position illuminates the debate along new pragmatic lines by refocussingattention on the underlying heuristics of learning about the natural world.

KEY WORDS: pragmatism, holism, reductionism, modelling, proof, heuristics,learning, reduction, philosophy, science

In the face of complexity, an in-principle reductionist may be at the same time apragmatic holist. (Simon, 1981)

1. INTRODUCTION

This paper is written from a practical perspective, namely theperspective of someone attempting to understand and controlthe world by modelling it using limited resources. From thisperspective, if an abstract ‘truth’ does not give its possessoranyleverage over the world then it is irrelevant. Also, if the applicationof this abstract ‘truth’ consistently requires a specific set of otherpractical considerations, then a practically orientated person wouldtend to use these considerations in place of the original ‘truth’.

The reductionist/holist debate is highly polarised. Many of theparticipants are not seekingusefulaccounts of modelling or under-standing phenomena but are more concerned with defending posi-tions. The two camps have adopted distinct terminology, styles,journals, conferences and criteria for success and thus are largelyself-reinforcing and mutually exclusive.

Foundations of Science4: 57–82, 1999.© 1999Kluwer Academic Publishers. Printed in the Netherlands.

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An outsider who is attempting to understand difficult domains(such as living organisms, social systems, or cognition) byconstructing formal and computational models will find littleguidance from this philosophical debate. Such persons may bewondering whether their objects of study are amenable to theirmodelling techniques, and feel that the inventiveness and effort thathas gone into this philosophical debate means that it should beable to provide new insights and perspectives as to this question.However, typically they come away only with a feeling of frustra-tion. I believe that this is not primarily due to the complexity of thephilosophical issues or the inevitable intricacy of the debate, but dueto a more fundamental misdirection in the debate away from realis-able processes of knowledge acquisition. This may be due to the factthat the primary effort in this field is not directed at understandingthe issues but in arguing against the opposition. Here I will arguethat the concentration on such dogmatic positions centred aroundlargely abstract arguments is unproductive and irrelevant to actualenquiry. In this way I hope to play a small part in refocussing thedebate in more productive directions.

The paper is divided into two main sections: the first presents thebroad arguments and the second some more detailed and technicalexamples to illustrate and support the arguments in the first half.

Section 1: The Main Arguments

2. THE REDUCTIONIST/HOLIST DEBATE

2.1 Versions of reductionism

The scientific method is not a well defined one, but one that hasarisen historically in the pursuit of scientific truth. From this prac-tice some philosophers have abstracted or espoused “purer” forms ofideal scientific practice, some of which is epitomized in the reduc-tionist approach. It is around this approach that debate has largelycentred. There are many formalisations of reductionism. Here aresome examples:

• Any phenomenon can be arbitrarily well approximated by anexplanation in terms of microscopic physical laws

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• Every definable process is computable• Every causal process is syntactically formalisable• Every problem is effectively decomposable into sub-problems• The explanation of the whole can be made in terms of its parts

All of these are subtly different. However, they all epitomise a singlestyle of inquiry, that any phenomenon, however complex it appears,can be accurately modelled in terms of more basic formal laws. Thusthey are rooted in an approach to discovering accurate models of thenatural world by searching for relatively simple underlying laws.These formulations of reductionism range from abstract questionsof whether all real systems can be modelled in a purely formal wayto more practical issues about the sort of reduction performed inactual scientific enquiry.

In this paper I aim to show the irrelevance of the abstract ques-tion; that when faced with a choice of action it is a very similarrange of issues that face both the in-principle reductionist and thein-principle holist. So for the purposes of this paper I will take themore abstract definitions as my target. These are the least practicaland thus represent more clearly what I am arguing against. The morea criterion is practical the happier I am with it.

2.2 Weaknesses in the reductionist position

Foremost in the weaknesses of the reductionist position is that theabstract reductionist thesis itself is neither scientifically testable noreasily reducible to other simpler problems. Thus, although manyscientists take it as given, the question of its truth falls squarelyoutside the domain of traditional science and the reductionist thesisitself. Its strength comes from the observation that much successfulscience has come from scientists that hold to reductionism – it isthus a sort of once-removed inductive confirmation. Such inductivesupport weakens as you move further from the domain in whichthe induction was drawn. This certainly seems true when applied tovarious “soft” sciences like economics, where it is spectacularly lesssuccessful. The current focusing on “complex systems” is anothersuch possible step away from the thesis’ inductive roots.

A second, but unconnected support comes from the Church-Turing Thesis (CTT). Following Turing’s definition of effectivelycomputable functions in terms of a Turing Machine, many others

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proposed their own definitions, including Church. It later transpiredthat all of these definitions were equivalent. In addition to this manyother, seemingly unconnected, problems also turned out to be equiv-alent (e.g. tiling the plane, solving integer equations using integersand the game of ‘Life’). Out of these sorts of discoveries camethe hypothesis thatall effectively computable processes could simi-larly be shown to be equivalent to a Turing Machine. This becameknown as the Church-Turing Hypothesis. A good introduction to thishypothesis may be found in (Cutland, 1980). Often the strength ofthis thesis within mathematics is projected onto physical processes,since any formal model of a process we care to posit is amenableto that thesis. If you conflate reality with your model of it then thethesis appears reasonable, but otherwise not.

Thirdly, attempts to formalise actual scientific reduction in set-theoretic or logical terms, have tended to be unsatisfactory (seeSarkar, 1992).

2.3 Weaknesses in the holist position

Holist literature abounds with counter-examples to the reductionistthesis. Some of these are seriously intended as absolute counter-examples. They tend to fall into two categories: the practical (andso unproved in an absolute sense) and the theoretical but flawed. Anexample of the former is Rosen’s example of protein folding whichhe justifies as a counter-example to the Church-Turing Thesis (CTT)on the grounds that

. . . thirty years of costly experience with this strategy has produced no evidence ofthis kind of stability. . . despite a great deal of work . . . the problem is still prettymuch where it was in 1960. . . this is worse than being unsuccessful; it is simplycontrary to experience. . . . (Rosen,1991: 270)

This is a perfectly valid pragmatic observation, justifying the searchfor alternative approaches to the subject. It does not, of course,disprove the CTT and in itself supplies only weak support forextrapolations of this argument to broader classes (e.g. all livingorganisms).

There are many such examples. All the arguments raised wouldtake too long and distract from the purpose of this paper. Sufficeto say that all of these (that I have seen) seem flawed if intended

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as absolutecounter-examples. Section 7.2 looks in more detail atseveral of them in order to support my contention of their futility.

The basic trouble that the holist faces in arguing against reduc-tionism is that any argument is necessarily an abstraction. Thisabstraction is to different degrees formal or otherwise. To the degreethat it is informal it allows equivocation and will not convince askeptic. To the degree that it is absolute/formal it comes into thedomain of mathematics and logic where the Church-Turing thesisis very strong. While informal arguments can be used with otherholists, in order to argue with a reductionist a more formal argumentseems to be required (despite the fact that this does not seem to berequired of the reductionist stance by reductionists).

It appears that it is a necessary limitation regarding the natureof expression itself that makes any suchcompletedemonstrationimpossible. For if a holist wishes to produce an argument whichis completely formalised, so that it can be demonstrated step-by-step to a skeptic, how can they avoid reducing their argument toa syntactic proof system? They are faced with presenting a systemwhose property of irreducibility is itself reducible – not an easy task!

2.4 Irrelevances to the debate

Associated with the debate on the absolute question, but not centralto it, are a host of old-chestnuts that have not been shown to berelevant, but are often assumed to be crucial. There is not spaceto deal with them all or thoroughly enough to convince a believerof their inadequacy, but I list some of the more frequent of thembelow. Despite their inability to determine the absolute question,they each have strongpractical consequences. I will argue that itis these practical consequences which are important.

2.4.1 DeterminismWhether natural systems are deterministic or not, in an absolutesense, seems to be an untestable question. Both the deterministicand non-deterministic viewpoints adequately describe the macro-scopic world. Thus the abstract question of whether such systemsare deterministic must be irrelevant to the abstract question of thevalidity of the reductionist thesis (unless this question is also asuntestable and irrelevant as that of determinism).

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Artificial situations can be categorised as deterministic or other-wise. For example in a board game (such as chess) your next movemay be completely determined by the rules or you may have achoice. Within the framework you are considering, there is eithera mechanism for determining your move (i.e. the player’s strategyfor the game) or not. If there is, then the move is determined bythat, if not, it is undetermined – i.e. there is simply not enoughinformation to determine this by any process (mechanical or other-wise). Note that whether the move is determined depends on theframework you are considering, but within any particular framework(however general) is not relevant to theabsolute question of whetherthe situation is reducible or not.

However, this does highlight the practical importance of choosingthe appropriate framework for a problem. The framework greatlyeffects the practicality of modelling a system, including themodel’s relevance, usefulness and complexity (Edmonds, 1999).In the above example in a framework which includes the players’strategies, it may be possible to model the game but this may beimpractical if you choose merely the rules as your framework.

2.4.2 Noise and randomnessNoise can be characterised as a random input into a system’sprocesses. Such randomness can be defined in several ways. It canbe any sufficiently variable data which originates from outside thescope of a system’s model of its world, and is thus unpredictable.It can be data which passes a series of statistical tests. It can be apattern which is incompressible by a Turing machine.

In any case there are fully deterministic processes which producesequences that are practically indistinguishable from random onesfrom almost any particular system’s point of view (it is possible toconstruct a Turing machine that simulates the series of other Turingmachines in order to induce the source code of any pseudo-randomsequence, but it is undecidable whether the TM that matches it isthe actual one or whether it might start to deviate in terms of outputsome time later). Thus a system with noise can be simulated by amodel with the addition of such a process, as long as that systemdoes not have access to the workings of that process. In the oppositedirection a noiseless system can be arbitrarily approximated by

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one that has noise, by suitable redundancy in its construction (VonNeumann, 1986). This is how we maintain information in digitalcomputers and our own genome.

Since both noisy and noiseless systems can effectively simulatethe other, any system could be implemented by either. Thus thequestion of the presence of noise can not be relevant to the absolutequestion of a system’s reducibility.

Of course, this does not prevent such noise having an immensepractical impact on any attempt at modelling. For example, Gaines(1976) shows that the size of required finite automata models of asimple Markov process can increase indefinitely in the presence ofnoise.

2.4.3 Particular formal languagesOne particular bugbear of holists is classical two-valued logic.This is criticised as not being expressive enough to capture all themeaning or reasoning necessary to model some systems. Some-times it is Zermelo-Frankel set-theory that is the target. For exampleRosen (1991) suggests an approach to modelling in terms ofcategory theory as a possible way forward in modelling complexbiological systems presumably because he thought this would bemore expressive in some way. Often it seems that the importance ofwhethera formal system is applicable is based on a shallow readingof the formal system’s immediate properties and passes over whatfurther expressive features can be formalised within it.

In fact the choice of formal system is not critical in absoluteterms, as long as the system is expressive enough. For examplecategory theory and set theory can each be used to formalise theother (Marquis, 1995). Similarly classical first order logic can beused to formalise almost any other logic (Gabbay, 1994). So thereis no fundamentalabsolute groundsfor preferring one such formalsystem to an other.

However, there will be very strong practical grounds for thechoice of system, like tractability or a wish for a natural inter-pretation. In practice choosing the appropriate formal language candetermine the success or failure of a modelling technique, since theease and naturalness of formalisations can greatly vary.

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2.4.4 Analogue vs. digitalThere is a basic difference between what can be theoreticallymodelled using analogue and digital formalisms. For example(Sieglmann, 1995), even here any practical computation whoseresult is measurable by us would not transcend a Turing Machine.Sometimes this is on the grounds of the importance of noise (for thissee Section 2.4.2 above).

Thereis an essential difference between analogue and digital inthe abstract but you can not encode all analogue values as digital,only some of them. This can be proved with a classical diagonalargument. This means that literally we can not talk about mostanalogue values, except as a collective abstraction (“let x be areal number . . .”), as there are no finite descriptions of them. Thedigital and analogue can arbitrarily approximate each other, thus thecolour of a pixel on a VDU is composed of different wavelengths(analogue), which are encoded by the computer as binary numbers(digital), which are encoded as voltages in circuits (analogue),which correspond to quantum energy levels (digital).

The natural world may, at root, be analogue or digital (or neither),we do not know. Even with matter and energy one could argue thatthe quanta are a result of observing a continuous wave function.Thus arguments which rely on a fundamental difference betweenthe analogue nature of reality and the digital nature of formal-isms and the modern computer, will not be relevant to the absoluteholist/reductionist debate. This is not to say that either simulatingthe analogue by the digital (or vice versa) does not present consider-able practical difficulties. Thus the choice of whether to simulate aparticular phenomenon by digital or analogue means can be crucialto the practical success of the enterprise.

3. PRACTICAL LIMITS TO MODELLING

In section 7.1 I give four examples to support my thesis that therewill be systems that are practically irreducible. The other half ofmy thesis is that one can not prove practical limits to modellingany specific problem (given an absolute characterisation of truth).However there are many general practical limitations, some ofwhich I discuss below.

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3.1 Finiteness

It seems we (us and our tools) are part of a finite universe, and arethus also finite. Any model we make, use or understand will alsobe finite (Although it may well be most useful to represent or talkabout an abstraction of this as infinite). Quite apart from this ourformal communications (written articles) are definitely finite. Thusanypracticallyuseful model that we want to share will also be finite.

In these circumstances the fact that a Turing machine (which isessentiallyinfinite in its definition) could compute something, maynot be relevant if the mapping from this abstraction to an actualcomputer may mean that the computation is impractical. Thus theabstract question of the CTT is superseded by the question of thepracticalities of modelling.

3.2 Limited computational resources

As well as limited memory we also have a limited time to do thecomputations in. It has been calculated that quantum mechanicsimposes a limit of bits/gram/sec on the amount of information thatcan be computed by each gram of matter per second (Bremermann,1967). Taking a conservative guess at the total mass in the universeas grams and the total time before the heat death of the universeas years this gives us an upper limit of about total bits of compu-tation that could possibly occur in this universe. This would beinsufficient to even investigate the possible colourings of a 12 by12 checker-board using just 10 colours.

Thus problems which take undue computational time come upagainst a fairly fundamental computational limit, even if they aretheoreticallycomputable. It is possible that quantum computationmay get around this limit, but the practical limitations of thisapproach are not yet clear.

3.3 Complexity

Computational Complexity is concerned with the computationalresources required, once the program is provided. It does not takeinto account the difficulty of writing the program in the first place.Experience leads me to believe that frequently it is the writing ofthis program that is the more difficult step.

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More fundamental than computational complexity is what Iwould call “analytic complexity”. This is the difficulty of analysing(producing a top-down model) of something, given a synthetic(bottom-up) model (Edmonds, 1999). Whether or not this difficultyis sometimes ultimate, few people would deny that such diffi-culties exist at arbitrary levels. Given that our analytic capabilitieswill always be limited, such complexity will always be a practicalbarrier to us (for a classic account of the inevitable emergence ofcomplexity in biological systems see Wimsatt, 1972).

3.4 Context

Not all truth can be expressed in a form irrespective of context.The very identity of some things (e.g. society) are inextricablylinked to context. Thus we will have to be satisfied that, for atleast some truths, it will not be practical to try and express themin a very general context and hence acquire the ‘hardness’ of more“analytic” truths (like “all bachelors are men”). It is true that we canlaboriously express larger and larger meta-contexts encompassingsub-contexts, but this will involve the construction of more andmore expressive languages (e.g. as found in Barwise and Perry,1983) and require disproportionately more computational power –this will make this sort of endeavour impractical, beyond a certainlevel (see Yates, 1978 for an overview of this problem in biology).Choosing an appropriately restricted context is one of the mostpowerful means at our disposal for coping with otherwise intractablesituations.

4. THE NUMBER – COMPLEXITY ANALOGY

Rosen introduces an analogy between what he calls “complexity”(i.e. things that aren’t mechanisms) and infinity; the reductionist/syntactic approaches to modelling correspond to finite steps. Heclaims that many systems (including all living organisms) areunamenable to such steps and qualitatively different – they corre-spond to infinity. Thus he postulates that to model these “complex”systems require some transcendental device, like taking limits orsome form of self-reference.

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I wish to alter this analogy and hopefully deepen it. I wishto draw an analogy between size as represented numerically andcomplexity. In this, size corresponds to the difficulty of modellinga system in a descriptive top-down fashion given a language ofrepresentation and almost complete information (model) from thebottom-up perspective of it components (Edmonds, 1999). Thusinfinite size would correspond to an infinite such difficulty – i.e.impossibility of such modelling (which roughly corresponds toRosen’s “complexity”). The abstract debate would then correspondto the question “Are there systems with infinite complexity?”.

Here we need to examine what we mean by theexistenceofsuch systems. The problems of showing that such systems exist areremarkably close to those involved in showing that infinityexists.You can not exhibit any real manifestation of infinity, since theprocess of exhibiting is essentially finite. Even if we lived in auniverse that was infinite in some respect, you could not show acompleteaspect that was infinite, only either that an aspect appearedunboundedor that a reasonable projected abstraction of some aspectwas infinite.

Note that I am not saying that infinity ismeaningless, merelythat it is always an abstraction of reality and not a direct exhibitableproperty of any thing (for a discussion of this issue see Rotman,1993). That infinity is a very useful abstraction is undeniable – itmay be possible to formulate much of usable mathematics withoutit, but this would surely make such symbolic systems much morecumbersome. So when we say something is infinite, we are talkingabout an abstract projected property of our model of the item,evenif the thing is, in fact, infinite. It is just theexhibiting that is anessentially finite process.

I suspect that the same is true of the irreducibly complex. Alanguage of irreducible “wholes” is useful in the same sense thatinfinity is useful, but only as an abstraction of our model,irre-spective of whether these “wholes”, in fact, exist. If they do notexist, the language of the holist is still useful as an abstract short-hand for systems whose complexity is potentially unbounded. Ifthey do exist the language of “wholes” would still be necessarilyabstract, i.e. not referring to direct properties of real things,even if

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the systems referred to were irreducible. It is that exhibitingsuchsystems (especially formally) is essentially a reductive process.

5. FORMAL MODELLING AND REDUCTIONISM

The computability of a well-defined function is a purely formalquestion. A function is computable if an indexexistssuch that auniversal Turing machine (UTM) with that index calculates thatfunction. For example, consider an enumeration of halting comput-able programs in order of size. Every function represented byprograms in this class is computable without us being able tocompute (or “write”) the programs for these functions, otherwisewe could use this enumeration to compute the “halting problem”.See Turing (1936) or Cutland (1980) for a more accessible account.

This shows that the theoretical computability of a function doesnot imply that we have the means to find the program to computeit. In other words, the characterisation of reducible as computabilityis too strongfor actual use in reducing a problem. To be able toreally computesomething we have to be able to follow the instruc-tions (or have a machine do it)andwrite the program to do this (orreally computethat). This can form a very long chain (the programthat computes the program that computes the program that . . . ),but eventually it has to be grounded in a program we are ableto write ourselves,if the final program is to carry out our inten-tions. I call this “directed modelling”. This is what we can compute(using computers as a tool), as opposed to whatcan theoreticallybecomputed by a Turing machine.

If we take a pragmatic view of reductionism only as suchdirectedmodelling, then we come to the surprising conclusion that theremay be some computable functions thatwe can’t compute (inten-tionally), even by using a computer. We may, of course, comeacross a few of these programsaccidentally, for example by geneticprogramming, but we can not, in general, be certain that these‘ready-made’ programs are the ones we wanted since verifyingthat these programs meet a specification is itself uncomputable ingeneral (although you may be able to analyse a few such particularexamples sufficiently).

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6. HEURISTICS IN THE SEARCH FOR TRUTH

So whichever is our belief about the abstract reductionist/holistquestion, we are left with very similar pragmatic choices of actionwhen faced with an overly complex problem. Here reductionist tech-niques will be of little practical value forus as limited beings andwe have to look to other alternatives if we want to make progresson them. Whether you choose another (possibly less successful)approach, depends upon a trade-off between the difficulty of reduc-tion and the importance of progress (of what ever kind) being madeon that problem. In the end, the biggest practical difference betweena reductionist and a holist is often only that a reductionist thenchooses another problem (where the reductionist technique mighthave more chance of success) and the holist chooses alternativeavenues of attack upon the same problem.

The point is that there isno necessity to prejudge this decisionfor every case, neither toalwayssay that alternative types of knowl-edge are worthless, despite the importance of a problem nor to saythat it isneverworth abandoning a problem because of the type ofknowledge that is likely to be gained about it. I call these theextremereductionistandextreme holistpositions respectively.

To hold to the extreme reductionist position in a practical sense,one must surely claim that no problem is so much more importantthan other more susceptible problems to be worth swapping the sortof analytic knowledge that results from reductionist approaches forother types of knowledge. This can be a result of one of severalsubsidiary claims:

1. that all problems are practically susceptible – this would amountto denying any practical limitations upon ourselves at all;

2. that there is always a near indefinite supply of equally importantproblems – denying any real difference in the importance ofproblems, regardless of circumstance;

3. or that alternative forms of knowledge are always effectivelyworthless – presumably including the reductionist thesis itself!

To hold to the extreme holist position in a practical sense, onewould have to claim that either there was no advantage to reduc-tionist knowledge as compared to other types of knowledge inany circumstances or that a problem wasso important that it was

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not appropriate for anyone to research any other, more amenable,problems.

These would both be extreme positions indeed! I know of no onethat holds them in these forms. The rest of us fall somewhere inbetweenin practice: we accept that there are some worthwhile prob-lems where the reductionist technique works well and we also acceptthat there are problem domains where the chances of a reductionisttechnique working are so remote and the problem so important thatwe value other forms of knowledge about it.

This does not mean that we will all have the same priorities inparticular cases, just that these decisions are essentiallypragmaticones differing in degree only. Once attention switches from thesterile abstract question of whetherin principle all problems areamenable to a reductionist approach, we can start to consider therich set of possible strategies for making such choices in differentcases. Sarkar (1992) suggests that there are several different types ofmodels of reductionism, so our model of reductionism itself must bechosen pragmatically. This has been up to now a largely unchartedarea, but one that might pay rich dividends.

7. CONCLUSION – COMBINING A PLURALITY OF TECHNIQUES

The conclusion is thus merely that you need to think about theapplicability of reductionist techniques before using them (justlike other techniques)regardless of the answer to the abstractreductionist/holist question.

Such an awareness opens up the possibility of a more systematicstudy of ways of combining techniques for the successful elicitationof knowledge. Ways which would include developing more effectiveways of using different types of knowledge to further guide andrefine that search. Such practical heuristics are maybe all that wefinally have – rejecting both blinkered single-strategy approachesand the extreme relativism ofanything goes.

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Section 2: Illustrative Examples

7.1 Examples of practically irreducible systems

In this section I describe four examples, which aim to show thatregardless of theirtheoretical reducibility, that we often have toapproach systems holisticallyin practice. Each of these are prac-tically irreducible in a strong way – they each go beyond the merelydifficult, in that with each it is possible to see why the associateddifficulties might not be easily overcome even by new conceptualadvances.

7.1.1 Wolfram’s random number generator based on cellularautomataStephen Wolfram has presented the abstract systems of cellularautomata (CA) as archetypal models of complexity (Wolfram,1984). In CA simple rules act locally on a set of states to determinethe next set of states. This makes them credible approximationsof certain natural systems (e.g. spin glasses). Yet despite theirlocal simplicity they can produce intricate behaviour. One such CAproduces such complex behaviour that it has been put forward as aserious cryptographc mechanism (Wolfram, 1985). This CA has thesimple rule that acts on each neighbourhood of three states usingthe rule:x′i ← xi−1 XOR (xi OR xi+1). Yet when acting on a singlenon-zero bit (or almost any other initial configuration) it produces asequence of digits that are effectively random (Wolfram 1986). Thatis, it is postulated that there is no algorithm that would compute thesequence of resulting bits other than by effectively re-running thisCA on the initial configuration (or worse), and that there is no wayof inferring this initial configuration from the resulting sequence ofdigits short of running the CA on many initial configurations until amatch is obtained.

If these postulates are correct, then it is possible that a naturalsystem could be built of simple components such that it accuratelymimics the behaviour of this CA. The detailed behaviour of thissystem would be effectively irreducible, since any reproduction ofits behaviour in a formal or computational system would have to(essentially) mimic its behaviour. In other words no reduction of thesystem to a simpler model would be effectively possible.

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7.1.2 Chaitin’s irreducible polynomialChatin has produced an example that is similar to the above (Chaitin,1995). Chaitin defines a number,ϕ, that is defined as the ‘haltingprobability’ of a randomly selected Turing machine (TM). That is,it is the proportion of TMs that halt when randomly selected with aprobability that is inversely proportional to the power of its length.To calculate this number to, say,n places one can prove that onehas to essentially run all lengthn TMs overn steps (or do an equiv-alent and equally long calculation). This means that any programto calculateϕ is effectively random (e.g. it passes all the Martin-Löf tests for randomness). Then, using well-known computabilityresults he exhibits a specific polynomial with a single parameter,c,such that it has a finite number of solutions precisely when thecthdigit of ϕ is 1 (when written as a binary expansion).

The upshot of this is that he has constructed a polynomial whichhas anessentiallyrandom property (i.e. this property is algorith-mically random) – you can actually download the polynomial andprint it if you wish! In this way he wishes to show that at leastsome mathematical truths are not reducible to axiomatic simplifica-tion and categorization but may only be investigated by empiricalmethods. This is an indication that some natural systems mightbe at least as impervious to formal simplification as the abovepolynomial.

7.1.3 Computer Programs that Cannot be ProgrammedThe third example is a demonstration that not all computer programsare effectively programmable. This is due to the fact that the defi-nition of computability is not a constructive one, that is to say theusual meaning of “a function is computable” is merely that thereexistsa TM that will compute it, but it does not guarantee that thereis an effective mechanism to construct this TM (in other words noeffective means of constructing the program from its specification).

This can be demonstrated by considering a version of Turing’sfamous ‘halting problem’ (Turing, 1936). In this version theproblem is parameterised by a number,n, to make the limited haltingproblem. This is the problem of deciding whether a TM with lessthann states, and an input of less thann will terminate. This ‘length’is the base 2 logarithm of the sum of the TM index in a suitable

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enumeration of machines and the input given it. The definition ofthe limited halting problem ensures that for any particularn it isfully decidable (since it is a finite function {1 . . .n} × {1 . . . n} →{0,1}), so there is a TM to calculate it, call this TM(n).

However there is not a general and effective method of findingthe TM(n) that corresponds to a givenn. Thuswhat ever method(even with clever recursion, meta-level processing, thousands ofspecial cases, combinations of different techniques etc.) we have forconstructing TMs from specifications there will be ann for whichwe can not construct TM(n), even though TM(n) is itself comput-able. If this were not the case we would be able to use this methodto solve the full halting problem. Full details and a more completeformal proof may be found in (Edmonds, 1997).

An intriguing corollary of this is that if autonomous artificial soft-ware life ever does evolve within our computers (or on the internet),then it is possible that it will be at least as difficult to reduce asits more squashy cousins, despite the fact that it is composed ofprogram code. This possibility certainly seems to relate strongly tothe experience of trying to work out the explicit chains of causationthat occur in simulations of co-evolving agents (Edmonds, in press).

7.1.4 Social embeddednessElsewhere (Edmonds, in press), I have speculated that beingembeddedin a society might be bound up with aninability tocompletely model that society. In a society of individuals with aroughly commensurate ability to model each other, it can quicklybecome impossible for any member to understand that society in itsentirety. This is due to the fact that a complete understanding mustcover the other individuals’ understanding and only in exceptionalcases (such as those whereeverybodyhas a perfect and symmetricunderstanding) will that be possible. In the more usual circum-stance where an individual’s understanding of each other is limitedand uneven, it will often be advantageous for them to adopt lessperfect but more efficient modelling strategies. One such strategy isto use other individuals’ communications and actions as proxies foraspects of the society they inhabit. When a substantial proportionof the individuals in a society base their decision-making on others’actions and utterances (at least in part), the chain of reference and

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causation within that society can become partially self-referential.The original grounding of the individual’s models become lost in thesociety’s shared history, so that they only make sense in that context.These aspects of a society may, in time becomeentrenchedin thatsociety, due to the fact that agents use their shared recognition ofthem as a basis for further action (and so they are further delineatedas important modelling referents etc.). In this way these aspects thenacquire a recognised identity and a causal role in that society. Thisis the social equivalent of the generative entrenchment described byWimsatt (1972)

In such a context shared constructs that an individual holds abouttheir society may simultaneously beessentialin characterising thatsociety (in the sense that no good understanding of that society couldbe made without them) as well as having the property that if theycould be reduced (in the sense of completely understood) in terms oftheir historical origins they would cease to be characteristic of thatsociety. In other words, an essential property of that society mightbe aresultof its practical irreducibility.

This possibility has been demonstrated in a computational simu-lation of social agents. Thus it is possible that there are similarlyirreducible aspects ofour society.

7.2 Examples of supposed counter-examples to the CTT

Here I exhibit some examples to illustrate how attempts to provethat irreducible wholes exist are futile and how they come down tomore practical modelling concerns. Of course, I am not attemptingto provethat all such proofs are doomed to be flawed – that wouldbe almost as Quixotic as the examples themselves.

7.2.1 Modifying hardwareThe ability of an organism to modify its own “hardware”, forexample when a protein acts on its own DNA (e.g. to repair it) oracts to affect the interpretation of that DNA into proteins, is some-times compared to a Turing machine which cannot directly affect its“hardware” (as usually defined).

This separation of hardware and software is arbitrary unless somesort of “physicality” can be shown to be an important attribute,affecting what can be computed. Otherwise, there is nothing to

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stop a TM simulating such a change in hardware (including itsown). It is this possibility of indirect simulation of “new” capabil-ities, that is frequently mis-understood. For example, imagine a TMwhich could execute an instruction-type that could change one ofits own instructions. It would seem at first glance that this newmachine “goes beyond” the usual version, but this is not so. Anormal Turing machine can compute exactly the same functions asthis new “enhanced” machine, because although it cannot changeits own instructions, those simple instructions can be combined ina sophisticated way to simulate the computation of the enhancedmachine.

Several such “essential” characteristics of such physicality arepossible.

1. The presence of noise in analogue systems.2. A fundamental difference between matter and symbols (as in

Löfgren, 1968). This is closely connected with the problem ofmeasurement.

3. That arbitrarily small changes in the initial conditions havesignificant effects on the outcomes. The significance of thisis either that noise can then be significant or that due to itsanalogue nature you can never know the initial conditionssufficiently.

We deal with noise and chaos in other sections (Section 2.4.4 andSection 2.4.2 respectively). We are thus left with the argument asto a fundamental dichotomy between matter and symbols. Whetheror not this turns out to be a fundamental distinction, it is not clearwhy this would effect of the ability to modify one’s own hardware(or simulate such a modification in software). Sometimes this islinked to the distinction between syntax and semantics (for whichsee Section 2.4.8 below).

Of course the presence of the ability to modify one’s own hard-ware does present practical problems. Such an ability can make theoverall behaviour much more unpredictable using simple models,forcing one to move to more sophisticated models which are farmore computationally intensive.

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7.2.2 Self-referenceAnother way in which holists claim that there are systems that areabsolutely unameanable to formalisation is by exhibiting those thatinvolve some form of self-reference. An example of this is found in(Kampis, 1995).

One can distinguish two different forms of self-referencetotalandgrounded. Total self-reference is completely self defining, if youfollow the causal (or formal) chain backwards, you do not come to afixed atomic starting place but find an infinite recursion of definitionin terms of itself. Grounded self-reference starts at a specific place(i.e. state or set of axioms), and then the next state is defined interms of the last state etc., so after a while the current state is almostcompletely defined in terms of itself and the origin is lost for allpractical purposes.

Most examples of self-reference in the natural world would seemto be cases of grounded self-reference: life itself presumably startedfrom some point, which arose from non-living state (even if oneconsiders the property of ‘living’ to admit of varying degrees thearguments still holds, suitably amended); language is ultimatelygrounded in our shared experience, either directly as a child learnsits first language, or indirectly in the evolution of language in ourspecies (as suggested in Harnad, 1990); even the universe itselfseems to have passed through an initial equilibrial stage (Weinberg,1977).

It would seem that total self-reference is difficult to embed in atraditional formalisation, if only because in formalising somethingyou need somewhere to start from. This should not be confused withthe consistency of self-reference, which is relatively easy to estab-lish, as in (Löfgren, 1968). It is possible to dissect such a totallyself-referential system so that it is representable within a traditionalframework (e.g. the technique described in Kampis, 1995), but onlyby effectively grounding it. The most common way of doing this isto represent it in a suitable meta-language. Thus even if total self-referential formal systems exist they are only usable in modellingand easily communicable if grounded.

Grounded self-reference is formalisable by traditional formalsystems, even though in some cases this may be a cumbersomeand “unnatural” way to proceed. It is true that some such formal-

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isations are either resistant to, or do not have, accessible analyticsolutions that allow effective prediction of future (or even descrip-tion of past) behaviour, but this has always been true of even themost classical of formalisms and thus is not relevant to theabsolutereductionist/holist question.

Again whether we use traditional styled or (grounded) self-referential systems for modelling, relates not to abstract but topragmatic considerations. Self-reference can vastly increase theexpressive power of a language, and hence its presence can forceone to use more expressive formalisms. These formalisms are likelyto be relatively intractable and difficult to use. On the other hand, itmay be far easier to model a grounded self-referential system usinga formalisation that has an explicit self-referential mechanism.

7.2.3 SimultaneityIn most existing computation devices, computation proceedssequentially. Even parallel devices are usually arranged so that theircomputations are equivalent to a sequential computation. Likewisein almost all formal systems, facts are derived via an essentiallysequential proof. Even when the proof is not sequential in nature,its verification is.

Natural systems, however, often seem to work in parallel. VonFœrster gives an example of a box with many block magnets init (von Fœrster, 1960), the box is shaken and when opened theyare arranged in a very non-random way, resulting in an attractivesculpture to an exterior observer. The two views of the box, internaland external are simultaneous and different. It is claimed that suchsimultaneous and (in some cases) irreconcilable viewpoints meanthat a single consistent formalism of a meta-model incorporatingboth viewpoints is impossible.

If you have a parallel system there will be either someconflict avoidance or a conflict resolution mechanism (where by“conflicting” I mean exclusive). Of course, it is quite possible tohave cases where (as in the above box of magnets example) thereare views that appear to be conflicting, but you won’t have conflictswithin the same context, this is impossible if a consistent languageis used. Here it is not the simultaneity that is the problem but the

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reconciliation (or lack of it) of the same thing from within differentframeworks (see Section 3.4 below).

On the other hand, the difficulties of reconciling different simul-taneous streams means that often the only practical option is toaccept such different views as complementary rather than to try andcombine these, which may cause more problems than it would solve.

7.2.4 Going beyond a Turing MachineAn example of this is Fishler and Firschein (1987), where they givethe spaghetti, the string, the bubble and the rubber-band computeras examples of machines that “go beyond” the Turing machine.These examples follow a section on the Busy-Beaver problem,which is interpreted as being a function that grows too fast for anymechanistic computation. The examples themselves do not computeanything a Turing Machine can not, but merely exploit some paral-lelism in the mechanism to do it faster (if you take into account thepreparation of these devices the speed-up is considerably reduced).

The implication is that since these “compute” these specific prob-lems faster than a single Turing machine, this is sufficient to breakthe boundaries of the Busy-Beaver problem. Of course, the speedup in these examples (which are of a finite polynomial nature) isnot sufficient to overcome the busy-beaver limitation, which wouldrequire a qualitatively bigger speed-up.

Of course, such parallel mechanisms might well be a morepractical way to implement certain computational processes, aphenomenon which has been repeatedly re-discovered by theprocess of evolution.

7.2.5 Arguments employing Godel’s TheoremAnother example is that used by Kampis (1991), that humans cantranscend the Goedelian limitations on suitably expressive formalsystems. He argues that because any such formal system will includestatements that are unprovable by that system but which an exteriorsystem can see are true, and humans can transcend this system andsee this, that they thus escape this limitation. He then cites Church’sexample of the conjuction of all unprovable statements as onewecan see is true but that is beyond any formal system.

The trouble with this is the assumption that humans can tran-scendany formal system to see that the respectively constructed

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Goedelian statement is true. Although we humans are quite goodat this, theassumptionthat we can amounts to a denial of the CTTalready, so this can not possibly be used as a convincing counter-example! If you state that the truth of the above is evident to usfrom viewing the general outline of Goedel’s proof, i.e. from a meta-logical perspective, then there will be other unprovable statementsfrom within this meta-perspective. Here we are in no better posi-tion than the appropriate meta-logic for deciding this (without againassumingwe are better and begging the question).

Church’s conjunction of unprovable statements gets us no further.We can only be certain of its truth as a reified entity in a very abstractlogic (which itself would then have further unincluded unprovablestatements at this level) – otherwise we are merelyinducing thatit would be true based on each finite example, despite the factthat such a trans-infinite conjunction (since “formal systems” wouldinclude models with sizes of all the hierarchy of ordinals, the sizeof the conjuction would be strictly bigger than any infinite ordinal!)is qualitatively different from these and undefinable in any of thelogics that were summed over.

Recently Penrose has put forward a similar argument (re-presented in a corrected form in Penrose, 1996), attempting to showthat some mathematical thought is beyond a Turing machine. Thisargues from an assumption: “. . .that the totality of methods of (unas-sailable) mathematical reasoning that are in principle humanlyaccessible can be encapsulated in some (not necessarily compu-tational) sound formal system F” to a contradiction using Gödel’sincompleteness theorem and the ability of humans to reflect on whatthey are doing/arguing. The trouble is that there is more than oneinterpretation of the result of this, depending on what assumptionone takes to have been contradicted. Thus instead of concludingthat this “totality of humanly accessible methods of mathematicalproof” is beyond any such formal system F, one could alternativelyconclude simply that there is no such totality, and that the contradic-tion arises from assuming the existence of this totality in a similarway to assuming that there exists a universal set.

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ACKNOWLEDGEMENTS

I would like to thank the members of the Principia Cyberneticamailing list (http://pespmc1.vub.ac.be/MAIL.html) with whom Ihave had many useful discussions on this subject, especially DonMikulecky, Jeff Prideaux, Cliff Joslyn, Francis Heylighen and OnarÅm. Also John Shand for the original impetus and Helen Gaylardfor her helpful comments.

REFERENCES

Barwise, J. and J. Perry: 1983,Situations and Attitudes. Cambridge, MA: MITPress, 352 pp.

Bremermann, H.J.: 1967, Quantal Noise and Information.5th BerkeleySymposium on Mathematical Statistics and Probability4: 15–20.

Chaitin, G.J.: 1995, Randomness in Arithmetic and the Decline and Fallof Reductionism in Pure Mathematics. In J. Cornwell (ed.),Nature’sImagination. Oxford University Press, 27–44. http://www.cs.auckland.ac.nz/CDMTCS/chaitin/unm.html

Cutland, N.J.: 1980,Computability. Cambridge: Cambridge University Press.Edmonds, B.: 1997, The Possible Irreducibility of Software Artificial Life. CPM

Report 97-14, MMU, UK. http://www.cpm.mmu.ac.uk/cpmrep14.htmlEdmonds, B.: 1999, What is Complexity? – the Philosophy of Complexity

per se with Application to Some Examples in Evolution. In F. Heylighenand D. Aerts (eds.),The Evolution of Complexity. Dordrecht: Kluwer.http://www.cpm.mmu.ac.uk/∼bruce/evolcomp/

Edmonds, B.: in press, Capturing Social Embeddedness – a Construc-tivist Approach. Adaptive Behavior 7(3/4). http://www.cpm.mmu.ac.uk/cpmrep34.html

Fishler, M.A. and O. Firschein: 1987,Intelligence in Eye, the Brain and theComputer. Reading, MA: Addison-Wesley.

Gabbay, D.M.: 1994, Classical vs Non-classical Logics (The Universality ofClassical Logic). In D.M. Gabbay, C.J. Hogger and J.A. Robinson (eds.),Hand-book of Logic, in Artificial Intelligence and Logic Programming, Vol. 2. Oxford:OUP, 359–500.

Gaines, B.R.: 1976, On the Complexity of Casual Models.IEEE Transactions onSystems, Man and Cybernetics6: 56–59.

Gödel, K.: 1931, Uber formal unentscheidbare Satze der Principia Mathematicaund verwandter System I.Monatschefte Math. Phys.38: 173–198.

Harnad, S.: 1990, The Symbol Grounding Problem.Physica D42: 335–346.http://cogprints.soton.ac.uk/abs/psyc/199803014

Page 25: Pragmatic Holism (or pragmatic reductionism)

PRAGMATIC HOLISM 81

Kampis, G.: 1991,Self-Modifying Systems in Biology and Cognitive Science: ANew Framework for Dynamics, Information and Complexity. Oxford: PergamonPress, 565 pp.

Kampis, G.: 1995, Computability, Self-Reference, and Self-Amendment.Communication and Cognition – Artificial Intelligence12: 91–109. http://www.c3.lanl.gov/∼rocha/kampis.html

Löfgren, L.: 1968, An Axiomatic Explanation of complete Self-Reproduction.Bull. Math. Biophys.30: 415–425.

Marquis, J-P.: 1995, Category Theory and the Foundations of Mathematics:Philosophical Excavations.Synthese103: 421–447.

Pattee, H.H.: 1995, Evolving Self-Reference: Matter, Symbols and SemanticClosure.Communication and Cognition – Artificial Intelligence12: 9–27.http://ssie.binghamton.edu/∼pattee/sem_clos.html

Penrose, R.: 1996, Beyond the Doubting of a Shadow: A Reply to Commen-taries on Shadows of the Mind.Psyche2(23). http://psyche.cs.monash.edu.au/volume2-1/psyche-96-2-23-shadows-10-penrose.html

Rosen, R.: 1985, Organisms as Causal Systems Which Are Not Mechanisms: AnEssay into the Nature of Complexity. In R. Rosen (ed.),Theoretical Biologyand Complexity: Three Essays on the Natural Philosophy of Complex Systems.London: Academic Press, 165–203.

Rosen, R.: 1991,Life Itself – A Comprehensive Enquiry into the Nature, Originand Fabrication of Life. New York: Columbia University Press.

Rosen, R.: 1993, Bionics Revisited. In H. Haken, A. Karlquist and U. Svedin(eds.),The Machine as Metaphor and Tool. Berlin: Springer-Verlag, 87–100.

Rotman, B.: 1993,Taking God Out of Mathematics and Putting the Body Back In.Stanford, CA: Stanford University Press.

Sarkar, S.: 1992, Models of Reduction and Categories of Reductionism.Synthese91: 167–194.

Siegelmann, H.T.: 1995, Computation Beyond the Turing Limit.Science268:545–548.

Simon, H.A.: 1981, The Architecture of Complexity. InThe Sciences of theArtificial. Cambridge, Massachusetts: MIT Press, 192–229.

Turing, A.M.: 1936, On Computable Numbers, with an Application to theEntscheidungsproblem.Proc. London Math. Soc.2(42): 230–265.

Von Fœrster: 1960, Self-Organising Systems and Their Environments. In M.C.Yovits and S. Cameron (eds.),Self-Organising Systems. Pergamon Press, 2–22.

Von Neumann, J.: 1956, Probabilistic Logic and the Synthesis of Reliable Organ-isms from Unreliable Parts. In C.E. Shannon (ed.),Automata Studies. Princeton:Princeton University Press, 43–98.

Weinberg, S.: 1977,The First Three Minutes. New York: Basic Books.Wimsatt, W.: 1972, Complexity and Organisation. In K. Schaffner and R. Cohen

(eds.),Studies in the Philosophy of Sciences, Vol. 20. Dordrecht: Reidel, 67–86.Wolfram, S.: 1984, Cellular Automata as Models of Complexity.Nature 311:

419–424. http://www.wolfram.com/s.wolfram/articles/84-cellular/summary.html

Page 26: Pragmatic Holism (or pragmatic reductionism)

82 BRUCE EDMONDS

Wolfram, S.: 1985, Cryptography with Cellular Automata.CRYPTO ’85 Proceed-ings: Advances in Cryptography, Santa Barbara, California;Lecture Notes inComputer Science218: 429–432. http://www.wolfram.com/s.wolfram/articles/85-cryptography/summary.html

Wolfram, S.: 1986, Random Sequence Generation by Cellular Automata.Advances in Applied Mathematics7: 123–169. http://www.wolfram.com/s.wolfram/articles/86-random/summary.html

Yates, F.E.: 1978, Complexity and the Limits to Knowledge.American Journal ofPhysiology235: R201–R204.

Centre for Policy ModellingManchester Metropolitan UniversityAytoun Building, Aytoun StreetManchester, M1 3GH, UKE-mail: [email protected]: http://www.cpm.mmu.ac.uk/∼bruce