Dummett's Intuitionism is Not Strict Finitism

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Dummett's Intuitionism Is Not Strict FinitismAuthor(s): Samuel William MitchellReviewed work(s):Source: Synthese, Vol. 90, No. 3 (Mar., 1992), pp. 437-458Published by: SpringerStable URL: http://www.jstor.org/stable/20117007 .

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SAMUEL WILLIAM MITCHELL

DUMMETT'S INTUITIONISM IS NOTSTRICT FINITISM1

ABSTRACT: Michael Dummett's anti-realism is founded on the semantics of naturallanguage which, he argues, can only be satisfactorily given inmathematics by intuitionism.It has been objected that an analog of Dummett's argument will collapse intuitionisminto strict finitism. My purpose in this paper is to refute this objection, which I argue

Dummett does not successfully do. I link the coherence of strict finitism to a view ofconfirmation - that our actual practical abilities cannot confirm we know what wouldhappen if we could compute impracticably vast problems. But to state his case, the strictfinitists have to suppose that we grasp the truth conditions of sentences we can't actuallydecide. This comprehension must be practically demonstrable, or the analogy with Dum

mett's argument fails. So, our actual abilities must be capable of confirming that we knowwhat would be the case if actually undecidable sentences were true, contradicting theview of confirmation. I end by considering objections.

DUMMETT'S CASE AGAINST REALISM

One reading of Michael Dummett's argument against realism inmathematics can be sketched like this:2(1) What our language means is an empirical matter.(2) The sole evidential basis for empirical hypotheses about whathumans mean by their words are the uses to which we can

put language on different occasions.(3) Hence, any hypothesis about meaning that goes beyond the

possibility of testing by human usage is illegitimate.On my reading of Dummett, point (3) should be interpreted as

claiming that no distinction in the meanings of expressions is possiblewhere no difference in usage by humans is possible, and correlatively,the claim that a sentence ismeaningful is empty if there is no possibleuse to which the sentence can be put by humans. The view requiresthat some specific kind of test be possible of comprehension of everysentence of the language. This condition is violated by theories ofmeaning like Donald Davidson's, in which no specific test of comprehension of an individual sentence is required, short of understandingthe whole language.3

Synthese 90: 437-458, 1992.? 1992 Kluwer Academic Publishers. Printed in the Netherlands.

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438 SAMUEL WILLIAM MITCHELL(4) Two kinds of evidence of comprehension are possible from

usage:(a) Evidence of a subject's verbal ability to use the sentence

appropriately, for example, by explaining what itmeans.(b) Evidence of a subject's practical ability to use the sen

tence, for example, in acting in appropriate ways uponrecognizing its truth.

(5) Only the practical evidence is non-circular. For linguisticevidence clearly requires that the speaker be credited withantecedently understanding the language used in displayingcomprehension. But this can only be shown by some furthertest that is not question-begging.

I have a very weak reading of Dummett, a practical test may be verydifficult to obtain, or require considerable effort and ingenuity to devise. But some test must be possible. As Dummett puts it, "theremust be an observable difference between the behavior or capacities ofsomeone who is said to have that knowledge and someone who is saidto lack it".4 It is particularly important that itmust be a genuine test;it cannot be ruled out by what we know about humans that anyonecould ever be recognized as having passed it. It cannot, for example,require an infinite amount of time.The case against classical logic and mathematics is that its insistencethat every sentence is either true or false leaves much of its languagebereft of any practical test of comprehension. What we test, when wetest comprehension, is grasp of truth conditions and, since accordingto classical logic every sentence is either true or false, this must be theability to recognize the sentence as having one or the other truth value.

(6) In finite domains, practical tests of the application of a decidable predicate to each object in the domain are always possible. It is therefore always possible to practically test anindividual's understanding of (for example) "Every evennumber less than amillion is the sum of two primes". Simplyset the individual a task in which he or she can do somethingpractical to discover the truth value. A rather cheap exampleof such an ability is the ability to create amachine to performthe calculation by programming a computer.

(7) Contrast this with the case where the domain is infinite.



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DUMMETT'S INTUITIONISM IS NOT STRICT FINITISM 439

Obviously, displaying knowledge of truth conditions is amatter of showing knowledge of what is the case if thesentence is true, and what is the case if the sentence is false.To display such a thing practically, the subject must dosomething which, putting the matter very loosely, is at leastrelevant to the truth value of the sentence. Dummett's caseis that it is ruled out, in the case of undecidable sentences,that anyone could do anything, under any possible circumstance, that is relevant to the truth value of the sentence.So, any theory of truth that retains bivalence cannot serveas an adequate theory of meaning.

This reading of Dummett may be very idiosyncratic. I hope, however,that it is neither uncharitable nor wildly inaccurate. It is a familiarstate of affairs to find Dummett confusing for those who read him. Ishould perhaps say, in defending the reading I have given, that Ithink this argument is actually compelling, whether or not it is actuallyDummett's argument.

A REALIST REPLY

I wish to focus on just one reply to Dummett's argument. The claim in(6), that we can always check the application of a decidable predicateto every element of a finite domain, is obviously false. For sufficientlylarge domains, or simply calculations involving sufficiently large num

bers, our access to practical tests of comprehension is as unobtainableas that which the realist postulates for infinite domains in (7). It is nogood replying that in principle we could discover whether every evennumber less than, for example, lO10000 is the sum of two primes, for,in the case of sufficiently large numbers, it is either highly probable oreven certain that we shall always lack the practical capacity to carryout the computation. Robin Gandy points out that spacetime maycollapse before certain computations can be carried out and, even ifthat does not happen, there are only a finite number of particles in theuniverse which can be put into only a finite number of discriminablestates.5 Therefore, we can in practice only represent a finite number ofnumbers. And, hence, it follows that there may be finite calculations

which the intuitionist regards as having a truth value, for which we



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440 SAMUEL WILLIAM MITCHELLcan only verbally affirm our grasp of their truth conditions, exactlyanalogously to the charge against the realist.The position that holds that practical limitations upon our ability torepresent numbers and perform proofs must be taken into account inthe claims of mathematics is called strict finitism by Dummett.6 Whatconcerns me is the use of strict finitism by realists against intuitionists.

My aim is to show that intuitionism is internally consistent and that theclaim by realists that the intuitionist complaints against them can beexactly duplicated by the strict finitist against the intuitionist is false.7I will argue that in the sense of 'practical demonstration' required bythe intuitionist, we can practically demonstrate that we understand alland only those sentences the intuitionist claims we can.Before turning to this, however, I want to look at Dummett's ownreply to this challenge, which I think is inadequate.

DUMMETT'S OWN REPLY

Dummett argues that, to make the case, the strict finitist must rely onvague predicates such as, "... is in practice countable", or

" . . . canin practice be proved". Dummett provides a variety of ingenious reasons for supposing that any vague predicate cannot be given an acceptable semantics. Therefore, the strict finitist cannot make the caseagainst the intuitionist, for there is no acceptable vocabulary in whichto do so.

I find the argument unsatisfactory. In the first place, it relies upon acertain lack of ingenuity on the part of the opponent. Perhaps, byrevising our view of what conditions a semantics should meet, an opponent may give a semantics for vague predicates.8 For example, in givinghis counterarguments, Dummett considers only briefly the possibilitythat truth itself may be vague. It might be difficult to spell this ideaout; one would need some notion of partial entailment for example,but Iwould not be willing to suppose that it is impossible. In the secondplace, vague predicates are very useful, and Dummett's argument appears to demand that we give them up. But, if we have to give upvague predicates, then much of the interest of semantics evaporates, formany philosophical and scientific problems concern vague predicates.Iwill not develop these objections because it seems to me there is afar cleaner reply to Dummett. Strict finitists are a rare breed. The strictfinitism objection is usually presented by realists who want to know



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why intuitionism does not collapse into strict finitism. Such an individualcan present the case by using no vague predicates.The Chinese superfactorial of PM I define to be the followingnumber:

Take the factorial of the G?del number of Principia Mathematica,superexponentiate it to its own power,9Chinese arithmetic the result with the decimal expansion of

pi-10A Big number is defined to be any number greater than the Chinesesuperfactorial of PM.This is clearly an intuitionistically acceptable definition, so Dummettmust claim that Big numbers exist. But we are almost certainly asincapable of actually going through these operations as we are of checking exhaustively whether every even number is the sum of two primes.How can the intuitionist claim both that grasp of these operations isdisplayed by our practical abilities and that we grasp the application ofthese operations to such huge numbers? Patently, we do not have suchpractical abilities.For this reason, I do not think Dummett's reply succeeds. In suggesting my own reply, Iwill first argue that strict finitism is incoherent. Anadequate theory of the practical abilities we actually possess requiresthat we be accorded possession of practical abilities we cannot actuallyexercise. Once our possession of these abilities has been justified in amanner I think Dummett would accept, I will argue that they aresufficient to establish that we could discover the truth values of thosesentences the intuitionists regard as having truth values, while they arenot sufficient to provide us with the ability to perform infinite tasks.

WHY I AM NOT A STRICT FINITIST

In this section, I argue that strict finitism is incoherent. In order tostate their position, the strict finitists must hold a theory of confirmationthat they deny to the intuitionists.The fundamental distinction between the intuitionist and strict finitistis that the intuitionist holds:



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442 SAMUEL WILLIAM MITCHELLA sentence has a truth value if and only if we could inprinciple discover which truth value it has,

and the strict finitist holds:A sentence has a truth value if and only ifwe can in practicediscover which truth value it has.

The strict finitist's argument is that since we are prevented from everactually arriving at the truth values for some sentences concerning verylarge numbers, we have no more practical abilities associated with thediscovery of such truth values than we do for sentences the intuitionistallows may be undecidable.

Of particular interest, for the argument Iwant tomake, is the positionof the strict finitist concerning sentences that the intuitionist thinks aredecidable, but which concern numbers that are too big to decide inpractice. An example Dummett gives is "1010a1? + 1 is either prime orcomposite", for brevity, refer to this sentence by V.11 An intuitionistclaims that the sentence is either true or false, in virtue of the fact that

we could decide it. It is irrelevant that the physical limitations of theuniverse may prevent us from being able to do so in practice, the onlything relevant to mathematical truth and falsehood are mathematicalmeans of establishing and refuting sentences.12 By contrast, the strictfinitist points out that the argument against the realist rested upon ourpractical abilities to understand the truth predicate, and we have nopractical ability to recognize a truth predicate as applying to a sentencewe cannot in practice verify or falsify. So, a is in the same boat as"Goldbach's conjecture is either true or false"; since we cannot discoverwhich alternative holds, we must allow that the disjunction might lacka truth value, just as the intuitionist allows in the case of Goldbach'sconjecture.The point I am emphasizing here is this: strict finitists hold sentence(A):

(A) No counterfactual is confirmable if the antecedent of thatcounterfactual cannot practically be brought about, due tothe physical constitution of the actual universe.Both the intuitionist and the strict finitist require that understandingmust be confirmed by our practical abilities if it is to be genuine. Thebite of the strict finitist arguments against the intuitionist stems from



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dummett's intuitionism is not STRICT FINITISM 443

the fact that the intuitionist wants to claim that we can confirm ourunderstanding of mathematical sentences in such a way that sentenceslike a must have a truth value even when we cannot in practice determine which truth value they possess. The strict finitists deny this claimbecause they require all practical abilities to be actually capable ofbeing implemented; we cannot confirm claims about what would happenin circumstances that cannot actually be brought about, so we cannotconfirm that we would arrive at a truth value for a. Sentence (A) is abit vague. Does itmean, "Given all the actual facts about the physicaluniverse . . .", or "Given the amount of matter in the actual universe . . .", or what? I will shortly argue that no strict finitist can holdany version of (A), so the reading given to it is irrelevant.The strict finitist is not alone in holding that we cannot confirmsomething, in this case our comprehension of a sentence, if that confirmation is ruled out by the physical contingencies of the universe. Ernst

Mach, charitably read, argues that Newton is not entitled to (putatively)distinguish absolute motion from relative by an experiment we cannotin practice carry out.13 Similarly, Bas van Fraassen holds that practicallypossible, but never performed, experiments cannot affect the choicebetween theories.14 We should, then, be at least agnostic about counterfactuals involving practically impossible experiments. On a closelyrelated topic, the same author points to the irrelevance of practicallyimpossible considerations to the practice of science.15 Admittedly, vanFraassen is not the exact corollary of a strict finitist, for he is simplyagnostic about the truth values of practically untestable sentences,rather than denying them a truth value or granting them some truthvalue apart from truth and falsehood. My point is that his theory ofconfirmation agrees with the strict finitist's in claiming that sentencesthat cannot in practice be confirmed through observation cannot beconfirmed at all. Though strict finitism is a prodigy in the philosophyof mathematics, it is a familiar beast in the philosophy of science; itsname is empiricism.To begin the argument, note the following: the strict finitist caserequires that we at least understand some sentences we are physicallyprevented from recognizing as true or false. (The disjuncts of a, above,are examples of such sentences.) The strict finitist cannot state his orher case without supposing we understand such sentences. Furthermore, our ability to identify such sentences counts as a practical testof our comprehension, for the physical limitations of our universe are



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444 SAMUEL WILLIAM MITCHELLan empirical matter, one we discovered by doing practical things, likebuilding telescopes and observing the red shift of galaxies.Now consider what must be established if the grasp of truth conditionsof any sentence is to be established, on both the intuitionist and strictfinitist accounts. It is that if the sentence in question has a truth value,then we would be able to recognize that truth value, if we were presented with a proof. So, what the strict finitist must hold, in order tohold that we grasp the truth conditions of a practically unverifiablesentence the intuitionist calls decidable, is that we can demonstrate thatwe could recognize the truth value of such a sentence if the universewere physically different. But, clearly, this violates condition (A).Let me spell this out. Let sentence s be any intuitionistically-decidablesentence that the strict finitist claims is undecidable for practical reasons. To understand s, our behavior must confirm:

(B) We could recognize the truth value of s, if the universe werephysically different.

Why does a strict finitist have to hold (B)? We must be able to understand s for the strict finitist to be able to state his or her argument.Our understanding consists in the ability to recognize the truth valueof 5, if it has one. But, s would have a truth value if and only if wecould calculate it. And, we could calculate whether s were true if theuniverse were physically different than it is. So, the strict finitist musthold that if the world were physically different, s would have a truthvalue which we could calculate. And, because we understand s, wewould then recognize that truth value.But (B) is a counterfactual the antecedent of which we are quiteevidently unable to bring about. Strict finitists cannot state their positionwithout identifying a sentence like s. So, they must claim we understands. As understanding must be confirmable, we must be able to confirmby their practical abilities that they do understand s. But, any theoryof confirmation that allows this also allows the confirmation that in sucha universe we would arrive at a truth value for s. And this the strictfinitist must deny can be confirmed, so strict finitism is incoherent.As already noted, Mach suggested that induction should be limitedto predictions about those cases we are physically able to test, and vanFraassen is agnostic about induction beyond the practicable. I think ananalogous argument to that just stated will show that the only optionavailable is to believe the issuances of the theories we hold for cases



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dummett's intuitionism is not strict finitism 445

that outrun the physical circumstances in which we find ourselves. Forthe physical limitations of the universe in which we find ourselves areempirical discoveries. If a physicist wishes to discover whether thefuture is open and infinite in time, or closed and doomed eventually tocollapse in on itself, he or she must do something practical, and thisactivity, being part of the physical universe, is something physics andthe other sciences are required to explain. Suppose one experiment todiscover our practical limitations is that of measuring the relative

Doppler shift of light from approaching and receding arms of galaxies,and suppose as a result of this experiment we conclude the universe isclosed.

In order to explain the activities of the physicist in performing theexperiment, we have to suppose he or she understood what would havehappened if the universe had been open, otherwise, we cannot explainwhy this conclusion was drawn from this experiment. But understandingthis involves knowing that a different conclusion would have beenappropriate if the results of the experiment had been different. Andthat, in turn, means that the experimenter, and the rest of us if weaccept the significance of the experimental result, must hold that thetheory makes predictions about what would have happened if the universe had been different. So, the option is not open to us of restrictingthe implications of theories we accept to those we could in fact encounter, given the universe we inhabit. Once the range of observable phenomena for which science must account is extended to include ourselvesand the phenomena of language use, there are good empiricist reasonsfor recognizing the truth of counterfactuals which our theories predicteven when those counterfactuals can only be indirectly established bythe success of the theory in other cases.This argument is relevant to my main concern. For which abilitieshumans possess is an empirical matter, and a theory of such abilitieswill issue in predictions for circumstances other than those we actuallyencounter. These include circumstances inwhich we are furnished withgreater resources for proving mathematical theorems than are actuallythe case.

The view I hold is that we know the Chinese superfactorial of PM iseither prime or composite because we can show that we could calculatewhich it was. We can actually calculate whether quite small numbersare prime or composite, and that demonstrates that we could decidethe matter for Big numbers if time and resources allowed. The strict



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446 SAMUEL WILLIAM MITCHELLfinitist reply, of course, is to deny induction from actual to impracticalcases. But, as I have shown, this reply is defective; it depends on anuntenable view of what actual experiments can confirm.

WHAT 'COULD IN PRINCIPLE' MEANS

Because I have shown, I think, that strict finitism is incoherent, I havedelivered myself from 'the frying pan' of having to say why I do nothold that position. But, obviously, unless I can show that the intuitionisthas a distinct consistent position, I shall simply fall 'into the fire' ofreducing my own position to absurdity. In this section Iwant to say whatexactly 'could in principle' means for the intuitionist in the sentence "Asentence has a truth value if and only ifwe could in principle discoverwhich truth value it has". Having said that, I will go on to argue thatjust those sentences the intuitionist believes to be decidable are sentences for which we could in principle discover the truth value.One common reading of the phrase is that "could in principle" means"could if our practical abilities were extended by a finite amount".

Crispin Wright, for example, takes this to be the correct reading.16 Idon't think this can be sustained. Recall that the important feature ofthe intuitionist criticism of the realist was that we must be able toconfirm our understanding of expressions by our practical abilities. Ifwe give up the claim that these practical abilities must be actuallypossessed by us, then the central premise of the challenge to realismdisappears. We can't confirm that we understand anything by exercisingpractical abilities we don't actually possess, because we can't exercisesuch abilities.

Such a concession, however, leaves open the issue of what it is thatthe abilities we actually possess show we could do if other factors werevaried. An ability is a particular kind of performance capability thatwill be evinced in the right circumstances. But those circumstances neednot actually be present for us to have excellent reasons for supposingthat, if they were present, the performance would take place. CouldDuns Scotus have understood Esperanto? Of course he could. Heunderstood Latin, and that is a much harder language. It is irrelevantthat Esperanto was not invented until centuries after his death. Iftime travel is physically impossible, then it is not only not actual, butphysically impossible, that he could have been exposed to the right



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conditions to demonstrate such a skill, yet that does not call intoquestion its ascription.We can be liberal about what is demonstrated by the abilities weactually do possess. Unlike the suggestion that we must vary our ownabilities, this does not undermine the intuitionist argument againstrealism. Furthermore, the argument against strict finitism shows thatwe simply must be liberal about what our actual abilities demonstratewe could do in other circumstances. For it can hardly be denied thatwe recognize that certain mathematical problems cannot be solved inpractice due to physical facts about the universe. To understand theseproblems is to know their truth conditions, which in this context meanswe would be able to recognize a proof of the result if one were presentedto us. But, for many of these problems, the only proof available willbe one that is in practice too long for us to comprehend. So, some ofour present practical abilities, those of discovering the limits of calculation our physical universe sets upon us, demonstrate we could recognize proofs that in practice we cannot. So we have some practicalabilities that show we possess skills that cannot be evinced except incounterfactual situations.

Therefore, my suggestion is this: "A sentence has a truth value ifand only if we are in principle able to discover which truth value ithas" is to be parsed: "A sentence has a truth value if and only if ouractual abilities would be sufficient for us to discover which truth valueit has, were the appropriate triggering circumstances present".What kind of abilities am I talking about here? In the mathematicalcase, these certainly include things like the ability to count, add, multiply and divide. They also include the ability to recognize proofs, anddifferentiate and integrate formulae. They also include abilities associated with the communicability of mathematics. One of the major differences between Dummett and earlier intuitionists is his explicit recognition of the public nature of mathematics and language. Thus, thepractical abilities we may exemplify in demonstrating our grasp of thetruth predicate are not those of an incommunicable mental conceptionof proof, but include things like the ability to communicate with oneanother and keep records.We possess these abilities because of the kinds of physical objects weare. But, the abilities can nonetheless be characterized independently ofthe physical details of their causes and instantiations. The point is afamiliar one from the philosophy of mind and computer science. We



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448 SAMUEL WILLIAM MITCHELLcan specify what something must do to approximate square roots usingNewton's method. Anything that actually does so is a physical object,but this way of describing what the thing does is supervenient on thephysical details. Similarly, all minds with which I am acquainted arephysical objects, but a statement that one of them is in a mentalstate is not a statement that uses a physical vocabulary. We are slow,inefficient, calculating objects. We have small memories, but can keeprecords or solve problems as a community. With the exception of theinclusion of the communal nature of mathematics, these kinds of abilities are exactly those the intuitionists used to found their mathematics,and are described in a way that is independent of virtually every physicalpredicate except time.It is because of this fact that only mathematical barriers to proof arerelevant to mathematical truth.17 The abilities we have to prove thingsare independent of the physical contingencies of the universe, and it isthe ability to provide a proof that is relevant to mathematical truth.For the same reason, the intuitionist may legitimately assert that werephysical circumstances different, or even physics itself different, theseabilities would issue in decisions for problems we are in practice unableto decide and, hence, that sentences stating such puzzles have a truthvalue, though we shall never know what that truth value is. And,because the exercise of such abilities constitutes our mathematical practice, it is illegitimate for the realist to answer intuitionist objections by

proposing that we could speak with a community of angels who tell usby exhaustively checking the even numbers that they have verified thateach is the sum of two primes. Even if, convinced of the abilities andhonesty of the angels, we would accept their word, this constitutes aradical change in our mathematical practice. Although what counts as aproof may be subject to evolution and, hence, impossible to characterizeprecisely, this constitutes abandoning any semblance of our currentnotion.

On Dummett's view, not all counterfactuals need have truth values,so it remains a question whether "If the universe were physically different we would conclude that '101Oa1? is prime' was either true or false"has a truth value. As an example of a counterfactual that probablylacks a truth value, Dummett gives that of the now-deceased Jones,who has never faced danger in his life. Dummett points out that wehave little inclination to think that "if Jones had faced danger he wouldeither have acted bravely or acted like a coward" must be either true



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or false.18 But as Dummett himself points out, our lack of inclinationto accord such a sentence a truth value stems from the fact that welack any evidence at all linking Jones to bravery or cowardice. If wehad overwhelming evidence that all those who enjoy anchovies on toastare invariably courageous, and that Jones enjoyed anchovies on toast,we shall know the truth value of the counterfactual. Counterfactualshave truth values if they are predicted to be true in the counterfactualcircumstances by a true theory. In the case in question we have everyreason to think that there is a true theory of human calculationalabilities that does bestow a truth value to the relevant counterfactuals.

DISCUSSION

I will conclude by considering several realist challenges to the view Ihave outlined.First, there is the objection that human abilities are too unreliableto support the kinds of conclusions I have rested upon them. If theseare our actual abilities that are being considered, then we don't havethe ability, even if physics were vastly different, to calculate whether

some vast number was odd or even because, using our abilities, we'dbe virtually certain to make an error in the calculation.To reply, intuitionists can draw upon a competence/performancedistinction. A practical ability can be meaningfully ascribed to someoneeven if they sometimes fail to perform accurately. They have otherabilities associated with the correction and recognition of such errorsthat show that their intention was infelicitously executed. A sentencehas a truth value ifwe could discover which truth value it has, and wecould always discover the truth value of decidable sentences even whenwe err in executing the algorithm, for we can always discover the errors.So, the fact that we are sometimes mistaken in thinking something wasa proof does not show we can't recognize proofs. An ability may becharacterized as one of being able to add one to any arbitrary number,and the characterization will be perfectly acceptable, even if in repeatedly exercising this ability upon the stones inWales, one loses countsomewhere inMid Glamorgan.19

Perhaps while statements concerning mathematical counterfactualsrequire for their confirmation that their antecedents be practically realizable, statements about empirical matters do not. Dummett sometimesdistinguishes mathematical from physical subject matters so, perhaps,



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450 SAMUEL WILLIAM MITCHELLthe strict finitist can do likewise. In which case, the strict finitists donot contradict themselves in accepting (A) for mathematical statements,but denying (B) for empirical ones.21In other parts of Dummett's work, however, and in the argumentsummarized at the beginning of this paper, Dummett clearly points outthat his argument applies equally to both mathematics and empiricaltopics.22 The challenge is to say why this argument does not collapseinto strict finitism and, since this argument applies equally tomathematics and empirical matters, it is illegitimate to introduce this distinctionin an attempt to rebut it.Another challenge: What theory of counterfactuals underlies theclaims made here? It would be unwise to tie my account to a specifictheory of counterfactuals or possibility, but I do want to point out thatthere is at least one anti-realist theory which is friendly to it. Recently,van Fraassen has suggested an anti-realist account of possible worlds.23Under this view, a possible world, allowed by a theory, is simply amodel of that theory. Applying this to the view I am proposing, asentence of mathematics has a truth value if and only if there is somemodel of a true theory of our actual practical abilities in which wediscover the truth value of that sentence. That makes the truths ofmathematics dependent upon the facts about human capabilities, butthe whole point of the intuitionist critique of classical mathematics wasthat the fact that mathematics is a human activity makes our capabilitiesrelevant. As I pointed out in the last section, the abilities we possesswe have in virtue of our physical structure, but those abilities may beinstantiated inmany other physical models. It therefore does not follow,from the fact that the universe would collapse before we could calculatethe truth value of a sentence, that the sentence might lack a truth value,or otherwise fail to be either true or false. In other physical models,beings with our abilities to calculate do discover its truth value.Let me show how such a theory might be used to reply to anotherobjection. I have emphasized that our actual practical abilities may beinstantiated in different physical systems. This suggests the challengethat if humans are just Turing machines, then many sentences willbe provable that are not intuitionistically provable.24 One example isMarkov's principle, which I will term 'M':

(Vx(Ax v ~iAx) & -i Vx~iAx) ?? 3xAxThe sentence is provable if there is a model of creatures with human



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dummett's intuitionism is not STRICT FINITISM 451

abilities who actually prove it. The objection is that there will alwaysbe such a model for this principle, since there is an algorithm forgenerating an instance of the consequent from the antecedent. The leftconjunct of the antecedent guarantees that creatures with our abilitiescan tell of an arbitrary number whether it satisfies the predicate or not;the right conjunct shows they can't all fail to do so. So, simply take auniverse in which time and resources do not place an upper bound onhow large a finite number we can check, and go through the numbersuntil an instance of the consequent is found. But, of course, the rightconjunct doesn't show what we need; all it shows is that we can reducedemonstrations that the predicate holds of an arbitrary finite numberto absurdity, so intuitionistically speaking, even amodel without a finiteupper bound cannot guarantee the needed instance. (Granted, for thesake of argument, the actual abilities we possess could show we canconceive of such a model.)

But, perhaps, this is not enough to turn back the objection. Afterall, the only case for taking our understanding of the quantifiers intuitionistically rests upon what we could do. In contrast to the realintuitionists, perhaps it is not available to me to refuse to affirm thatan instance of the consequent will eventually be found. Consider, forinstance, the following: on my view a sentence is decidable if and onlyif there is a model of creatures with our practical abilities who actuallydecide it.Well, then, cannot the above argument use amodel of humanabilities in which there is no upper bound? And, if I am allowed touse such models for impractical calculations, why can the Markovianconstructivists not press it into the service of making sense of their ownquantifiers?There is, however, a further restriction on our use of models: we

must be able to confirm by our actual demonstrable abilities that wegrasp the models in question.25 In order to genuinely grasp a model inwhich M, above, is calculated by beings like us, it is first necessary tograsp the meaning of the quantifiers inM. But, to reiterate Dummett'sargument, there is nothing we can do, save simply to claim we understand the quantifiers in the constructivist way and, so, no way to confirmthis claim.

A Markovian constructivist might argue against the application ofDummett's argument to this case. In the constructivist case, unlike therealist one, an existential generalization can only be proved if an instance of it is proved, just as in the intuitionist case. And it seems as



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452 SAMUEL WILLIAM MITCHELLif an instance of the consequent always can be proved, for in cases inwhich the antecedent of M is known to be satisfied, we can actuallyfind an instance of the consequent, and so demonstrate in a direct wayour understanding of the quantifier as it appears inM.But can we? We have no guarantee that in the actual cases weactually shall discover the instance of the consequent. In cases in whichwe lose interest, or funding, for the project before its completion, theMarkovian is reduced to the claim that, had we only gone on, we wouldhave discovered the instance. But, again, it seems to me that theunderstanding of the quantifier putatively embodied in this claim isillusory and that even the myriad cases in which we actually do find aninstance of the consequent do not in fact confirm what the constructivistwants. The reason is that even when we do not actually come upwith an instance of the consequent, the Markovian still claims theunderstanding is not refuted. No experiment can confirm anything unless there is the possibility that it could fail, and the understanding berefuted. The constructivist will allow no failure of the experiment torefute the proposed understanding of the quantifiers. So that understanding cannot be confirmed, and the process of going through thecalculations even when they succeed in finding an instance does notconfirm the understanding claimed.To make the point vivid, contrast it with the case of a sentence forwhich we possess a decision procedure in the intuitionistic sense thatwe are in practice unable to carry to completion because of contingentphysical facts about the universe. (For example, factoring some verylarge number.) In this case we can use the facts we know about howquickly humans can calculate in order to establish what kind of universewould allow the algorithm to be completed, for example, how old itwould have to be. The specification of such a universe involves expressions and knowledge that we can actually use, for example tooccupy someone for a fixed period of time by asking him or her to factorsome number, so there is some genuine check on the comprehensionof the expressions used. By contrast, we cannot guarantee, from thesatisfaction of the antecedent of the above schema, that we shall beable to specify any model in which the consequent is satisfied in a waythat brings such practical abilities in its train. We can't say simply, "themodel in which the computation proceeds far enough to actually findthe number", since that answer falls to Dummett's objection, citedat the beginning of this paper; there is no way to demonstrate our



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comprehension of that answer that is not question-begging. And,clearly, we cannot say, "the model in which creatures with our abilitiescheck the complete totality of numbers", for there is no such model.This raises another controversy: Could realists attempt to force intuitionists back to strict finitism by challenging that our practical abilitiesare independent of physics? I cannot claim to have a decisive argumentthat this cannot be done, but it seems to me to be very difficult. In thelast section I pointed out that everyone, even the empiricists, mustadmit that we have some practical abilities the physical world preventsus from exercising, namely, the ability to recognize experimental outcomes that would demonstrate that the physical world is otherwise thanit is. So, the realist argument cannot take the form of claiming thatintuitionists must only talk about physically actual circumstances. Themost promising line of attack seems to be that of denying the separability of physical and mathematical abilities by attacking the claim thatmathematical abilities are supervenient on physical ones.But consider how little the intuitionists need. Mathematics is foundedupon familiar mathematical abilities; we can count one syllable numbers at the rate of about one a second, we can add two one-digitnumbers, and so on. These statements of ability don't on the face of itrequire a specific piece of physics for their realization. The reasoncomputers are so useful is just that they do what we intend to do atmuch higher speed in a different physical way. All the intuitionists needof our abilities is to extend our present mathematical practice to newheights in non-actual cases. We might be a blind, paralyzed, agelesscommunity of mathematicians living in a vast cave, organizing ourselvesinto huge human computers, each of us entrusted with rememberingjust a part of the immense numbers with which the community deals,or calculating just a single operation in the Promethean proofs weperform. It is a short step from that to being completely disembodiedspirits, in which case we can dispense with the physics altogether (although it is hard tomake sense of practical abilities deriving from graspof mathematical concepts by a non-physical being).Alexander George distinguishes two senses of the sentence schema"X can in principle construct a proof that P".26 Under the first reading,called the narrow scope by George, the sentence means that X knowsa strategy that, if it were carried out, would establish the truth valueof P. For example, knowing how to generate prime numbers and dividea large number by them successively might be an in-principle-proof



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454 SAMUEL WILLIAM MITCHELLthat the Chinese superfactorial is either prime or composite. The widereading, emphasized here, is that X could construct a canonical proofthat P under certain circumstances.

I have several interconnected misgivings about the narrow scope.The first is that I cannot see how to prevent the realist from having anin-principle-proof of Goldbach's conjecture or its negation by sayingwe could check whether every even number is the sum of two primes.Second, as George himself cogently argues, the intuitionist cannot bythis means indicate what he or she means by 'finite' or 'effective procedure' in an acceptable way, and the narrow reading requires that theabilities be defined in terms of these expressions, rather than the otherway around, as I have suggested here. Third, and most important, theintuitionist demand that truth must be accessible is based upon therequirement that understanding must be non-circularly demonstrableby practical abilities. Under the narrow reading, our recognition of anin-principle-proof does not require that we practically be able to cashout the activity of proving in a practical way, even in counterfactualcircumstances. Under the wide reading, by contrast, our comprehensionof a sentence, and its possession of a truth value are always explained bythe abilities we actually possess. There could always be a discoverablepractical difference that hangs on the truth value, of a sentence, although our own world may be too sparse for us to be able to takeadvantage of it. So, the central question this anti-realist theory of truthis supposed to address goes unanswered on the narrow reading.Saul Kripke's Wittgenstein on Rules and Private Language addressesa closely related but distinct question to that raised here. Kripke askswhat it is that allows the projection from actually practically tested usesof a mathematical function to future actually practicable uses. I havebeen addressing the issue of how our present restricted abilities couldserve as a public test of what we would do in other circumstances.

Despite the divergence between Kripke's purposes and my own, he, atone point, appears to contest the view that the construction of a machine to calculate a function demonstrates that an individual knowswhich function she or he intends.27 These arguments might perhapsbe extended to vitiate the extension of other demonstrations of abilitiesto untested cases.

Kripke has three arguments.28 His first points out that a computeronly has a finite capacity to calculate, so there will still be untestedcases in which the skeptic can doubt the algorithm. Clearly this is true,



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dummett's intuitionism is not strict finitism 455

but the task is only to show practical abilities connected with valuesthe function is supposed to have beyond those so far explicitly calculated. The ability to construct a machine shows this even if it only hasa finite capacity. If a doubt arises beyond this capacity one need onlybuild a bigger machine.Kripke will perhaps reply that there will still be untested cases forthe larger machine. But that is entirely beside the point: the ability toconstruct a machine is a practical demonstration of what is intendedfor arguments that a machine can calculate, and the ability to construct

larger and larger machines shows what values are intended for largerand larger arguments. Doing this for indefinitely-large finite argumentsis identical to showing that we understand the function. We do nothave, nor do we need, a practical ability to show we know what all thevalues of the function will be for the entire totality of natural numbers,for that totality does not exist. I think the objector must retreat tonumbers so large that the physical universe prevents us from calculatingthe values, and I have already addressed such a position.

Kripke points out, secondly, that machines malfunction, so that amachine may fail to instantiate the procedure I intend. But this overlooks the fact that we have practical abilities to detect malfunctions.We can run the algorithm on two different machines, or use the outputas input to another program that should give easily surveyable andpredictable outputs, and so on. I can show that I know what it wouldbe for a machine to correctly instantiate my intent, often through myattempts to prevent errors. So, once again, my ability to build anautomatic calculator gives some evidence that I know what I'm talkingabout.

Kripke argues, thirdly, that ifwe say that the machine embodies thefunction we intend, we must explain its workings as an embodiment ofthe algorithm. To do this we have to use language, and the strict finitistcan ask whether we really grasp the meanings of the words we usewhen we refer to cases outside those we have explicitly calculated.Now here, I think, the divergent purposes of myself and Kripke cometo the fore. For my reply is simply that I do not have to explain themachine as an embodiment of the algorithm at all. The construction ofthe machine is, for me, a means by which a third party can test mycomprehension which does not depend on what I say. If those whodoubt my comprehension say to me, "Build us a machine which willcompute automatically what mathematicians mean by 'plus'", and a



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456 SAMUEL WILLIAM MITCHELLmachine they judge to be appropriate is produced, then as far as theyare concerned that is very good evidence that I understand mattersaccording to their lights without my saying anything else. Even if themachine computes only a finite initial segment of the plus function,there is a ready explanation for this in the materials available to thesubject, an explanation which can be tested by running the experimentagain with greater available resources. There is the further issue, ofcourse, about whether both they and I intend the 'right' function by'plus', but that is not my concern in this paper.

The point I've been arguing might be put this way: the practicalactivities of empirical hypothesis formation, testing and confirmationentail that the only available hypotheses about how we would behavein practically inaccessible circumstances generalize in the way we intuitively accept as correct from those we can actually test. In this light, thestrict finitist challenge becomes: say what is wrong with a practice ofsminduction, smonfirmation, etc., which predicts the intuitively acceptable result for hypotheses we can practically test, but which makesno predictions, or different predictions, about practically inaccessiblecases.

As I've argued, one must make some predictions about the inaccessible cases, or certain actual pieces of behavior become unintelligible.And as soon as one tries to give an example of a possible exception toan intuitively acceptable generalization about how we would go oncalculating, one has to presuppose that one understands the example.As this is an example in a practically inaccessible realm, this attributionof comprehension requires that the practical abilities in which comprehension consists can be generalized to the inaccessible case. But, ofcourse, that presumption is a refutation of the challenge that is supposedto be presented. Dummett argued in his review of Wittgenstein's philosophy of mathematics that itwas the essence of language that we shouldunderstand new sentences and, hence, that understanding acquired ina restricted range of examples should be generalized to new cases.29

My argument has been that we are forced to generalize even beyondcases we shall ever encounter and, so, we would carry on calculatingand proving in the same way as we now do if the universe were physically richer than it is.I think these are the most pointed objections to the position I havedeveloped. While I retain a disquiet at whether I have correctly interpreted Dummett, I think his philosophy does have the resources to



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reply to the charge that it collapses into strict finitism, which I think isthe most common and robust charge leveled against it.

NOTES1 I especially want to thank Alex George and Philip Kitcher for their help on this paper.I'd also like to thank the members of the Propositional Attitudes Task Force, Jane

Braaten, Jay Garfield, Lee Bowie, Murray Kitely and Tom Tymoczko. My thanks alsoto Peter Godfrey-Smith and the anonymous reviewers of Synthese, one of whom wasparticularly helpful.2 Dummett (1978, pp. 216-20) is clearest. See also Dummett (1976, pp. 80-101).3 See Davidson 1984, pp. 22, 133.4 Dummett 1978, p. 217.5 Gandy 1982, p. 131.6 Dummett 1978, pp. 248-68.7 See, for example, P. Bernays, 'On Platonism in Mathematics' (in Benacerraf andPutnam, 1964 (1983), p. 265).8Wright (1982, reprinted inWright 1987, pp. 167-75) gives such a semantics, whichcontains the vague predicate "... is actually verifiable" in the metalanguage.9 m superexponentiated to the n is m raised to its own power n times.10

Chinese arithmetic is the operation of summation without carrying; if one of thearguments is unpatterned, so is the value.11Dummett 1978, p. 239. I use '10*10' to write '1010' as an exponent. If you are nothappy with this example, just imagine some much bigger number is used.12Dummett 1977, p. 19.13Mach 1893 (1960), pp. 281, 284, 340-41.14Van Fraassen 1980, p. 60.15 Churchland and Hooker (eds.), 1985, p. 255.16Wright 1987, pp. 107-75.17Dummett 1977, p. 19.18Dummett 1978, pp. 15-16.19Adams 1988, p. 220.20 For example, in 1977, pp. 55-56.21 This argument was suggested by a helpful reviewer for Synthese, whom I should liketo thank.22 For example, Dummett 1978, pp. xxix, 16-17, 157, 227; Dummett 1982, pp. 55-56.23 Van Fraassen 1989, pp. 92-93.24 This useful objection was suggested by an anonymous reviewer, to whom I am grateful.25 This is the burden of The Philosophical Significance of G?del's Theorem' (Dummett1978, pp. 186-201).26 George 1988, p. 152.27 Kripke 1982. Cf. Dummett 1978, pp. 171-72.28 Kripke 1982, p. 32ff.29 Dummett 1978, p. 177.



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458 SAMUEL WILLIAM MITCHELL

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Adams, D.: 1988, The Long Dark Tea Time of the Soul, Pocket Books, New York.Benacerraf, P. and Putnam, H.: 1964 (1983), Philosophy of Mathematics, Selected Read

ings, Cambridge University Press, Cambridge.Churchland, P. and Hooker, C. A. (eds.): 1985, Images of Science: Essays on Realism

and Empiricism, University of Chicago, Chicago.Davison, D.: 1984, Inquiries into Truth and Interpretation, Clarendon Press, Oxford.Dummett, M. A. E.: 1973 (1981), Frege, the Philosophy of Language, Harvard University

Press, Cambridge MA.Dummett, M. A. E.: 1976, 'What is a Theory of Meaning? IF, in M. G. J. Evans and

J. McDowell (eds.), Truth and Meaning: Essays in Semantics, Clarendon Press, Oxford,pp. 67-137.Dummett, M. A. E.: 1977, Elements of Intuitionism, Oxford University Press, Oxford.Dummett, M. A. E.: 1978, Truth and Other Enigmas, Harvard University Press, Cam

bridge MA.Gandy, R. O.: 1982, 'Limitations to Mathematical Knowledge', in D. van Dalen, D.

Lascar and J. Smiley (eds.), Logic Colloquium '80, North-Holland Publishing Company, Dordrecht, pp. 129-46.

George, Alexander: 1988, 'The Conveyability of Intuitionism, an Essay on MathematicalCognition', Journal of Philosophical Logic YJ, 133-56.

Kripke, Saul: 1982, Wittgenstein on Rules and Private Language, Harvard UniversityPress, Cambridge MA.

Mach, Ernst: 1893 (1960), The Science of Mechanics, Open Court Classics, La Salle IL.Newton, Isaac: 1729, Principia, trans. Andrew Motte, University of California Press,

Berkeley.Van Dantzig, D.: 1956, 'Is 1010'10 a Finite Number?', Dial?ctica 10, 273-77.Van Fraassen, B.: 1980, The Scientific Image, Clarendon Press, Oxford.Van Fraassen, B.: 1989, Laws and Symmetries, Clarendon Press, Oxford.

Wright, C: 1987, Realism, Meaning and Truth, Basil Blackwell Ltd., Oxford.

Dept. of PhilosophyMount Holyoke CollegeSouthHadley, MA 01075U.S.A.

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Dummett's Intuitionism is Not Strict Finitism