franzen provability and truth

7/30/2019 Franzen Provability and Truth

1/94

ACTAUNIVERSITATIS STOCKHOLMffiNSISSTOCKHOLM STUDIES INPHILOSOPHY 9

PROVABILITY AND TRUTHTorkel Franzn

ALMQVIST & WIKSELL INTERNATIONALSTOCKHOLM


2/94

" " ' , ^ ^ ^

^-S i iSsfeS e

n-'^l i


3/94

Provability and TruthAkademisk avhandling som fravlggande av filosofie doktorsexamen vidStockholms Universitet, offentligenfrsvaras i hrsal 7, hus D, Frescatifredagen den 18 september 1987kl 10.00av

Torkel FranznFilkandFilosofiska institutionen, Stockholm 1987sid81, ISBN91-22-01158-7, ISSN0491-0877

ABSTRACT

According tomathematical realism, the truths or facts of mathematics are not dependenton human knowledge, and in particular not on being provable. In the words of themathematician G.H.Hardy, "Mathematical theorems are true or false; their truth or falsity isabsolute and independent of our knowledge of them." Mathematical anti-realism has it thatall mathematical truth must in some way or another come down to theconcrete realities ofrules, meaning, human practices and inclinations: "For after all, in the end every questionabout the expansion of V2 must be capable of formulation asa practical question concerningthe technique of expansion." (Ludwig Wittgenstein)

The thesis poses two questions: what is the role in our thinking of mathematical realism,and how are we to understand the natural appeal of the anti-realistic view? These questionsare considered in relation to elementary arithmetic, where m athematical realism is generallyrecognized as having a particularly strong hold on our thinking. The answers given in thethesis are based on a characterization of realism as consisting in a certain use ofmathematical statements in non-mathematical contexts, rather than on the usual associationof mathematical realism with the use of classical logic in mathematical reasoning. It isargued that mathematical realism thus conceived does play an essential role in our thinkingabout rules, machines,possibilities, formal systems. The appeal of anti-realism is held tospring from certain natural metaphysical predilections which we cannot expect to eliminate,but on which we may profitably reflect

Theexposition draws a great deal on the work of Ludwig Wittgenstein, although with noexegetical ambitions. It is also indebted to the work by Michael Dummett and Dag Prawitzon meaning theories, and contains a number of critical remarks on the association of realismwith the concept of a theory of meaning.


4/94


5/94

88STOCKHOLMIENSISSTOCKHOLM STUDIES IN PHILOSOPHY 9

PROVABILITY AND TRUTH

Torkel Franzn

ALMQVIST &WKSELL INTERNATONALSTOCKHOLM


6/94

ABSTRACTAccording to mathematical realism, the truths or facts of mathematics are not dependent

on human knowledge, and in particular not on being provable. In the words of themathematician G.H.Hardy, "Mathematical theorems are true or false; their truth or falsity isabsolute and independent of our knowledge of them." Mathematical anti-realism has it thatall mathematical truth must in some way or another come down to theconcrete realities ofrules, meaning, human practices and inclinations: "For after all, in the end every questionabout theexpansion of ^2 must be capable of formulation as a practical question concerningthe technique of expansion." (Ludwig Wittgenstein)

The thesis poses two questions: what is the role in our thinkingof mathematical realism,and how are we to understand thenatural appeal of theanti-realistic view? These questionsare considered in relation to elementary arithmetic, where mathematical realism is generallyrecognized as having a particularly strong hold on our thinking. Theanswers given in thethesis are based on a characterization of realism as consisting in a certain use ofmathematical statements in non-mathematical contexts, rather than on the usual associationof mathematical realism with the use of classical logic in mathematical reasoning. It isargued that mathematical realism thus conceived does play an essential role in our thinkingabout rules, machines, possibilities, formal systems. The appeal of anti-realism is held tospring from certain natural metaphysical predilections which we cannot expect to eliminate,but on which wemay profitably reflect

The exposition draws a great deal on the work of Ludwig Wittgenstein, although with noexegetical ambitions. It is also indebted to the work by Michael Dummett and DagPrawitzon meaning theories, and contains a number of critical remarks on the association of realismwith theconcept of a theory of meaning.

1987, Torkel FranznISBN91-22-01158-7ISSN 0491-0877Printed in Sweden byAkademitryck ABEdsbruk


7/94

ContentsI Introduction Proofs and provabilityinRealismIV Realismand meaningV Anti-realism


8/94


9/94

The metaphysically-minded person feels that the actual world is made upsolely of positive, specific, determinate, concrete, contingent, individual,sensory facts, and that theappearance of a penumbra of fictional, negative,general, indeterminate, abstract, necessary, super-individual, physical facts issomehow only an appearance due to a lack of penetration upon our part

John Wisdom: Metaphysics and Verification


10/94


11/94

INTRODUCTION

In the philosophy of mathematics, classical mathematics is often associated withmathematical realism or Platonism, according to which the numbers, functions, and soon, of mathematics belong to a mathematical reality existing independently of humanthought. A closely related view, also called Platonistic, is that mathematical truth isindependent of our mathematical knowledge, so that e.g. Poincar's conjecture has adeterminate truth value, whether ornot the conjecture isever settled.

There are obvious grounds for associating some such metaphysics with classicalmathematics. In analogy to the designation "constructive mathematics", classicalmathematics may be called "descriptive mathematics". Thereal numbers, for example, aredescribed or characterized as constituting a completeordered field. The quantifiers us ed inthis description - "for every real number", "for every set of real numbers" - are not givenany special interpretation or justification. Apparently it is assumed that real numbers andsets of real numbers are objects to which we may refer using ordinary language with asmuch justification as when we speak of past events, molecules, or far galaxies. Wereason about the real numbers in the same descriptive spirit. Two examples of suchreasoning will be noted here for later reference. If M is an infinite subset of a closedbounded interval I, then I contains an accumulation point of M. We can see this bysubdividing I=Io according to the following prescription: let I^+i be the left half of 1^ i fthis half contains infinitely many points of M, otherwise the right half. A sequence < X n >where x^ lies in 1 ^ will converge to an accumulation point of M. The existence of thissequence is plain only i f we grant that which of the two cases in the prescription obtains isdetermined, as it were, by the mathematical facts. Our knowledge, if any, of the matterdoes not enter into the reasoning. Indeed there is no assumption in the argument that M isin any way describable ordefinable. The second example is the use of the axiom of choicein the argument proving that there is a non-measurable subset of the interval I=[0,1], Wepartition the interval by putting and y in th e same equivalence class if x- y is rational.N ow let M be a set containing a representative from each of th e uncountably manyequivalence classes: M cannot be measurable, since I is the union of th e countably manydisjoint translations M + r (mod 1) of M . The existence of this set M is not at all eviden t ifwe take th e existence of sets of real numbers to depend on their being in any way definedor constructed.


12/94

It seems, then, that to accept at face value the language and modes of reasoning usedin ordinary mathematics is to take mathematics to be the study of an objective realm ofinfinite objects. Although somemathematicians - e.g. Hermite and Hardy - haveexplicitlyembraced this Platonistic view, it is probably true that few mathematicians regardthemselves as committed to it. And indeed the use of classical logic and classicalinfinitistic concepts and methods in mathematics is perfectly compatible with verydifferent views of the subject matter of mathematics. We may for example take the viewthat mathematics is, in the words of Paul Bemays, "the theoretical phenomenology ofstructures"^ and justify our modes of reasoning in terms of the way in which weconceive of these structures, without thereby denying that the structures conceivedof arephenomenal or fictitiousor merely posited, and statements made about them true only inso far as they can be seen to be true. The question of the truth or falsity of Poincar'sconjecture, for example, is then a question of what we can make convincing or evencompelling to ourselves and others concerning these entirely fictitious or imaginaryobjects.

Such a view of classical mathematics Iwill call a "distancingview". This term meanssimply that the language and modesof reasoning of classical mathematics are not taken atface value outside mathematics itself. A n extreme distancing view is that the things we sayin mathematics are strictly speaking meaningless, that only statements of the results offormal computations or derivations can be taken at face value. This view, althoughoccasionally expressed, is generally recognized as inadequate and will not be consideredhere. I believe, however, that the distancing view sketched in the preceding paragraphcorresponds rather closely to the attitude of many mathematicians.

A comparison with intuitionistic mathematics is instructive. Intuitionistic mathematicsis explicitly separated from the idea of a mathematical reality analogous to the physicalworld. Brouwer puts it as follows: in intuitionistic mathematics "the criterion of truth orfalsehood of a mathematical assertion is confined to mathematical activity itself, withoutappeal either to logic or to a hypothetical omniscient being".^ We may apply this viewto classical mathematics as well, without precluding our using classical logic (or anyclassical mathematical methods) in attempting to solve a mathematical problem:intuitionistic and classical problems differ only in that different notions of proof andevidence are involved, and not in the ontological status of their subject matter. Forexample, the truth or falsity of the axiom of constructibility is, on this view, no more andno less objectively determinate than that of the intuitionistic version of Church's thesis.

It is true that the concepts of intuitionistic logic and mathematics are systematically^ Bemays [1], p.528.^ Brouwer [1], p.552.


13/94

explained and motivated in terms of proofs in an abstract and theoretical sense (and thusin terms of hypothetical mathematical activity), whereas there is no systematic explanationand motivation in such terms of the methods and notions of classical non-finitisticmathematics. The obstacles to such an interpretation in the case of classical mathematicsare well known. The ordering relation between real numbers, for example, cannot beexplained by saying how a


14/94

From the point of view of classical eclecticism, proofs, or"mathematical activity", donot enter into theexplanation of mathematical statements at any level. The assumption thatPoincar's conjecture is true is understood, in a mathematical context, as an assumptionconcerning the phenomenal or imaginary world of mathematics. A very different contextis created if we put the following question: supposing the conjecture to be true, is thereany guarantee that it can be shown to be true? A common reaction among mathematiciansis to shy away from such questions. Often the question will be met with misgivingsconcerning the use of the word "true" (misgivings which do not arise in a mathematicalcontext). The nature of these misgivings is best brought out by eliminating the word"true" and putting the question in the form "assuming that..., is there any guarantee that itcan be shown, i.e. proved, that...?" (where a formulation of Poincar's conjecture takesthe place of the dots). From the distancing point of view of classical eclecticism, thisis anodd question since it apparently presupposes an ontological or metaphysical criterion forthe truth or falsity of the conjecture, one that is independent of provability. Theformulation of this criterion is no different from the formulation of the conjecture itself:the difference is that we are invited to take this formulation at face value outsidemathematics. We are asked toassume that Poincar's conjecture is "in fact" true, withoutbeing asked to assume anything concerning the provability of the conjecture. Ifprovability here has its most general sense, this is unacceptable. The world ofmathematics being phenomenal or imaginary, there is no other fact of the matterconcerning the truth or falsity of Poincar's conjecture than that which has been or (moreproblematically) can be arrived at by mathematical reasoning.

Similar situations abound in and outside philosophy.A particularly transparent case isthat of discussions of indubitably fictitious worlds.The assumption that Sherlock Holmeswas bom in Shropshire makes good sense in the context of Sherlockian studies. In awider context, this assumption can only be understood as tantamount to the assumptionthat it is stated in,or can be inferred from, the canonical writings thatHolmes was bom inShropshire. That is, the general view of things, even among Sherlockians, is such thatthey take a distancing view of their investigations outside certain recognizably specialcontexts.

Less transparent cases are usually more controversial.In ethics, it isa traditional viewthat questions of what is morally right or wrong can be meaningfully posed and answeredonly within a cultural setting; that it makes no sense, in a wider context, to assume thatthis or that ethical doctrine is tme and others false. Equally traditional is theopposite viewthat there is no context so wide, no perspective so objective or scientific, that ethicalquestions and distinctions do not arise with the same force as in our ordinary parochialperspective. In physics, some take a strongly distancing view of theoretical speculationsconceming e.g. strange particles or the geometry of space-time, and explain that such


15/94

questions pertain only to a mathematical model and have no direct bearing on what thenon-mathematical world is like. Othersdo not hesitate to translate the theories, questions,and speculations into non-mathematical termsand relate them to concems outside physics.

The view that mathematical questions and assertionsdo not refer to any mathematicalreality, but rather to the products of the fertile mathematical imaginationof human beings,leaves largely open the question of the nature of mathematical problems and mathematicalknowledge. In particular nothing follows concerning the sources or the scope of thisimagination. The remarkable fact that the generally accepted methods of classicalmathematics are formally encompassed within such systems as ZFC has prompted theattitude (which seems fairly widespread among mathematicians) that no mathematicalproblem of truth or falsity remains when a question is known to be undecidable in thesesystems. The continuum problem is a well-known example. The contrary view that thisproblem has n ot been solved is often assumed to go together with a Platonistic convictionthat the continuum has a determinate cardinality. But (as was emphasized by Gdel) noPlatonistic conviction need be involved.The attitudeof classical eclecticism is that there isno reason to regard certain formal systems as laying down limits for mathematicalknowledge. It is not surprising that most mathematicians prefer to leave aside problemsknown to be unsolvable in current mathematics. Long before the independence results.Hardy contrasted Goldbach's conjecture that every even number greater than two is thesum of two primes - "a strictly mathematical question to which all questions of logic orphilosophy seem irrelevant" - with the continuum hypothesis:^

This again appears to be a mathematical question; one would suppose that, i f aproof were found, its kernel would lie in some sharp and characteristicallymathematical idea. But the question lies much nearer to the borderline oflogic, and a mathematician interested in the problem is likely to hold logicaland even philosophical views of his own.

Hardy's suspicions have since been formally verified. Whether we still regard thecontinuum hypothesis as something to be settled depends on our degree of confidence inthe concepts of set theory, and optimism concemingthe possibility of extending the rangeof acceptable principles. There is of course no guarantee that the continuum hypothesiscan be settled. Indeed in this context the very notions of solution, decision, and proof arelargely indeterminate. The simple reason why this is not in itself discouraging is that theintroduction of new convincing principles and definitions has been a part of mathematicsso far.

The attitude of classical eclecticism as regards the justification of the present methodsand concepts of mathematics must also be briefly considered. How do we explain or4 Hardy [l],p.2.


16/94

justify the use of classical logic and of such concepts as "set of real numbers"? Whatgrounds do we have for using the axiom of choice and other set-theoretical principles inproofs? Essentially, we can at present do no better than present these methods andprinciples as convincing in the light of our informal descriptions or visualizations of theworld of mathematics. Classical logic isused as a matter of course, since th e mathematicalworld is conceived of as analogous to the physical world. We may or may not go on toclaim (with Hardy and Hermite) that we are describing or investigating an objectiverealm, but such claims add nothing immediately apparent to our explanations orjustifications.

The axiom of choice illustrates several points concerning explanation and justification.Having accepted the conception of sets as extensional totalities, independent ofconstructions and definitions, we find this axiom eminently convincing and pleasing tothe intellect. This can be elaborated in different ways. We may for example say that wecan imagine picking outone element from each set in the collection, or that any possibilityof selection is realized as a set in the full set-theoretical universe. There is nothing eitherprofound or compelling in such justifications. Those who do not find the axiom (as usedin the proof above) at all convincing will be far from satisfied. Mathematicians, whenpressed for comments, usually play down the intuitivelyevident character of the axiom ofchoice and emphasizeinstead its demonstrable consistency (relative to other axioms) andusefulness in proofs. This attitude, it has often been remarked, does not seem to be inharmony with mathematical practice, in which the also demonstrably consistent and evenmore useful axiom of constructibility occurs only as an explicit hypothesis,if at all. Alsoit has been proved (by Solovay) that the countable axiomof choice is compatiblewith thenon-existence of non-measurable subsets of R, but this is essentially a curiosity and hasnot prompted the development of alternative forms of analysis. It isof course difficult todraw any definite conclusions from such observations concerning current practice, or topredict the futurecourse of mathematics. Classical eclecticism, at any rate, recognizes thedistinction between evident principles such as the axiom of choice and highly doubtful(even if powerful and interesting) ones such as the axiom of constructibility. To give atheoretical analysis of this distinction is another matter.

The emphasis so far has been on the distancing aspect of classical eclecticism. Themathematical reality described and investigated by mathematicians has been viewed asphenomenal or fictitious. No other criterion for the truth of a mathematical statement hasbeen invoked than its acceptability to mathematicians - whatever the nature of theprocesses by which statements are accepted as evident, plausible, or proved inmathematics. The point has been made that Platonism is neither necessary nor in anyobvious way helpful in explaining, understanding, or justifying classical mathematics.

On the other hand we find on reflection that we do not in fact take adistancing view of


17/94

all mathematics. In particular, the properties of the natural numbers are normally regardedas objectively determinate. We don't know whether there are finitely or infmitely manytwin primes, but the actual number of such primes will, in any ordinary context, be takento be ontologically determined. It takes an effort to consider the view that the number oftwin primes, if at all determinate, depends on what is or can be made convincing to thehuman intellect. It is not surprising that Hardy and others us e arithmetical examples w h e npresenting a realistic view of mathematics.

This realistic view of an indeterminate ran^e of elementary mathematics will be takento form part of classical eclecticism. As a consequence, we can no longer express adistancing view of highly infinitistic or abstract mathematics in the easy terms usedpreviously. The fact that such mathematics has consequences in the realm of elementarymathematics means that there is after all an ontological condition to impose on abstractaxioms, namely that these elementary consequences should be true.

The following philosophical aside n a paper on set theory i l lustrates:^Of course, as with any axiom, an initial act of faith is required conceming theconsistency: we assume that the existence of, say, an inaccessible cardinaldoes not lead to a contradiction with ZFC... It is the author's viewpoint thatconsistency is the only point at issue here, and that the question as to the"existence" of inaccessible cardinals is totally meaningless. To us, largecardinal theory is a (worthwhile) structure theory, no more.

The author's attitude towards the question of consistency is notably realistic: we assume,as an act of faith, that e.g. the theory ZFI (ZFC plus the axiom "there is an inaccessiblecardinal") is in fact consistent. "In fact" because even i f we are correct in this assumption,there is no guarantee that a consistency proof can be given. The question whether thereexists an inaccessible cardinal, on the other hand, is dismissed as "totally meaningless".Perhaps what is intended is only the natural and reasonable view that questionsconceming the existence of large cardinals of various kinds are not questions of fact, butquestions of what can be made plausible or convincing on the basis of our (developing)conception of the set theoretical universe in combination with technical results. We know,however, that large cardinal axioms have consequences for sets of low rank, and even forthe hereditarily finite sets. It therefore seems arbitrary to say that consistency is the onlypoint at issue when the question of the acceptability of inaccessible cardinals is raised. If,for example, ZFI is consistent but it is provable in ZFI that ZFI is inconsistent, then thereis no inaccessible cardinal.

On a resolutely formalistic view of set theory according to which it has noepistemological significance there is of course no reason to reject axioms with false^ Devlin, p.93.


18/94

arithmetical consequences, since questions in set theory are regarded as posing onlycombinatorial problems of derivability or pseudo-algebraic ones concerning models oftheories. It isdifficult to say what interest the study of large cardinals has on such a view.The attitude of classical eclecticism is very different. Even "theological" set theorybelongs with other extensions of the methods and concepts of mathematics, for examplethe introduction of the real and complex numbers, functional analysis, non-Euclideangeometry. By these extensions we seek to deepen our knowledge and understanding ofolder parts of mathematics as well as introduce new fields of thought and topics ofinvestigation. Both new proofs of known theorems and proofs of new theorems inknown fields using these extensions have very great interest, whereas isolating a part ofmathematics and attempting to invest it with merely formal significance is sterile andboring. Of course this attitude is not tied to realismor the particular methods of classicalmathematics but applies equally tofinitistic or intuitionistic mathematics. My present pointis only that given this attitude (which is that of classical eclecticism) and a realistic view ofelementary mathematics we find that the existence of large cardinals - whatever the natureof the concepts involved - is not wholly a matter of what is "pleasing to the intellect" orotherwise accepted by or acceptable to mathematicians, but is linked to arithmeticalmatters of fact.

The description of the world of mathematics as a product of the imagination, a creativefield in which the mathematician need not defer to any external reality, captures animportant aspect of mathematics. In Brouwer's resounding phrase, mathematics is an"autonomic interior constructional mental activity"; Cantor more simply characterizes puremathematics as free mathematics. In its distancing view, classical eclecticism emphasizesthis aspect of mathematics. But of course this is far from being the whole story. Both thedescription of mathematics as the language of nature and the observation that the world ofmathematics has a reality of its own are alsofounded in the experience of mathematiciansand users of mathematics. The patent objectivity of an elementary part of mathematics hasbeen introduced above as a particularly simple corrective to the distancing view. It is apart of mathematics that allows us to indulge wholeheartedly in the traditional terminologyof Platonism. It does not necessarily follow that restricting Platonism to arithmetic is away of capturing itsessential insights.

3

To complete the description of classical eclecticism, it remains to comment on theeclectic attitude towards non-classical mathematics. Non-classical mathematics in practicemeans some kind of constructivist mathematics. Intuitionistic and constructivisticconcepts and theories are associated with interpretations and justifications which are


19/94

inappropriate in the case of classical mathematics. In particular, proofs enter into theinterpretation of mathematical statements, and the objects of mathematics are thought of asconstructed rather than described in our mathematical activity. Traditional (Brouwerian)intuitionism introduces objects and principles that have no counterpart in classicalmathematics (choice sequences, continuity principles); all forms of systematicconstructivism introduce a reinterpretation of logic.

According to one well-represented tradition there is aconflict, in some formulations a"battle", between classical and non-classical mathematics. This isoften the view of thoseconstructivists who regard classical mathematics as ill-founded or senselessor a "scandal"or otherwise unsatisfactory and look forward, in Errett Bishop's words, to "the inevitableday when constructive mathematics will be the accepted norm".^

The obvious contrary view, which is that of classical eclecticism, is that the battle isone-sided. Classical and non-classical mathematics both make mathematical sense.Indeed, to those who areeven moderately pluralistically minded it seems that new formsof mathematics should, generally speaking, be sought and encouraged.

In one sense the conflict is undeniable. "There is a war between the ones who saythere is a war and the ones who.say that there isn't." There are many points of conflictbetween the attitudes associated with classical eclecticism and the viewsof e.g. Bishop orBrouwer. But this is not a conflict between classical and non-classical mathematics assuch, and the eclectic view is that there is no obstacle to our feeling at home in differenttypes of mathematics.

This is not to deny that there are genuine issues regarding the place of constructivismin mathematics. One type of issue concems the usefulness and efficiency for the purposesof computer programming or numerical mathematics of (i) the consistent use ofconstructive logic or other forms of non-classical mathematics and (ii) the constructivetradition of classical mathematics. For of course the interest of constructive methods andresults (sharp bounds, explicit solutions, algorithms, etc.) has always been recognized inclassical mathematics. Philosophical disputes over meaning and existence in mathematicscontribute nothing to these issues and no position on them will be counted as part ofclassical eclecticism.

^ Bishop, Preface.


20/94

The views and impressions presented above as "classical eclecticism" are, I believe,both natural and plausible, and they will on the whole be upheld in the following. Clearlythey do not amount to any "philosophy of mathematics". In the following chapters, theywill serve as background and starting point for an examination of certain aspects ofrealism and anti-realism. Indeed the term "classical eclecticism" was introduced only forconvenience in sketching this background.

By (mathematical) realism, I mean the metaphysical doctrine that the statements ofmathematics have adeterminate truth value, or that they refer to an objective mathematicalreality, or that mathematical truth is independent of provability. "Doctrine" is not perhapsan apt word, since there is no canonical formulation or detailed development associatedwith mathematical realism. It lives rather as a view or attitude: like other traditionalphilosophical or metaphysical views it is rooted in the strong impressions of manypeople. The range of these impressions varies: some take a realistic view of largecardinals, nearly everybody is a realist (in the present sense) where elementarycomputations are concerned.

By (mathematical) anti-realism I mean theno less profound and influential impressionthat there can be no other truth of the matter when a mathematical question is raised thanthat which is explicit or (i n a moreor less problematicsense) implicit in rules, in meaning,in human actions and inclinations.Again the range of impressions of this kind varies fromone individual to another.

The terms "realism" and "anti-realism" are perhaps not understood in quite the sameway by any two writers on this topic. It should be noted in particular that I have takenover the convenient term "anti-realism" from the work of Michael Dummett withoutfollowing his usage in any detail. Some such convenient term is needed to denote theattitude of mathematical anti-realism. Thereare traditional philosophical terms with a morespecific content which are usually associated with anti-realistic views, e.g."verificationism", "reductionism", "phenomenalism", "conventionalism". But these morespecific terms are not applicable in this context. On an anti-realistic view of mathematics,we cannot accept the apparent facts, truths, objects, questions of mathematics at facevalue: whatever facts, truths, objects, questions are associated with mathematics must insome way or another come down to realities: human institutions, inclinations, actions,rules. "For after all in the end every question about the expansion of V 2 must be capableof formulation as a practical question concerning the technique of expansion.Thisdoes not imply that mathematical truth is in any sense a matter of convention, or that it isat all possible toreduce mathematics to anything else. Anti-realistic views may or may not^ Wittgenstein [1], V-9.


21/94

lead to more specific schemes and doctrines as to the nature of mathematics, but suchschemes and doctrines are not my concem here.

Realism and anti-realism, as these terms are understood here, are standard ingredientsin everybody's everyday thinking, although not necessarily in connection withmathematics. In accordance with Bradley's description of metaphysics as the finding ofbad reasons for what we believe upon instinct, realism (i n mathematics and elsewhere)rests very largely on shared inclinations and natural predilections. We take a realistic viewof past events, molecules, and far galaxies as it were by default. At the same time we arevividly aware of the metaphysical distinctions without which a realistic view would haveno leverage: we are all to some extent "metaphysically-minded" in Wisdom's sense.Realism and anti-realism can only be understood as two sides of the same coin.

Two questions concerning mathematical realism and anti-realism will be raised andanswered in the following chapters. The first question is:

What essential role, if any, does realism play in our thinking in or aboutmathematics?Since Brouwer, one influential view has been to ascribe to mathematical realism theimportant role of supporting or justifying the use of classical logic and other non-constructive notions in mathematics. In accordance with the ideas of "classicaleclecticism", this view will be rejected below. Chapter 4contains a critical examination ofanother view which has been influential in recent years, viz. the association of realismwith a "realistic theory of meaning", and hence with large claims concerning meaning andunderstanding in mathematics. Another widespread opinion is that the realistic conceptionof mathematical reality has only heuristic value and importance and that realism as ametaphysical doctrine has no essential role to play.

Here a different view of the matter will be argued. Again in accordance with thegeneral impressions sketched above, it will be argued in chapter 3, which is devoted tothis issue, that realism does play an important, indeed essential, role in our thinking inand about elementary mathematics and its associated concepts such as rules, machines,and possibilities. This should not be a surprising claim since the controversy over realismhas always centeredon just these elementary mathematical concepts, where realism has itsstrongest hold on our thinking. To describe or pinpoint this role is not a simple matter.One of the useful aspects of Wittgenstein's writings, on which I draw throughout theessay, is that they are concerned with bringing out the influence of realism (Wittgensteindoes not use this word, but speaks e.g. of "the extensional viewpoint") in contexts whereit is not obvious.


22/94

The second question to be dealtwith is the following:How are we to understand themetaphysical allure of anti-realism?

Chapter 5 isdevoted wholly to this question. The answer given consists in an expositionand examination of certain ideas and trains of thought which can be held to underlie anti-realism. The view taken in chapter 5 is that these ideas arise naturally and are at least asrewarding to study as, say, such traditional philosophical topics as our ideas of time andspace or of the nature of knowledge. Again Wittgenstein is an important source, andmuch attention is given to his reflections on rules, although without any claims toexegetical correctness. The ideas underlying anti-realism are usually presented ascorrecting misconceptions or unjustified beliefs inherent in realistic views: the treatment inchapter 5 is critical of anti-realism in the sense that such claims are rejected. As has beenemphasized above, there is no necessary connection between anti-realism and the viewthat mathematics can or should be pursued along constructivistic lines. This latter viewwill not be taken up forconsideration.

The choice of questions is guided by the general attitudes of "classical eclecticism".No attempt will be made to justify by arguments a realistic view of elementarymathematics. In my opinion there can be no stronger justification than a convincingdemonstration that it does play an essential role in "our" thinking, i.e. in the mainstreamof at least some significant areas of human thought. Naturally such a demonstration is noproof of the correctness or inescapability of a certain view, but it is easy to underestimatewhat is required of a truly "revolutionary" change of ideas.

Proofs and provability enter into the twoquestions in several ways, and chapter 2 hasbeen set aside for anumber of comments on these notions preliminary to the discussion ofthe main questions in later chapters. There is no recondite, detailed, or technicalinvestigation of proofs or the concept of proof in chapter 2. In fact the whole of thechapter is directed towards substantiating a claim which is probably not verycontroversial, viz. that provability, taken in its most general epistemological sense, is anessentially open notion.


23/94

PROOFS A N D PROVABILITY

1

In the year 1900, David Hilbert made a famous aff irmation:^Take any definite unsolved problem, such as the question as to theirrationality of the Euler-Mascheroni constant C, or the existence of aninfinitenumber of prime numbers of the form 2n+l However unapproachable theseproblems may seem to us, and however helpless we stand before them, wehave, nevertheless, the firm conviction that their solution must follow by afinite number of purely logical processes. ...This convictionof the solvabilityof every mathematical problem is a powerful incentive to the worker. Wehearwithin us the perpetual call: There is the problem. Seek its solution. You canfind it by pure reason, forin mathematics there is no ignorabimus.

This is one formulation of Hilbert's "non ignorabimus".The role and interpretation of the"non ignorabimus" in Hilbert's thought varied over the years, and my comments in thefollowing are not to be taken as exegetical.

The "non ignorabimus" is most naturally read as non-theoretical, as expressing andinstilling an optimistic attitude. Then the f irm conviction is irrefutable (though it may peterout) and its justification consists in pointing out, as Hilbertdid, that many long intractableproblems have eventually been solved. On this reading there is no question of Hilbertbeing right or wrong in his affirmation, only of his attitude being vindicated to a greaterorlesser extent by the developmentof mathematics.

But the "non ignorabimus" has also been regarded as acontroversial claim which m aybe open to theoretical justification or refutation. Brouwer in particular has emphasized,with reference to the "non ignorabimus", that "there is not a shred of proof for theconviction, which has sometimes been put forward that there exist no unsolvablemathematical problems."^

Before considering the "non ignorabimus" from this latter point of view, we must ask^ Quoted in Browder.9 Brouwer [3], p.109.


24/94

to what kind of problem it is to be taken to apply. Most of the famous open problems inmathematics are of the "true or false"-variety. That is, the problem is todecide whether ornot a certain statement or conjecture is true. To solve such a problem is to prove ordisprove the statement; the solution is correct if the proof is correct. The problemsmentioned by Hilbert are of this kind, and I shall assume in the following that the "nonignorabimus" refers only to "true or false"-problems and to definite "what is"-problems.By the latter I mean a problem of the form "What is...?"" which calls for an answer in asharply defined range of answers. For example, "What is the number of Fermat primes?"calls for either theanswer "There are infinitely many Fermat primes" oran answer of theform "There are k Fermat primes".

There are mathematical problems of many other kinds. Onemay for example pose theproblem of finding useful sufficient conditions for something, of explaining a strangecoincidence or phenomenon, of characterizing or classifying mathematical objects, ofextending a theory, improving an approximation, finding a definition. Solutions to suchproblems are not just proofs, and it is not a straightforward matter to say what makes asolution correct or adequate. There is little hope of making any theoretical observationsconceming the solvability or unsolvability of mathematical problems in this general sense.

Can the "non ignorabimus" be justified? A mathematician's conviction that a problemis solvable may be based onexperience. The problem does not look intractable, he has aninkling, or even a good idea, of how to attack it. In such a case he has good grounds forhis conviction even if he cannot articulate them, so he is not just expressing optimism orgeneral confidence. Or perhapshis conviction that a particular problem issolvable may bewell-founded even though he has no idea how to solve it. The "non ignorabimus",however, being abstract and general, cannot be justified by invoking the impressions ofexperts.

It may also be the case that a given problem is seen to belong to a class of problemsfor which there is a known uniform method of solution. Such methods - algorithms -form part of mathematics since antiquity. For example, the problem of finding the greatestcommon divisor of two integers is solvable for any pair of integers using Euclid'salgorithm. More generally, uniform methods apply - at least in principle - to any explicitlycomputational problem involving only finite domains and objects and algorithmicallydefined operations and relations. There are also algorithms for classes of problemsconceming infinite objects and domains - e.g. Sturm's algorithm (1829) for finding thenumber of real zeroes of a polynomial withreal coefficients.

The formalization of mathematics led to the possibility of radically extending the classof problems demonstrably solvable by specified means. If a formalized theory T isrecognized as sound, i.e. if the formal derivations in T have an interpretation as valid


25/94

proofs, then one way of proving that a problem is solvable is to prove that a solution tothe problem is derivable in T. If the theory T is proved to be complete, the "nonignorabimus" will have been proved for the class of problems that can be formulated inthe language of T. Sincestrong theories T adequate for the formulation and derivation ofexisting mathematics were known around 1910, the idea of establishing the "nonignorabimus" in this way for allordinary problems was not unreasonable.

Today we know that such a mathematical justification of the "non ignorabimus" can begiven for some quite extensiveclasses of problems - an example being Tarski's extensionof Sturm's algorithm to the elementary theory of the field of real numbers - but not(according to Gdel's incompleteness theorem and later refinements) for elementaryarithmetic, and in particular not for the class of problems of the form "Does theDiophantine equation p(xi,...xji)=0 have a solution?".

O f course, Gdel's theorem applies only to recursively axiomatizable theories, and theconclusion that there is no uniform method for solving all arithmetical problems dependson Church's thesis. It has been suggested that Church's thesis is incontrovertible only i fwe require a uniform method to be mechanically applicable, that there may be otheruniform (and uniformly successful) methods that require non-mechanizable insight orintuition. Thispossibility cannot be ruled out, butit is at present wholly insubstantial.

It must be noted that the kind of justification of the "non ignorabimus" provided by inprinciple applicable algorithms or complete formalizations does not in fact adequatelysupport Hilbert's declaration quoted above. Forexample, the solution to a problemof theform "Is 2^+1 a prime?" follows by a finite number of purely logical processes - ifcomputations are counted as such - but not necessarily in any way that is of any use to usi f we want to solve the problem. The problem is only known to be solvable in principle.The conviction that a problem is solvable in this sense can hardly serve as a powerfulincentive to somebody who wants to actually solve the problem. But we have no theory atall of actual solvability, so in seekingto make a theoreticalclaim of the "non ignorabimus"we must inte ret it as aclaim that every problem is solvable in principle.

2It is generally agreed that the "non ignorabimus" has no known justification if it is

taken to include the problems of elementary arithmetic. But does it follow (as Brouwer'sremark suggests) that some such problems may be unsolvable in an absolute sense? Theview to be argued here is that this suggestion cannot at present be taken seriously - notbecause there are grounds for optimistic faith in the solvability of all arithmetical


26/94

problems, but because of the lack of substance of the notion of absolutely unsolvableproblems.

A "true or false"-problem is unsolvable if the statement or conjecture at issue isundecidable: neither the statement, nor its negation can be proved. All proofs in currentstandard mathematics are believed to be formalizable in theories such as ZFC. If we hadno idea how to extend ZFC, we would have at least some grounds for speaking ofstatements undecidable in ZFCas absolutely undecidable. But we doknow how to extendZFC. Gdel's proof introduced one method of extending any formal axiom systemrecognized as sound, viz. by reflection principles. Another open-ended series ofextensions of the axioms of set theory is given by the strong axioms of infinity. Indeed ithas been suggested that any statement in arithmetic is decidable by some recognizableaxiom of infinity. This is aptly characterized by Cohen as "a rather vague article offaith" but there is nothing absurd about it. More generally, we have no idea of anykind of theoretical bound on what can be achieved by way of extending the methods,principles, and conceptsof mathematics.

This openness or indeterminateness of the notion of proof has another aspect. There issurely nothing impossible about our regarding any given problem as solved. If weenvisage the possibility of an arithmetical statement being absolutely undecidable, whatwe have in mind is that no valid or corrector conclusive proof can be given by which thestatement is settled. Butit is highly doubtful whether thedistinction between valid proofsand inconclusive arguments can carry the weight put on it when we speak of absolutelyundecidable statements. To take a simple case: if a recognizable axiom of infinity isshown to imply an arithmetical statement, the statement need not thereby be proved, forwe may have noreason or inclination toaccept the axiom. On reflection, we may come toregard the axiom as convincing and the statement as proved. Talk of absolutelyunsolvable problems presupposes an absolute and general distinction between axioms thatare really evident or well-founded and such as are accepted merely on the basis of"familiarity passing for evidence".One need not hold that the distinction between theevident and the merely familiar is illusory or radically subjective to recognize that thispresupposition is highly problematic.

In the literature one finds the following argument for the existence of absolutelyCohen [1], p.l52.Berkeley [1], 21:

M en learn the elements of science from others: and every learner hath a deferencemore or less to authority, especially the young learners, few of that kind caring todwell long upon principles but inclining rather to take them upon trust: And thingsearly admitted by repetition become familiar: And this familiarity at length passethfor evidence.


27/94


28/94

arithmetical statement, an obstacle other than the statement being false. If we take arealistic view of arithmetic ("mathematical truth is independent of provability") there is noabsurdity in speaking of Goldbach's conjecture as possibly true but in this senseunprovable. Indeed the possibility of there being unprovable mathematical truths of thiskind has sometimes been regarded as an important consequence of a realistic view.According to the comments above, however, talk of absolutely unsolvable arithmeticalproblems or, equivalently, absolutely unprovable arithmetical truths, has no substance atall, whether or not we take a realisticview of arithmetic. The formula "mathematical truthis independent of provability" will be given a different explanationin chapter 3.

3Two notions of proof, with corresponding notions of provability, have figured in the

above discussion of the "non ignorabimus": actual proofs (just "proofs" for short) andformal derivations.

By "actual proofs" I mean written or spoken proofs such as those given in textbooks,papers, and talks. This is not to say that proofs are to be identified with written or spokenpresentations - we speak of different presentations of the same proof, of the history andessence and analysis of particular proofs, and so on, and it also makes sense to speak ofproofs that exist only as a train of thought in somebody's mind. In these respects proofsare no different from ideas, arguments, theories, stories, and other such abstract objects.The essential point is that actual proofs are theactual arguments and procedures which weuse to prove mathematical statements. For a statement to have been proved, there mustexist a written or spoken or mumbled presentation of an actual proof of the statement; fora statement to be actually provable (just "provable" for short) it must be actually possibleto produce such a proof. This is the provability Hilbert had in mind in his exhortation.

Proofs were described above as "arguments and procedures". In fact it is not a simplematter to say what goes into an actual proof. For example, proofs often contain reports onthe results of computations ("Solving this equation, we obtain..."). These reports cannotnowadays always be replaced by actual protocols of the computations. As is well known,this is not feasible in the case of the Appel-Haken proof of the four color theorem.Keeping in mind that the actual proof is the actual mathematical verification of a theorem,what is the proof of the four color theorem? It is not just (the contents of) a number ofpages of print, but includes the actual procedure of performing lengthy calculations usinga computer. Or we may count the calculations not as part of the proof, but as part ofchecking the proof. In the present context it is not necessary to decide just how the term"proof is to be applied, aslong as it is clear that to ask for the actual proof is to ask how


29/94

we actually come toregard a statement as mathematically verified.

My use of the word "we" calls for a comment. Proofs have different audiences. Aproof in a mathematical joumal may well be comprehensible only to a handfulof experts;the rest of us accept the theorem as proved on their authority. By definition, actual proofsare relative to a mathematicalcommunity: they are the proofs pointed to or dragged out forinspection when the assertion that a certain statement is a theorem is to be justified.

From the epistemological point of view, it should be noted, checking proofs is animportant aspect of theorem-proving. Checking and cross-checking proofs serves both toincrease our confidencein the conclusion and to improve the epistemological standing ofthe proof. The role and character of such checks and cross-checks has not beeninvestigated in traditional philosophy of mathematics, but must be taken into account inany genuine mathematical epistemology. Again the Appel-Haken proof provides anexample: to dispose of the rumor that there is something wrong with the proof, itsoriginators "try to explain the intuition of the proof and let the reader understand how theproof was obtained and why the type of errors that crop up in the details do not affect therobustness of the proofThisexplanation is not part of the proof in the ordinarysense, and traditionally has no place at aU in mathematical epistemology; but to those whohave tried to follow the proof it may clearly be of crucial importance in helping toestablish the theorem.

The difficulties and uncertainties associated with actual proofs are illustrated by theproposed proof of Poincar's conjecture announced in September 1986. One of theoriginators of the proof held it to be "cut and dried" and took the view that it was aquestion of waiting for the proof to "sink into people's subconscious", whereas accordingto another mathematician "he's given some evidence that the proof is truly there" and yetothers were convinced that the proof was inconclusive.^"^

Corresponding to actual proof is the notion of actual provability. In the following theword "provable" will be used in a way that does not accord with ordinary mathematicalusage. In mathematics one would not ordinarily speak of axioms or logical principles asprovable. In the present context, however, no distinction wil l be made between statementsprovable in a respectable mathematical way and trivial consequences of logical ormathematical axioms, or those axioms themselves. Thus "provable" in the followingstands for "provable or evident". Another way of putting it is that "provable" stands for"mathematically verifiable". The use of the word "evident" here is noncommittal: nothingis presupposed concerning the nature of such mathematical evidence. The evidentprinciples and axioms are merely those which are accepted as correct and used in proofs

Appel & Haken, p.lO.Article in the New York Times, September 30, 1986.


30/94

without themselves admitting orrequiring mathematical proof.

What is assumed is that we may appropriately speak of proofs and statements ascorrect, evident, valid, true, acceptable, and so on - i.e. apply the terminology oftraditional epistemology (without necessarily having any opinion or theoryconcerning thenature of mathematical knowledge or mathematical truth). If, on the contrary, we regardsome part of mathematics as having no more than formal significance, we cannot easilyapply the above terminology. For example, many would hold that it makes little sense toinquire into the provability in the sense of "mathematical verifiability" of statements inabstract set theory: we can only ask whether they are derivable using specified rules andaxioms, which themselves lack epistemological significance. On the other hand,practically everybody would be at ease with thequestion whether the Appel-Haken proofis corrects not in the sense of conforming to some set of formal rules, but as ademonstration of the truth of the four-color conjecture. In the following the range of suchepistemologically meaningful mathematics will beleft open.

The first notion of provability to emerge, then, is actual provability. The onlyindisputable principle as regards actual provability is that if a statement A has beenproved, then A is actually provable. Whetheror not a statement A has been proved is, asnoted above, sometimes hard to say. Also, as the case of the four color theorem shows,technology is not irrelevant to actual provability. For theseand other reasons it is standardpractice in philosophy to consider instead the "provability in principle" that has alreadybeen touched on in connection with the "non ignorabimus". To understand this notion,we must first look atformal derivations.

Formal derivations are certain sequences, terms, trees, and so on, associated withformal systems of various kinds. Predicate logic theories such as ZFC are the primeexamples. Thedistinguishing property of formal derivations in this context is that they arethemselves mathematical objects - they are sequences, terms, trees in the mathematicalsense. That a statementA is formally derivable in a theory T is accordingly a mathematicalexistence assertion.

Strictly speaking, not statements, but some kind of formulas - mathematical objects -are derivable in formal systems. However, for a formal system to have anyepistemological significance, at least some of the formulas must be interpreted asmathematical statements. Inte reted formal systems will be referred to as "theories", andthe notation #A will be used for a formula expressing the mathematical statement A. A isprovable in a formal systemT i f there exists a derivation of #A in T.

T will be assumed to satisfy further the condition that formulas and derivations arefinite objects, and the set of derivable formulas recursively enumerable. Systems which


31/94

do not satisfy these conditions may well have epistemological relevance, but the relationbetween proofs and formal derivationswill be more remote.

This gives us the second notion of provability: provability in T, for a specified theoryT. In contrast to actual provability, this is a mathematical concept: in fact, "A is provablein T" will be a statement of elementary arithmetic, by the conditions imposed on T. Theepistemological significance of such provability will depend on the theory T. If T is aformalization of mathematical and logical principles (in the form of rules and axioms)which we accept as valid, the statements provable in Tare, in the usual phrase, "provablein principle". The idea behind this terminology is that only limitations of time and spaceand various human frailties prevent statements provable in principle from being actuallyprovable. This observation is in itself incontrovertible. Still, from an epistemologicalpoint of view such provability in principle is of doubtful interest. This isapparent alreadyin the fact that every computational statement is in principle decidable, whereas from thepoint of view of human knowledge such statements are no more easily decided that anyothers. That is, a field of knowledge with a rich and intricate structure collapses intotriviality in a proposed idealization. Thestriking and useful kind of application of formalderivability in epistemology israther the negative one in which statements are establishedas not decidable (provable) in certain theories, and therefore not actually decidable(provable) by certain specified means.

The third and last notion of provability to be introduced is theoretical provability, bywhich I mean provability in some provably valid theory. This again isnot a mathematicalconcept, since there is no mathematically definable range of provably valid theories.Theoretical provability is the notion commonly employed in philosophical discussions anddoctrines concerning "provability in principle", although not perhaps defined in just theseterms. The essential point is that theoretical provability refers to what lies within thetheoretical range of principles "potentially evident to the human mind", to borrowPutnam's phrase. The weaknesses of this notion, already touched on above, will becommented on further in 5.

4

In the literature on intuitionistic mathematics occurs a variously conceived notion of"canonical proof. Canonical proofs are neither actual proofs, nor formal derivations.Instead they enter into the very meaning of mathematical statements: it is usually said thatto explain a statement is to say what counts as a canonical proof of it. As a consequenceany true statement will have a canonical proof. An actual proof is, in this terminology,normally a "demonstration" showing how (i n principle) to obtain a canonical proof.


32/94

The relation between canonical proofs and actual proofs orformal derivations dependson how the canonical proofs are conceived. In one version, the canonical proofs form anopen and evolving family of (idealized) arguments and procedures, asdo actual proofs; asa consequence, the meaning of mathematical statements will be similarly open andevolving. In another version, the canonicalproofs are essentially mathematical constructs,the existence (in an appropriate sense) of which guarantees the truth of the correspondingstatements. To ti e themeaning of mathematical statements to canonical proofs of the latterkind is not in itself to say anything about the acceptability of realism or anti-realism. Toclarify this point, I will give an analog of intuitionistic canonical proofs for classicalelementary arithmetic. We are toexplain the statements of elementary arithmetic by sayingwhat is meant by a (canonical) proof and a (canonical) refutation of an arithmeticalstatement. This explanation proceeds by induction on the complexity of the statement(assumed to be formulated, as usual, in predicate logic), as follows (s and t stand forclosed terms, and is the numeral with value n):

A proof of s= t is a computation reducing s an d t to th e same numeral.A refutation of s= t is a computation reducing s and t to different numerals.

A proof of A=)B is a function taking proofs of A to proofs of B.A refutation of A=)B is a pair consisting of a proof of A and a refutation of B.A proof of -lA is a refutation of A.A refutation o f - is a proof of A.A proof of VxA(x) is a function taking each to a proof of A(n).A refutation of VxA(x) is a refutation of A(n) for some n.

A proof of 3xA(x) is a proof of A(n) for some n.A refutation of 3xA(x) is a function taking each to a refutation of A(n).

On th e basis of these clauses, we ca n establish by induction that th e rules of first orderarithmetic are correct in th e sense that all theorems of first order arithmetic have canonicalproofs. For example, th e classical rule of negation elimination is trivially correct since acanonical proof of - is th e same thing as a canonical proof of A. To verify th e rule ofnegation introduction, we need to establish by induction that A is provable if an d only if itis not refutable (using, of course, classical logic in the inductive argument). An actualproof of an arithmetical statement is, from this point of view, not a canonical proof, but a


33/94

demonstration showing that the statement has a canonical proof.

Nothing is gained by this explanation of arithmetic. Nevertheless it is perfectlyanalogous to similar intuitionistic schemes. The analogy extends to the relation betweendemonstrations and canonical proofs: in both cases the demonstration establishes theexistence of a canonical proof. There is a difference in that the intuitionistic demonstrationgives a recipe for constructing or obtaining (in principle) a canonical proof of thestatement. This difference in the notion of mathematical existence is not anepistemological difference which breaks the analogy, since the canonical proof, e.g. of anexistential arithmetical statement, has as a rule noepistemological significance at all .^^

I have introduced these classical canonical proofs only to emphasize that the use ofcanonical proofs to explain mathematical statements is compatible with a realisticinterpretation of those statements. For example, if Goldbach's conjecture is true in itsintuitionistic interpretation, then the conjecturehas a canonical intuitionistic proof. Indeed,to assume that the conjecture is true can only mean, from the intuitionistic point of view,to assume that thereis such acanonical proof. But the same thing can be said on a realisticinterpretation: the conjecture is true i f and only if it has a canonical proof. In order tobring out the difference between a realistic and a non-realistic interpretation, we must gobeyond the notion of canonical proofs as truth-guaranteeing constructs. In particular,there is nothing in this notion to support the idea that Goldbach's conjecture istheoretically provable if true - themotivation for this idea must be sought elsewhere.

5The notions of actual and theoretical provability have, according to the preceding

argument, an essential opennessor indeterminateness. This point will be important in laterchapters and deserves closer consideration. The following comments will be restricted totheoretical provability and to problems of elementary arithmetic such as those mentionedby Hilbert in the quoted passage.

In the philosophical literature it is widely agreed that theoretical provability cannot beidentified with provability in any particular formal system. The usual argument is that an yprovably valid theory can be extended, e.g. by reflection principles as shown by Gdel,to yield a stronger provably valid theory. The basic reflection principle for a theory T is

There are nowadays striking illustrations of the gulf between actual proofs andcanonical proofs. A theorem proved by Friedman (see Harrington et al.) states that a certainexistential arithmetical statement which is provable in a few pages in ZFC has no proof inpredicative analysis (and a fortiori no computational proof) shorter than 21000] symbols,where the brackets indicate an exponential tower of twos of height 1000.


34/94

the assertion that all statements A (here:of elementary arithmetic) provable in T are true.This is not ametaphysical but a mathematical principle, expressible in the schema

(R) if #A is derivable in T,then A.Each instance of (R) is a mathematical statement, in fact a statement of elementaryarithmetic, given the restriction to arithmetical statements A and the conditions imposedonT in 3 above. One formof Gdelian extension of T is obtained by adding the instancesof (R) as new axioms. Assuming the theory T to be a formalization of evidently validprinciples - an example would be Peano arithmetic - the validity of (R) follows from theobservation that itonly makes explicit our acceptance of the principles embodied in T.

It is notable that mathematicians arereluctant to accept arguments along these lines -call them arguments by reflection - which have no ordinary mathematical contentwhatsoever, but yield mathematical conclusions. They are certainly not proofs in theordinary sense, and if we call (R) provable, the word must be understood in the generalsense of 3, i.e. as"mathematically verifiable" or "mathematically knowable". Even so, itis common enough for mathematicians to claim that not even arithmetic is known to beconsistent. Since the rules and axioms of Peano arithmetic are not in the leastcontroversial, there is at least an apparent disagreement here concerning theepistemological status of Gdelian extensions.The extent of this disagreement isdifficultto estimate because of the prevalence of ritualistic attitudes regarding consistency.Generally speaking, it is notunreasonable to accept a theory in a tentative or moreor lesspragmatic way without being prepared to assert (R). The reflection principle makes a largeglobal claim for the theory Tand is evident only i f we have a corresponding global insightinto the theory. What I mean by this is illustrated by the axiom schemas of ZFC. Theevidence for these schemas derives from thevery abstract second order principles: i f wehave no confidence in these principles, the first order schemas will appear as logicallyimmensely complicated assumptions. On the basis of our familiarity with the use of theschemas we may well regard them, in a tentative way, as acceptable mathematical tools,without being prepared to hold that they are in any sense demonstrably consistent.

From a formal point of view, nothing in Gdel's theorem excludes the possibility thatthe set of theoretically provable statements of elementary arithmetic is recursivelyenumerable. In the terminology of the debates where this point arises: we may yet beTuring machines. However, the indeterminateness of provability means that there is nowell-defined "set of theoretically provable statements", and this is the aspect of provabilitywhich is of present concem.

The possibility of new concepts and methods being introduced into mathematics is thefirst source of the indeterminateness of theoretical provability. That thispossibility is real


35/94

is shown by the growth of mathematics e.g. since Goldbach's conjecture was firstformulated. We cannot circumscribe these concepts and methods in any way; they havethe same characterof invention, creation, and revelation as work in other fields of art andscience. This does not by itself imply any indeterminateness in the notion of theoreticalprovability: we need to invoke Gdel's theorem again toestablish that not only actual buttheoretical provability has all the indefiniteness of the possible paths of human invention.

To speak of the range of mathematical invention is to draw immediate attention to thesecond source of the indeterminatenessof theoretical provability: the non-existence of an yabsolute standards of evidence for rules and axioms. This point is fully illustrated by thedivergent attitudes towards currently formulated principles. There is a standard spectrumof formalized mathematical theories, of increasing degrees of abstraction. At one end wehave primitive recursive arithmetic which formalizes "finitary" reasoning; at the other endwe have set theories with strong axioms of infinity, often spoken of as "theological".Considering one of these theories T as a machinery for producing arithmeticalconclusions, we ask: how convincing is the reflection principle (R), and what kind ofjustification can we give for it?This question can occasionally be answered by a Hilbertstyle reduction of T to a weaker theory (at least for some range of arithmetical statementsA), the eliminability of the axiom of choice being a well-known example. Anotherinstance of this is the eliminability of classical logic from proofs of arithmetical V3-statements. In general, however, (R) has no technical justification, and the informal("philosophical", "semantical") justifications are rudimentary even when (R) is whollyconvincing. On anybody's view, there will be cases where (R) is wholly convincing, orfairly convincing, or not implausible, or highly problematic: there is no theory ofmathematical evidence to apply to such cases, and no apparent way of resolvingdifferences of opinion as regards what is oris not evident or acceptable.

The point here at issue is whether such differences of opinion can be held to concernquestions of fact. For example, is it a question of fact whether we know that ZFC isconsistent (albeit without an y apodictic certainty) ormerely believe it to be soon the basisof "familiarity passing as evidence", a mistaken assimilation of the infinite to the finite, orsomething similar? Without arguing the point I submit that it is not; that standards ofknowledge, in mathematics no less than elsewhere, can and must be understood anddiscussed without presupposing a timeless or absolute distinction, founded in the natureof things, between what is known and what is merely accepted. This being so, thequestion whether a statement A can be known, i.e. is provable in the widest sense, neednot in general have any determinate answer, known or unknown.

It was suggested in 1.4 that it is uncontroversial that provability is an essentiallyopen notion. This remark must now be qualified. There is general agreement thatprovability is an open concept in the sense that there is no well-defined totality of


36/94

allowable methods and principles to be used in proofs. This corresponds to what I calledthe first sourceof the indeterminateness of theoretical provability. In a similar vein, it hasoften been observed in other contexts that we cannot inany informative way circumscribethe possibilities of solving theoretical or technical problems, or of obtaining newknowledge. There is alsowide agreement that there areno absolute standards of evidencein mathematics. On the other hand it is not unusual for philosophersor mathematicians tospeak about proofs in notably objectivistic and absolute terms, especially when puttingforth the view that truth in mathematics means provability. ThusPrawitz suggests that wemay reasonably assume an "objective realm of proofs" in terms of which we canunderstand ordinary references to the truth or falsity of mathematical statements. Forexample, the hypothesis or postulate that the arithmetical theorems of ZFC are true istaken to mean that each such theorem "can be proved" in the sense of having a proof inthis objective realm. The proofs are here "canonical proofs", and as noted in 4, thephilosophical import of this scheme dependson how the canonical proofs are conceived.If they are themselves formally defined mathematical objects, nothing in my previouscomments contradicts the description of the totality of canonical proofs as forming an"objective realm of proofs". Prawitz's canonical proofs are not of this kind; they are"related to ourrecognitional capacities". In particular, a proof of an arithmetical statementVxP(x) is not a mere function or algorithm which yields a proof of P(n) given n, but a"method for which it recognized that" it yields a proof of P(n) given n. As an example,consider the proof by reflection that every numerical equation provable in ZFC is true.The function or algorithm involved is trivial: given a derivation in ZFC of a numericalequation, we obtain a proof of its truth by carrying out the computation. To have a"method" we must recognize that this function does what is required; i.e. the wholesubstance of the canonical proof lies in the unspecified "recognition". Hence the idea thatsuch proofs can be thought of as forming an "objectiverealm", and that the truth orfalsityof mathematical statements, understood in such terms, is timeless depends heavily on(among other things) the notion that there is an objective and timeless distinction betweengenuine insight orrecognition and mere conviction oracceptance.

In support of the view that it is makes good sense to quantifyover a timeless domainof canonical proofs, Prawitz cites as an example of similar quantification over "methods"the observation that there may exist "a specific method for curing cancer, which we maydiscover one day, but which may also remain undiscovered". Now in practice it is a verydelicate and difficult question whether ornot something is a method forcuring cancer, bu tthe natural view is certainly that it is a question of fact. In justification of this natural viewI would point to the distinction between having cancer and not having cancer: this is aperfectly objective distinction (in the metaphysical sense of being independent of what, i fanything, we think about the matter) and a genuine method for curing cancer is a

Prawitz, 7.


37/94

procedure by which an organism is transformed from the first state to the second. In thecase of the "methods" that constitute canonical proofs the situation is very different. Agenuine method would serve to transform an organism, e.g. me, from a stateof ignoranceor unjustified affirmation to a state of knowledge and insight: the distinction betweenthese states is not (according to the remarks above) an objective one in the strongmetaphysical sense here at issue.

6The considerations of 5 have no application in problem solving. Take for example

the classical problem mentioned by Hilbert regarding the existence of infinitely manyFermat primes (primes of the form 2n+l). In seeking a solution to this problem we haveno need to stop and reflect on the lack of any absolute standards of evidence, or on theessential openness of mathematical proof. Certainly we do not, in seeking a proof,presuppose that the existence or validity of proofs is an objectively determinate matter.Our business in solving the problem is to find, or invent, or construct - each term isequally acceptable - a proof by which the problem is settled. Once a proof has been foundwe may appropriately consider its epistemological merits or problematic aspects. It wouldbe surprising if the problem of the Fermat primes turned out to require problematicprinciples for its solution, but there is no point in speculating on this interestingpossibility in advance of having any proof.

A practically minded mathematician may well wonder, therefore, what is supposed tobe the point of emphasizing the openness and indeterminatenessof provability. The pointis a simple one: I want to bring out our realistic preconceptions or attitudes bycontrastingtruth and provability. Whether or not it is provable that there are infinitely many Fermatprimes is, in the sense indicated above, a question which may have no other answer thanthat which we decide to give it (and thus perhaps none). Whether or not there areinfinitely many Fermat primes, on the other hand, isquite independent of the obscure andmore or lesssubjective questions of what isevident or valid: it is, quite simply, aquestionof mathematical fact, and there is nothing subjective orindeterminate about itSo, at least,goes the realistic view. And to the extent that we incline to this view, we distinguishbetween truth and provability in mathematics in a realistic way. The role and importanceof this realistic attitude inour thinking is the subject of the next chapter.


38/94

mREALISM

In speaking of realism and anti-realism as metaphysical doctrines or attitudes, I usethe word in an imprecise but I believe standard sense. Metaphysics, particularly in itsontological aspect, is concerned with the question of what there is, what is real orobjective, what belongs to the nature of things and what is human invention, imaginationor convenience. Today few philosophers seek or expect a comprehensive answer to thismetaphysical question, and many prefer to frame their metaphysical concerns in moresober and perhaps scientific terms. Still, I believe most people would agree thatrecognizably metaphysical views play an important role in both everyday andphilosophical or scientific concerns. In this chapter an attempt will be made to substantiatethe claim that (amodest and limitedform of) mathematical realism plays such a role.

The words "true" and "truth" figure very largely in debates over realismand indeed inphilosophical discussion in general. In this essay the words "true" and "false" will alwaysbe understood as though explained through Tarskian equivalences

#A is true if and only i f A,#A is falseif and only if it is notthe case that A,

where A is a statement and #A an expression designating A. This isa kind of minimal useof "true" and "false". In any standard use of these words, the Tarskian equivalences willhold; in the minimal use, they fully explain the terms. In so far as the word "truth" is notinterpretable as "true statement" it will be explainedin context in the following.

This policy is not based on any "redundancy theory of truth" or on any kind of theory.My motives are wholly practical. The philosophical use of the terms "true" and "truth" isvery convenient, very natural, and imbued with countless associations. It is never veryclear just what is at issue when one relies on these associations in philosophicaldiscussions. To avoid the free use of this terminology is not to dismiss or resolve any ofthe "problems of truth", "theories of truth", "notions of truth", etc, but to try to force thetreatment of these problems, theories, notions into a lessfacile mold.


39/94

I will give a fewexamples of what I regard as the drawbacks of the free use of "true"and its cognates. Theconvenient term "truth conditions" is often used. When one speaksof the "truth conditions" of a statement "obtaining" or "not obtaining" there is thesuggestion of some background matrix of factual circumstances or states of affairs interms of which the statement is to be understood. In the Tractatus the states of affairs areexplicitly postulated, and every meaningful statement has truth conditions, determined byits relation to elementary propositions. In ordinary philosophical usage, the term is usedwithout any explicit commitment to a metaphysics modelled on the propositional calculus,but with a definite ontological slant Inspection reveals that the use of the term is guidedby tacit restrictions on the allowable "conditions" together with the requirement that aspecification of the truth conditions of a statement should allow us to make immediatesense of the negation of the statement in terms of those same conditions. This requirementis not explicitly stated, but is implicit in the terminology: "the truth conditions of A do notobtain" is the obviousnegation of "the truth-conditions of A obtain". Thus one "specifiestruth conditions" by giving equivalences subject to these restrictions, and when suchequivalences are hard tofind there is said tobe a problem concerning the truth conditionsof the statements in question (e.g. ascriptions of responsibility or belief statements). Whatis usually missing is any explicit consideration of what restrictions apply to the"conditions" and why the statements at issue should have truth conditions. Thus guidingassumptions are left unarticulated and unexamined and act through a terminology which isused as though it were self-explanatory.

Again, consider the debates over whether "the notion of truth" has a place in physics,or ethics, or mathematics. Such debates often rely heavily on unstated formal ortraditional attributes of "the notion of truth". Simply because of the form of "A is true",itis apparently incumbent on us, if we use this locution, to be able to make equally goodsense of "A is not true", "if A is true...", "A is known to be true" etc. Also there is atraditional suggestion that "truths" properly so-called are in somesense statements of fact;that the question "is A true?", if at all admissible, must admit a clearcut answer. Here tooa number of ill-defined and unargued requirements are expressed in a too convenient way.

It is not surprising that scientists tend to shy away from talk of truth. Bondi's remarkthat "my own inclination is that science has nothing to do with truth" is not untypical, andsometimes such declarations are made with a great deal of satisfaction in spite of beingquite absurd. If science has nothing to do with truth there can be no conflict - so onewould imagine - between e.g. astronomical and geological history and cosmology andother stories of what the world is and was like. This is not in practice the attitude ofscientists, but the easy dismissal of "truth" is not usually accompanied by any moreilluminating (and demanding) reflections on what is and is not claimed for science. Thisnegative reliance on the terms "true" and "truth" is just as unsatisfactory as the idea that

Bondi, p.3.


40/94

the mere use of the term "true" allows one to put meaningful and important questionsabout scientific theories ("is the theory of relativity true"?).

I am not proposing to reject such questions as "What is mathematical truth?" orformulations of metaphysical views such as "mathematical truth is independent ofprovability". My point is that we need to unpack these formulations and be explicit bothabout the unstated constraints and assumptions suggested by the terminology, and aboutthe problems addressed.

2The more specific definition of what constitutes a realistic interpretation or use of

mathematical statements to be given in 3 has three general characteristics: (i) Theemphasis is on undecided statements, rather than on theorems or assertions; (ii) realismistaken to begin with computational (finite) mathematics; (iii) realism isnot taken to be tiedto any particular analysisor explanation of mathematical statements. In thenext few pagessome motivation will be given for thesecharacteristics.

In his Apology,Hardy comments as follows:It may be that modem physics fits best into some framework of idealisticphilosophy -1do not believe it, but there are eminent physicists who say so.Pure mathematics, on the other hand, seems to me a rock on which allidealism founders: 317 is a prime, not because we think so, or because ourminds are shaped in one way rather than another, but because it is so,because mathematical reality is built that way.

These comments are open to criticism. No idealist would claim that 317 is a primebecause we think so; that 317 is a prime "because it is so" is on the face of it a vacuousobservation. But Hardy's remarks have a point that has been frequently overlooked.Consider the classical question - intensively treatedin the philosophical literature - of thestatus of the equation "7+5=12". Is this equation conventional or factual, analytic orsynthetic, a truth of logic, a regulative principle, a rule concerning the use of signs? Thelengthy and intricate investigations of such questions can presumably be extended tocover "317 is a prime". Clearly these investigations are a great deal more sophisticatedand informative than Hardy's remark. Yet it seems to me that they fail to come to gripswith the substance of the observation that 7+5 equals 12 or 317 is a prime "because it isso, because mathematical reality is built that way". The reason for this failure is that theyignore the question of the status of extremely lengthy computations or of computationalstatements not known to be decidable. "7+5=12" and "317 isa prime" are special in being

Hardy [2], p.l30.


41/94

easily verified - as ise.g. the statement "the Statueof Liberty is larger than a banana". Inpractice such statements may function as "rules of language", but does this tell usanything about the metaphysical status of arithmetical equations or size comparisons? Idon't believe it does: it is only when we turn to contexts where the truth or falsity of acomputational statement is moreor less an open question that we can properly appreciateHardy's remark.

For a formulation of a realistic view, therefore, we replace "317 is a prime" with thestatement

5 75 7r - R N 5 75 +7 + 1 is a prime.

The realistic attitude expressed by Hardy must now be worded a bit differently: whetheror not (B) is true is quite independent of what we believe about it; it is a question ofmathematical fact. If we ask what makes it true (i f true) or false (i f false) we can only givesome variant of this trivial answer: the mathematical facts. "Mathematical reality is builtthat way." This is the realistic attitude towardscomputational statements. It has nothingtodo with any insistence on theimpossibility or irrationality of denying arithmetical truths,or on the certainty or incorrigibility of mathematical knowledge. It is largely independentof whether the rules of arithmetic are characterized as conventional or factual, analytic orsynthetic, since the truth or falsity of (B) is an open question whatever the character ofthose rules.

According to a widespread line of thought there is no metaphysically essentialdifference between (B) and "317 is a prime", since (B) is after all decidable in principle ifnot in fact. This view is associated with the traditional doctrine that there is a greatmetaphysical gulf between the finite and the infinite. The accompanying tendency todismiss distinctions within the realm of the finite as inessential makes it difficult toappreciate the highly abstract character of (B). R.O.Gandy highlights this abstractcharacter in dramatic terms in his remark that "statements such a

franzen provability and truth

Documents