Formalism In Mathematics

[Denisha Padley]

Alongwith realism and intuitionism, formalism is one of the main theories in the philosophy of mathematics that developed in the late nineteenth and early twentieth century. Among formalists, David Hilbert was the most prominent advocate.[2]

The early mathematical formalists attempted "to block, avoid, or sidestep (in some way) any ontological commitment to a problematic realm of abstract objects."[1] German mathematicians Eduard Heine and Carl Johannes Thomae are considered early advocates of mathematical formalism.[1] Heine and Thomae's formalism can be found in Gottlob Frege's criticisms in The Foundations of Arithmetic.

According to Alan Weir, the formalism of Heine and Thomae that Frege attacks can be "describe[d] as term formalism or game formalism."[1] Term formalism is the view that mathematical expressions refer to symbols, not numbers. Heine expressed this view as follows: "When it comes to definition, I take a purely formal position, in that I call certain tangible signs numbers, so that the existence of these numbers is not in question."[3]

Thomae is characterized as a game formalist who claimed that "[f]or the formalist, arithmetic is a game with signs which are called empty. That means that they have no other content (in the calculating game) than they are assigned by their behaviour with respect to certain rules of combination (rules of the game)."[4]

Frege provides three criticisms of Heine and Thomae's formalism: "that [formalism] cannot account for the application of mathematics; that it confuses formal theory with metatheory; [and] that it can give no coherent explanation of the concept of an infinite sequence."[5] Frege's criticism of Heine's formalism is that his formalism cannot account for infinite sequences. Dummett argues that more developed accounts of formalism than Heine's account could avoid Frege's objections by claiming they are concerned with abstract symbols rather than concrete objects.[6] Frege objects to the comparison of formalism with that of a game, such as chess.[7] Frege argues that Thomae's formalism fails to distinguish between game and theory.

A major figure of formalism was David Hilbert, whose program was intended to be a complete and consistent axiomatization of all of mathematics.[8] Hilbert aimed to show the consistency of mathematical systems from the assumption that the "finitary arithmetic" (a subsystem of the usual arithmetic of the positive integers, chosen to be philosophically uncontroversial) was consistent (i.e. no contradictions can be derived from the system).

The way that Hilbert tried to show that an axiomatic system was consistent was by formalizing it using a particular language.[9] In order to formalize an axiomatic system, you must first choose a language in which you can express and perform operations within that system. This language must include five components:

Gdel's conclusion in his incompleteness theorems was that you cannot prove consistency within any consistent axiomatic system rich enough to include classical arithmetic. On the one hand, you must use only the formal language chosen to formalize this axiomatic system; on the other hand, it is impossible to prove the consistency of this language in itself.[9] Hilbert was originally frustrated by Gdel's work because it shattered his life's goal to completely formalize everything in number theory.[10] However, Gdel did not feel that he contradicted everything about Hilbert's formalist point of view.[11] After Gdel published his work, it became apparent that proof theory still had some use, the only difference is that it could not be used to prove the consistency of all of number theory as Hilbert had hoped.[10]

Hilbert was initially a deductivist,[citation needed] but he considered certain metamathematical methods to yield intrinsically meaningful results and was a realist with respect to the finitary arithmetic. Later, he held the opinion that there was no other meaningful mathematics whatsoever, regardless of interpretation.

Haskell Curry defines mathematics as "the science of formal systems."[13] Curry's formalism is unlike that of term formalists, game formalists, or Hilbert's formalism. For Curry, mathematical formalism is about the formal structure of mathematics and not about a formal system.[13] Stewart Shapiro describes Curry's formalism as starting from the "historical thesis that as a branch of mathematics develops, it becomes more and more rigorous in its methodology, the end-result being the codification of the branch in formal deductive systems."[14]

Care must be taken, however. Wittgenstein distinguishes utteranceswhich are sinnlos, which lack sense (including logicaltautologies and contradictions here) from those which areunsinnig, nonsensical; it is not clear into which classmathematical utterances fall. One might well think that the gameformalist should treat mathematical utterances, on that view juststrings of meaningless marks, as unsinnig, not justsinnlos. One clear difference from game formalism however isthis: for Wittgenstein mathematics should not be conceived of as acalculus separate from other uses of language. Rather he attempts toshow that parts of arithmetic, at least, can be seen as grounded innon-mathematical uses of language. Frege by contrast, whilst arguingthat a proper account of arithmetic (and analysis) should show how itsgenerality enables one to give a uniform account of multifariousdifferent applications (cf. Dummett, 1991 Chapter 20) also arguedstrongly for the view that mathematical utterances have a meaningindependent of, conceptually prior to, their use in applications.

A number of concerns arise here. How can Carnap distinguish betweenempirical, scientific theories, and mathematical ones? Secondly, ifpragmatic utility is primarily a matter of empirical applications, howdoes the Carnapian formalist know that a given calculus willconservatively extend empirical theory, how can this be known withoutappeal to meaningful mathematical results? Carnap writes:

That is, we link the numeral for zero with a sentence stating there isexactly one entity of the appropriate type, the numeral for one with asentence stating there are no such entities. If we do so, add therules for standard decimal arithmetic, and then try to apply thiscalculus, disaster will ensue; but do we not need acontentful conservative extension result to show that for thecalculi we do use, no disaster can occur?

The connection with intuitionism, then, is clear: but what is therelevance of the CH correspondence to formalism? There are, in thefirst place, clear overlaps between some forms of intuitionism andcertain formalist positions. Not the philosophical intuitionism of thefounding father Brouwer, of course, with its ontology of mathematicalobjects as mental constructions and an epistemology in whichmathematical knowledge is based on internal reflection on thesuccession of ideas; this is a mathematical metaphysics far removedfrom formalism. But many constructivists have embraced, withoutaccepting his metaphysics, the Brouwerian identification, or closelinkage, of mathematical correctness (truth, if one is prepared tospeak of mathematical truth) with provability. This sort ofidentification is more than congenial to a certain brand of formalism,one which rejects the idea that mathematical theses represent amind-independent reality and which also divides the mathematical sheepfrom the goats on the basis of those which are provable, in someformal system, versus those which are disprovable.

But there are also substantial differences between the intuitionistand the formalist. For one thing, not only Brouwer but also many laterconstructivists refuse to identify provability with provability insome formal system. For another, formalists have generally felt freeto help themselves to classical logic, and have emphasised thefree creativity of the mathematician: she should be free togenerate whatever mathematical theories she wishes, subject only towithdrawing them if they turn out to be inconsistent (in the chosenbackground logic).

On the first point the formalist will, of course, be a formalist! Shewill link correctness, at least at the most fundamental level, toformal proof. Here, then, the CH correspondence, or bettercorrespondences, are surely very attractive to the formalist. Thelinkage between propositions and computations, algorithmic reductionsof terms coding proofs to irreducible normal forms in particular, fitsvery snugly with those versions of formalism which take mathematics tobe, at heart, shuffling of symbols with no external reference.

What of the free creativity the formalist cherishes? Constructivisttype theory has, of course, been extended well beyond Heytingarithmetic; particularly ambitious extensions are to be found in theunivalent foundations project based on homotopy type theory (Awodey,2014). This, then, is an avenue a formalism based on Formulae-as-Typesmight pursue. But for a formalist who wishes to be non-revisionistabout non-constructivist mathematics the prospects are perhaps lessclear. It is not enough just to add in the extra axioms or inferencerules which yield the particular theory, in a standard framework, e.g.of a first-order or higher-order language. For one needs to do thefurther work needed to show that an extension of the CH correspondenceobtains in this system.

There is also the problem of applicability, which Frege thought aninsuperable one for formalists. What can the meaning be of appliedmathematical notions, such as the number of \(\phi\)s,where mathematical and non-mathematical discourse is mixed together?Unless the CH formalist wishes to go down the Dummettian anti-realistroute and generalise the notion of proof to a notion of verificationappropriate for empirical language, she will have to find a way ofcombining, without too much ad hocness, a proof-theoretic semanticsfor pure mathematics with a different, perhaps a realist,truth-conditional semantics, for empirical language.

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