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Philosophy of Mathematics

A Neo-Formalist Approach to Mathematical Truth

Alan Weir
The Queen's University of Belfast

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ABSTRACT: I outline a variant on the formalist approach to mathematics which rejects textbook formalism's highly counterintuitive denial that mathematical theorems express truths while still avoiding ontological commitment to a realm of abstract objects. The key idea is to distinguish the sense of a sentence from its explanatory truth conditions. I then look at various problems with the neo-formalist approach, in particular at the status of the notion of proof in a formal calculus and at problems which Gödelian results seem to pose for the tight link assumed between truth and proof.

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§O. A great many philosophers, including some of a generally realist outlook, feel strongly attracted to anti-realism in the philosophy of mathematics because of the well-known epistemological difficulties with mathematical realism. Those whose scepticism regarding mathematical realism derives from specific features of the mathematical case rather than a general anti-realist rejection of unverifiable truths will tend to eschew constructivist anti-realism, especially of the highly revisionist form found in intuitionism. For such philosophers, modal reconstruals of mathematics or fictionalist denials that mathematics comprises a body of truths hold greater attractions. Very few mathematical anti-realists now view formalism as a viable account of mathematics, however.

It is not hard to see why formalism has fallen out of favour- the standard objections seem insuperable. (1) The formalist claim that mathematical utterances cannot be used to make assertions and that mathematical theorems are not true is both grossly counterintuitive and leads to huge difficulties in explaining why mathematics is so useful in the physical sciences. Even worse, if formalism is not a form of strict finitism and the language of mathematics is therefore taken to consist of infinitely many expression strings then the formalist seems committed to an ontology every bit as abstract as the platonist's (perhaps even the same ontology, if one identifies formal languages with sets of abstract objects). So either the formalist embraces a bizarre conviction that spacetime contains infinitely many concrete utterances or else lapses into self-refutation.

These objections do indeed seem to me to be conclusive, as pressed against textbook 'formalism'. What I want to argue is that there is a variant position, recognisably akin to formalism, but which evades those objections and deserves serious consideration. I will call this view 'neo-formalism'.

§1. Neo-formalism takes as its starting point that distinction between the sense and the explanatory truth-conditions of a sentence familiar from such programmes as those of giving a precise theory of meaning for vague language, or a context-independent theory of meaning for context-dependent language (or at any rate showing how such theories are possible). Although any particular such programme is contentious, the general idea is, I think, relatively uncontentious, at least for anyone to whom the idea of a systematic semantic theory is not entirely hopeless. The idea is, then, that e.g. 'It is raining in Boston Mass. 13.00 EST, 14th August 1998' though certainly not having the same truth conditions as the sentence 'its raining', a fortiori not having the same sense, may nonetheless play a key role in explaining what makes the sentence true or false, on a particular occasion. More generally, appeal to 'pegged sentences' of the above relatively context-free type may play a crucial role in explaining how it is we understand the context-dependent 'it's raining'. Similar remarks apply to the relation between vague sentences and the precise language which will feature in any explanation of how we understand the vague language, according to those who believe in the existence of such precise explananda.

If a sceptic asks why the explanandum sentence has the explanatory truth-conditions it does without actually meaning the same as the explanans, two reasons can be given: (a) speakers may modulate their opinions on the sentence so as to settle on the verdict that it is true just when the explanatory truth-conditions say it is true (allowing for explicable error) but lack reflective grasp of some of the concepts in the explanans; (b) the sentence and its explanatory truth-conditions may behave very differently in modal or other such intensional contexts: 'I believe it's raining' can be true even though 'I believe it's raining in Boston Mass. 13.00 EST, 14th August 1998' is false.

This sense/explanatory truth-conditions distinction is even more plausible from within a minimalist perspective according to which, roughly speaking, any utterance which satisfies the surface criteria for being an assertion (2) is an assertion, and is thereby truth-evaluable, that is satisfies the Tarskian truth scheme for utterances. (3) On such a minimalist position true assertions can be non-representationally true, full representationality holding only of those sentences whose sense just is their explanatory truth-conditions.

The neo-formalist programme then aims to give a non-representational account of mathematics by distinguishing the sense of mathematical sentences (there may be no non-trivial way to give this in general, as far as neo-formalism is concerned) from the explanatory truth-conditions. Adhering to minimalism, the neo-formalist affirms that mathematical theorems express true assertions but denies that they are made true by virtue of correctly representing some mind-independent realm and rejects a referential semantics as not suited to form part of an explanation of how we understand them. If this programme can be made out then the double talk on existence deprecated by Quine (4) can be justified; we can hold that, yes there are infinitely many prime numbers while denying prime numbers really exist, meaning by this that the existence claims are made true non-representationally by the holding of some suitably non-platonistic set of explanatory truth-conditions.

What could these truth-conditions be? The basic idea is as follows. Take the standard formalist analogy with games and consider in particular linguistic games, games in which moves are utterances, or can only be effected by means of utterances. A hackneyed but useful example is postal chess. The (textbook) formalist would surely be right to say that such utterances are not assertions and just as surely wrong to assimilate mathematical sentences to non-truth-evaluable moves in games. But though we make no assertion in such contexts, we can of course make assertions about games. We can say that such and such a move is legitimate, is in accord with the rules; or that such and such a state of play cannot be reached from the current position. These assertions have, I take it, a straightforward representational meaning and are made true or false by the facts about the game. Though one might adopt a platonistic metaphysics towards such facts- actual chess events and pieces are mere instances of the abstract game- or a platonistic construal of the notion of possibility which is used in saying that such and such a move is legitimate, I take it that there is no great plausibility in such a position. At any rate, a naturalistic, anti-platonist account of the metaphysics of games is surely not nearly so problematic as anti-platonism in mathematics.

How, then, do we get mathematical anti-platonism out of such considerations? Imagine that as well as making moves in the game players also start to make what superficially seem to be assertions by means of declarative sentences closely linked to the utterances which form part of the game. As well as posting their move, 'Be3', say, they say things like 'Bishop moves to e3'. Suppose they are disposed to say such things when and only when they judge (perhaps rather unreflectively) that the move in question is legitimate at that stage of (5) The suggestion then is that any such declarative sentence expresses a truth-evaluable assertion whose explanatory truth-condition is that the move in question (i.e. the related non-assertoric linguistic utterance) is legitimate in the current context (a stage of the game say). But there are, as before, grounds for denying that the explanatory truth-condition gives the sense of the sentence. For, firstly, the utterers may play the game in an unreflective fashion; they may obey the rules, and tie their declarative utterances such as 'Bishop moves to e3' to situations where they feel the move is acceptable, without having an articulate grasp of the rules and so without grasping the concept of being a legitimate move according to the rules of chess. Secondly, the sentence and its explanatory truth-conditions may behave very differently in complex contexts such as modal or intensional contexts. Different behaviour in belief contexts and the like is secured by the first point, the fact that they may tie the assertoric utterances to the merely formal moves in the game unreflectively.

Now the games analogy is very limited as a means of explicating a notion of non-representational, non-realist discourse which is nonetheless truth-evaluable. This is because games such as postal chess do not satisfy even the minimal syntactic criteria for assertoric discourse, in particular closure under logical constructions such as negation. (6) What we need, then, is a game in which ordinary words, such as 'exists' and 'not' can occur, with their ordinary sense, alongside the special game symbols.

The neo-formalist claim is that mathematics constitutes such a practice. The conception here is of a two-tiered use of language:- at the bottom level mathematical sentences are used to make non-assertoric moves in a formal calculus. This is exactly the status of formal sentences in elementary logic classes and also, arguably, the status of elementary arithmetical sentences for children learning mathematics. (7) But speakers can in addition make assertions which are keyed to their (doubtless often rather inchoate) beliefs as to the provability, according to the rules of the formal calculus, of the related sentences. Thus, the claim is that the assertion 'sixty eight plus fifty seven equals one hundred and twenty five' is true just when '68+57=125' is provable according to 'correct' norms of proof (I will return to the question of correctness of proofs). But, appealing to the sense/explanatory truth-conditions distinction, the neo-formalist denies that 'sixty eight plus fifty seven equals one hundred and twenty five' means that '68+57=125' is provable. Rather it has a non-representational sense, for which no non-trivial synonym need exist; it is not made true or false by any external reality, whether that of a structure of abstract objects or by the corpus of actual concrete utterances of moves in non-assertoric mathematical calculi.

§2 One obvious problem for neo-formalism is its apparent conflict with Gödel's first incompleteness result showing that not all mathematical truths are provable, under a certain conception of provability. Even though the neo-formalist makes no synonymy claim between 'sixty eight and fifty seven equals one hundred and twenty five' and '"68+57=125" is provable', this result seems to rule out any tight equivalence between truth and proof of the sort envisaged.

However Gödel's result appeals to a very special notion of proof, one in which the class of (number-theoretic codes of) proofs is recursive and the derivability relation, over recursive classes of premisses, is recursively enumerable. This is generally thought to be a plausible constraint on proof, given the epistemic role it plays. There are two reasons why this orthodoxy should be challenged: a) firstly it assumes mathematical proof is an entirely formal, syntactic notion; b) secondly it assumes all genuine proofs are finite objects, or at least that all can be coded by the natural numbers and in such a way that there is an algorithm for determining the overall premisses and conclusions of each proof.

As to the first point, we should note that the proofs which actually convince us, in number theory, analysis, set theory, indeed proof theory, are all written in natural languages, augmented with some special mathematical notation. It is not at all obvious that proofhood in mathematical English, German, Chinese or whatever is accurately represented as being effectively decidable- such language contains, for example, ambiguity and context-dependence, albeit on a smaller scale than everyday language. Even in the case of computer-generated proofs, what convinces a mathematician is not the wad of computer print-out but, among other things, the account of the software used in the proof-searching program (supposing we do not take these things on the authority of experts who are convinced of the soundness of the program); and this account, where it convinces, will be written in technical English, Chinese or whatever. Hence it may well be that a more accurate idealisation of actual mathematical proof should incorporate a non-formal element, perhaps even a semantic element.

Regarding the second point, Gödel's conception of proof as essentially finitary quickly become the dominant one from the 1930's onward. The systems of infinitary proof which emerged thereafter were generally viewed as of merely 'technical interest', in investigating problems involving large cardinals, for example; but infinitary proofs are not 'real' proofs since it is not even possible 'in principle' for us to grasp infinitely long proofs, it is widely believed. This position was not the view of a great many of the founding figures of modern logic. (8) And the matter is certainly not beyond reasonable doubt. Is there really an interesting sense of 'in principle possible' (one which is more than merely a rhetorical embellishment on the claim that a given type of infinite structure exists) according to which it is in principle possible to grasp finite wffs or proofs with more symbols than the estimated number of quarks in the observable universe, but not in principle possible to grasp infinitary wffs and proofs?

§3 A second important question is how one can extend the neo-formalist idea from very elementary arithmetic, where it is plausible to suppose we all have familiarity with a practice which can reasonably be thought of as merely a formal manipulation of symbols, to mathematics in general. Only mathematical logicians engage in formal manipulation of symbol strings which can be thought of as related to assertions in analysis, topology, set theory or whatever, and even then very rarely.

However it is not essential that the syntactic string named in the provability claim- the claim which forms part of the explanatory truth conditions- be distinct from the sentence which expresses the mathematical assertion. (9) What matters is that speakers implicitly understand the string- 'for every set there exists its power set', say- in two distinct ways: firstly as a purely syntactic object upon which certain inferential transformations are legitimate; and a second derivative sense under which one uses the string exactly as one would an assertion but affirming it just when one would judge that the string as a formal expression is derivable according to the legitimate transformations. To be sure, whenever someone engages in genuine deductive reasoning she can be thought of as implicitly treating sentences as formal objects exemplifying syntactic patterns abstracted from their determinate content. The difference in the mathematical case is that there is no such representational determinate content to abstract from. The sense of the assertion that for every set there exists its power set is distinct from but dependent on that of the claim that the formal string 'for every set there exists a power set' is provable; and 'provability' here means derivability in a certain practice in which that string has no meaning other than that given by the transformation rules of the practice.

But what rules are these? We need a distinction between legitimate and illicit transformations, if neo-formalism is to avoid the consequence that in mathematics there is no distinction between truth and falsity. Moreover, it cannot be that a string is provable if derivable in the one true logic from some consistent set of axioms or other. Even if there is only one true logic it would still follow that any logically consistent sentence, relative to that logic is, for the neo-formalist, a mathematical truth. Is 'provability', then, to mean derivability from special axioms such as the Peano-Dedekind axioms or those of standard set theories such as ZFC or NBG? But once we abandon mathematical realism, what is so special about these sentences? They do not depict the structure of some realm of abstract entities so why should the consequences of such axioms enjoy any special status or necessity as compared with the consequences of any other set (at least in the practice of speakers who are mathematically competent but unacquainted with formal axiomatisations)?

The neo-formalist answer is that provability in a practice means derivable using only inference rules which are in some sense analytic, constitutive of the meaning of our logical and mathematical operators. Nothing in Quine's critique of the concept of an analytic sentence shows the concept of an analytic rule of inference to be incoherent. Arguably, indeed, the notion of an analytic inference rule is essential, if one is to make sense of the objectivity of inferential norms. (10) The neo-formalist programme, then, will require an adequate elucidation of the notion of a meaning-constitutive inference rule. Paradigm cases which should be captured by the elucidation will be examples such as conjunction elimination in logic. But we will need also specifically mathematical rules. One natural example is the rule which George Boolos, following Frege, called "Hume's principle": (11) from [property F is 1:1 bijectable onto property G] conclude [the number of F's = the number of G's]; and conversely. Of course you need not be an expert in the second-order logic needed to formalise this principle in order to grasp the concept of a number; still it seems reasonable to think of this principle as an articulation of one implicit in our numerical practice. A child has grasped a fragment of the number series only when, if presented with a comprehensible grouping of n objects, for n in that fragment, she can pair off the objects one-to-one with the first n numerals in some canonical sequence of numerals.

However Boolos raised an objection along the following lines:- Hume's principle is formally very similar to the naïve rules for class, i.e. the rule-form version of Frege's notorious Axiom V. (12) Two responses are open to the neo-formalist. Firstly, that no rule can be meaning-constitutive if it is trivial, cf. the introduction and elimination rules for tonk. A more radical response, though, one which perhaps holds out better prospects for a neo-formalist account of set theory than looking for consistent weakenings of Axiom V, is to deny that naïve set theory is inconsistent. The inconsistency and indeed triviality of 'classical' naïve set theory is a product of three things: the classical operational rules (of some given proof-architecture), the classical structural rules and the naïve rules or axioms, such as Axiom V or naïve comprehension. The conventional response of blaming the last feature rather than, for example, the classical structural rules, is not beyond question the right one.

§4 To conclude: the neo-formalist agrees with the strict finitist that the only objects with a title to being called mathematical which exist in reality are the presumably finite number of concrete mathematical utterances. Some of these utterances, however, are used to assert that infinitely many objects- numbers, sets, strings of expressions, abstract proofs, etc.- exist. For the neo-formalist these utterances express genuine expressions which are true just if that string (or one linked to it in the utterer's practice) is derivable (13) using meaning-constitutive rules implicit in the utterer's practice. Mathematical truth is thus linked, though not as part of the meaning of mathematical assertions, with provability in formal calculi, as the formalists thought, and in such a way as to be perfectly compatible with the claim that all that exists in mind-dependent reality are (perhaps finitely many) concrete objects together with their physical properties.

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(1) The textbook formalist derives largely, perhaps, from the targets of Frege's anti-formalism such as Thomae- cf. Michael Resnik: Frege and the Philosophy of Mathematics (London: Cornell University Press, 1980) Chapter Two. The position of twentieth century figures associated with formalism, such as Hilbert, is considerably more subtle and complex. See e.g. M. Detlefsen: Hilbert's Program: An Essay on Mathematical Instrumentalism (Dordrecht: Reidel, 1986).

(2) Here, admittedly, there is plenty of room for contention as to what these criteria are.

(3) This does not preclude a truth-evaluable utterance or sentence s lacking a truth value:- this will be so if excluded middle fails with respect to (True s v ~ True s) whilst the semantics for the biconditional occurring in the Tarskian scheme renders the instance True s <-> p nonetheless true.

(4) Word and Object (Cambridge: MIT Press, 1960) §49.

(5) In this relatively simple context, judging thus may amount to making the move just when it is legitimate, modulo explicable error.

(6) Cf. Crispin Wright, Truth and Objectivity (Cambridge: Harvard University Press, 1992).

(7) The formal calculus of elementary calculation which a child masters is, of course, not a precisely characterised system such as is found in proof theory; it is formal in the sense of lacking assertoric content.

(8) See Gregory H. Moore, 'Beyond First-Order Logic: The Historical Interplay between Mathematical Logic and Axiomatic Set Theory', History and Philosophy of Logic 1 (1980) pp. 95-137 and especially 'The Emergence of First-Order Logic' in History and Philosophy of Modern Mathematics (Minnesota Studies in the Philosophy of Science No. 11) (Minneapolis: University of Minnesota Press, 1988) pp. 95-135. See also Stewart Shapiro, Foundations without Foundationalism: A Case for Second-Order Logic (Oxford: Clarendon, 1991) Chapter Seven.

(9) The use of English versus Arabic numeral strings in the arithmetic example above was largely for purposes of clarity.

(10) See, e.g. Michael Dummett: Frege: Philosophy of Language (London: Duckworth, 2nd Edition 1973) p. 596.

(11) See "The Standard of Equality of Numbers" in Meaning and Method: essays in honour of Hilary Putnam ed. G. Boolos (Cambridge Eng.: Cambridge University Press, 1990) p. 267. The Frege reference- taken in turn from Baumann's Die Lehren von Raum, Zeit und Mathematik Berlin 1868- is from the Grundlagen §63- see The Foundations of Arithmetic trans. J.L. Austin (Oxford: Blackwell, 2nd Edition, 1980) p. 73.

(12) ibid. p. 273. See also his 'Basic Law (V)' in Proceedings of the Aristotelian Society, Supplementary Volume LXVII (1993) pp. 213-233. Also Hartry Field Realism, Mathematics and Modality (Oxford: Blackwell, 1989) pp. 157-158. Michael Dummett also advances a criticism along similar lines; see his Frege: Philosophy of Mathematics (London: Duckworth, 1991) pp. 188-189, p. 208.

(13) The modality here is a natural one: there could have existed a concrete object which could have been interpreted so as to count as a proof on our actual criteria of proofhood. Must this object be a comprehensible one, or is it enough that each small patch of the proof could have been checked by us for correctness (in which case, large, computer-generated structures or enormous molecular chains are potential proofs)? Neo-formalists will answer differently here depending on whether or not they have sympathy with verificationism in general.

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