## 1. Introduction



The *locus classicus* of game formalism is not a defence of the
position by a convinced advocate, but a demolition job by a great
philosopher, Gottlob Frege. Not that he was attacking a straw man
position: the highly influential diatribe by Frege in volume II of his
*Grundgesetze Der Arithmetik* (Frege, 1903) is an attack on the
work of two real mathematicians, H. E. Heine and Johannes Thomae.
Moreover, philosophers of mathematics are wont to claim that the
position is still widely adopted by mathematicians. It must be
emphasised, however, that "formalism" in this
sense--the Heine/Thomae position as interpreted by Frege, and its
descendants--is to be distinguished from a more sophisticated
position (it is claimed), namely Hilbertian formalism. For more
information on the latter see Detlefsen (1993) or consult the entries
on
 Hilbert's program
 and
 the Frege-Hilbert controversy.)
 Detlefsen (2005) also provides a detailed historical treatment of
formalist themes in thinkers from the ancient Greeks up to the period
of Frege and Hilbert and beyond which is the focus here. 



Although it is the non-Hilbertian approach we will be concerned with
in this entry, we briefly discuss the Hilbertian approach. The
Hilbertian position differs because it depends on a distinction within
mathematical language between a *finitary* sector, whose
sentences express contentful propositions, and an *ideal*, or
*infinitary* sector. Where exactly Hilbert drew the
distinction, or where it should be drawn, is a matter of debate.
Crucially, though, Hilbert adopted an instrumentalistic attitude
towards the ideal sector. The formulae of this language are, or are
treated as if they are, uninterpreted, having the syntactic form of
sentences to which we can apply formal rules of transformation and
inference but no semantics. Nonetheless they are, or can be useful, if
the ideal sector conservatively extends the finitary, that is if no
proof from finitary premisses to a finitary conclusion which takes a
detour through the infinitary language yields a conclusion we could
not have reached, albeit perhaps (herein lies the utility) by a
longer, more unwieldy proof. The goal of the Hilbert programme was to
provide a finitary proof of this conservative extension result; most,
though not all, think this goal was proved impossible by
Godel's second incompleteness theorem.



Returning now to our non-Hilbertian focus, the earlier formalism which
Frege attacked does not divide mathematics into the aforementioned
dual categories of the finitary/contentful, and the
infinitary/essentially meaningless but, on the contrary, treats all of
mathematics in a unitary and homogeneous fashion. From now on I will
use 'formalism', to refer to the non-Hilbertian positions
and will start with an account of the formalist views distilled by
Frege from Heine and Thomae and the criticisms he made of them. These
criticisms are widely believed now to contain conclusive refutations
of the Heine/Thomae approach. But there are a number of post-Fregean
views which seem heavily influenced by, or strongly analogous to,
formalism. I will go through these in turn:


* Wittgenstein's views on mathematics, primarily the
conception to be found in his *Tractatus
Logico-Philosophicus*;
* formalism as found in the Logical Positivists, particularly
Carnap;
* Goodman and Quine's nominalist formalism;
* Haskell Curry's version of formalism and
* formalist interpretations of the Curry-Howard correspondence.



I will conclude with a look at more recent formalist philosophers and
an overall assessment of the prospects for formalism in contemporary
philosophy of mathematics.


## 2. Game and Term Formalism



Frege does not extract a unified, consistent position from the work of
Heine and Thomae, and much of his criticism is devoted to showing that
they inconsistently slip into modes of thought which are only
appropriate for 'contentful' arithmetic, which Frege takes
to be a body of truths expressed by utterances in which numerical
expressions designate abstract referents independent of the mind (or
at least any particular individual's mind). Heine and Thomae
talk of mathematical domains and structures, of prohibitions on what
may be uttered (e.g. against writing '\(3\div 0\)' which is
deemed, in some special sense, meaningless), of numbers being greater
than or less than one another (rather than physical marks being larger or
smaller, darker or lighter)--all things which make no sense if
arithmetic is a theory of marks and their physical properties or is
just a body of transformations of referent-less symbols. (Heine
however reserves, with Kronecker, a special place for arithmetic,
treating it in non-formalist fashion; this position may thus be more
coherent than Thomae's. See Simons, 2009 especially
293-6.)



Nonetheless two distinct positions emerge from the material Frege
works over, doctrines which Resnik (1980: 54), and likewise Shapiro
(2000: 41-48) describe as *term formalism* and *game
formalism*, respectively. The term formalist views the expressions
of mathematics, arithmetic for example, as meaningful, the singular
terms as referring, but as referring to *symbols* such as
themselves, rather than numbers, construed as entities distinct from
symbols. Thus Heine writes:



> 
> I define from the standpoint of the pure formalist and *call
> certain tangible signs numbers.* (quoted in Frege, 1903 [1980
> SS87: 183]).
> 



The game formalist sticks with the view that mathematical utterances
have no meaning; or at any rate the terms occurring therein do not
pick out objects and properties and the utterances cannot be used to
state facts. Rather mathematics is a calculus in which
'empty' symbol strings are transformed according to fixed
rules. Thomae puts it this way:



> 
> For 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) (Frege, 1903/80
> SS95: 190).
> 



Thomae also remarks 'the formal standpoint rids us of all
metaphysical difficulties' (*ibid*: 184). One main
motivation then seems to be to block, avoid, or sidestep (in some way)
any ontological commitment to a problematic realm of abstract objects.
For standard mathematics entails a plethora of theorems affirming the
existence of infinite realms of entities--numbers, functions,
sets and so forth, entities which do not seem to be concrete.
Formalists, in general, wish to divest themselves of any commitment to
these domains which do, indeed, seem hard to fit into a thoroughly
naturalistic conception of reality.



Frege concentrates most of his fire on the term formalist
pronouncements of his targets; but game formalism is the only game in
town for the anti-platonist worried about the ontological commitment
to a realm of abstract objects. For term formalism treats mathematics
as having a content, as being a kind of syntactic theory; and standard
syntactic theory entails the existence of an infinity of
entities--expression types--which seem every bit as abstract
as numbers. Indeed, as Godel's arithmetisation of syntax
showed, the elements and inter-relationships of standard formal syntax
can be modelled as an infinite substructure inside the standard model of
arithmetic.



Frege mercilessly exposes the inadequacies of Heine and Thomae's
position&dash;their confusions as they slip from term to game
formalism; their conflation of sign and signified; the fact that they
do not set out an account of the syntax and proof theory which is
remotely adequate as an account of the mathematics with which they
deal; their hopeless attempts to extend their position from arithmetic
to treat of analysis and real numbers, by this stage in mathematical
history construed by Weierstrass, Cantor and others not geometrically
but as infinite sequences. Thus Frege writes:



> 
> In order to produce it [an infinite series] we would need an
> infinitely long blackboard, an infinite supply of chalk, and an
> infinite length of time. We may be censured as too cruel for trying to
> crush so high a flight of the spirit by such a homely objection; but
> this is no answer. (219)
> 



Now Frege, himself, ironically, had revolutionised mathematics by
introducing hitherto unprecedented standards of rigour in the
formalisation of mathematical theories. He recognised (SS90:
185-6) that one could improve vastly on Heine and Thomae by
treating mathematical theories, their language, axioms and rules, as
formal, mathematical objects in their own right. This is exactly what
the Hilbert programme set out successfully to achieve, creating the
new disciple of metamathematics.



However Frege lays down very stiff challenges even for a rigorous game
formalist fully equipped with the techniques and results of
metamathematics. Such a theorist gives us a characterisation of a
language, for example by setting out what the basic elements
are--primitive symbols, and strings thereof--and then gives
a recursive specification of which strings count as well-formed.
Similarly we will be given a rigorous specification of which
arrangements of well-formed formulae count as proofs in a given
system, and of what theorem they prove in each case. If strings such
as '\(3+1=0\)' or '\(3 \gt 2\)' come out as provable
in the system (arithmetic modulo 4, say) then that is enough to count
them as correct utterances of the system. No further issue of truth
need arise; nor do we need to assume that there is only one system for
a given set of symbols. Nor, furthermore, need we assume that each
such system is complete (though Frege took Thomae to task for the
incompleteness, massive though readily rectifiable, of his arithmetic
calculus). We need make no assumption that the numerals in these
strings refer to anything outside the system, indeed we need not
assume they refer to anything at all. (This game formalist, then, is
not subject to the objection that '3 \(\gt 
2\)' should come out as false, on any legitimate formalist reading
of '\(\gt\)'; no need to think of the numerals as referring
to concrete marks and '\(\gt\)' meaning physically greater in
size.)



Such a game formalist is a more worthy opponent for the platonistic
Frege to tackle, but there are two main objections he sets out which
still apply to this more sophisticated position. The first is the
*question of applicability*: if mathematics is just a calculus
in which we shuffle uninterpreted symbols (or symbols whose
interpretation is a matter of no importance), then why has it been
applied so successfully--and in so many ways, to so many
different things--ordinary physical objects, sub-atomic objects,
fields, properties, and indeed from one part of mathematics to another
(we can count the number of dimensions in a pure geometric space)?
Frege writes:



> 
> It is applicability alone which elevates arithmetic from a game to the
> rank of a science. (SS91 in Frege 1903 [1980: 187])
> 



Secondly, Frege quite rightly and insistently distinguishes on the one
hand the 'game'--arithmetic, set theory, topology or
whatever--treated simply as a mathematical object in its own
right, a formal system, and, on the another, the theory of the game.
'Let us remember that the theory of the game must be
distinguished from the game itself' (SS107, p. 203). Thus in
the 'game' of trigonometry we might derive



\[
\sin^2\theta +\cos^2\theta = 1
\]


from the Pythagorean Theorem. In the metatheory we can prove:



\[
\vdash \langle \sin^2\theta + \cos^2\theta = 1\rangle,
\]


the claim that the formula with such and such a code in the
mathematical representation of the syntax (the code represented in the
meta-meta-theory here by '\(\langle\sin^2\theta +\cos^2\theta =
1\rangle\)') is provable. Likewise in the meta-theory we can
prove lots of other things about proof and refutation, for instance we
may be able to show that lots of sentences are neither provable nor
refutable.



The problem this raises for the formalist is this: the metatheory is
itself a substantial piece of mathematics, ostensibly committed to an
infinite realm of objects which are not, on the face of it, concrete.
Tokens of the expressions of the object language game calculus may be
finite--ink marks and the like; but since there are infinitely
many expressions, theorems and proofs, these themselves must be taken
to be abstract types. At best, the formalist can achieve no more than
a reduction in commitment from the transfinite realms of some
mathematical theories, such as set theory, to the countably infinite,
but still presumably abstract, realm of arithmetic, wherein the syntax
and proof theory of standard countable languages such as those of
standard set theory, can, as Godel showed, be modelled.



Can formalism be developed in such a way as to surmount these two
crucial objections, the problem of applicability and the problem of
the metatheory, as I will call it? (Not that these are the only
objections to formalism, but they are two fundamental ones.) Because
Frege's critique did not quash all formalistic impulses in later
philosophers of mathematics, we shall look, now, at future
developments, to see how they fare.


## 3. Tractarian Formalism



Wittgenstein was a keen student of Frege's work, directed to
further his studies under Bertrand Russell by Frege himself in the
course of a visit he made to see Frege in Jena. One might, then, think
him inoculated against formalism. But definite formalistic elements
surface in Wittgenstein's *Tractatus*.



True, the *Tractatus* is a notoriously difficult work to
interpret. Even the question as to whether the main portion of the
book, essentially all of it other than the 'frame' of the
preface and the end, is to be taken as a serious attempt to present a
metaphysics is controversial. If we leave that hermeneutic controversy
aside, and consider the metaphysics which is offered to us, the
formalistic aspects one finds are twofold. Firstly, mathematical
sentences are said to express 'pseudo-propositions', and
so are devoid of truth value (only contingent propositions have a
truth value). Secondly, mathematics is described as a
'calculus' which is not to be used to represent the world
as it is in itself but whose value is entirely instrumental. To be
sure, the most explicit statement of this is *not* in the
*Tractatus* but in comments Wittgenstein wrote on
Ramsey's copy of it:



> 
> The fundamental idea of math. is the idea of *calculus*
> represented here by the idea of *operation* The beginning of
> logic presupposes *calculation* and so number. Number is
> *the* fundamental idea of calculus and must be introduced as
> such (Lewy, 1967: 421-2).
> 



In the *Tractatus* proper we do get the idea that mathematical
propositions are mere instruments (all of mathematics, not just an
'ideal' fragment, as in Hilbert):



> 
> Indeed in real life a mathematical proposition is never what we want.
> Rather, we make use of mathematical propositions only in inferences
> from propositions that do not belong to mathematics to others that
> likewise do not belong to mathematics. (In philosophy the question,
> "What do we actually use this word or this proposition
> for?" repeatedly leads to valuable insights.) *Tractatus*
> P6.211.)
> 



This idea is prefaced by the statements:



> 
> Mathematics is a logical method. The propositions of mathematics are
> equations, and therefore pseudo-propositions. (*ibid*, P6.2)
> 



> 
> A proposition of mathematics does not express a thought. (*ibid*,
> P6.21)
> 



Care must be taken, however. Wittgenstein distinguishes utterances
which are *sinnlos*, which lack sense (including logical
tautologies and contradictions here) from those which are
*unsinnig*, nonsensical; it is not clear into which class
mathematical utterances fall. One might well think that the game
formalist should treat mathematical utterances, on that view just
strings of meaningless marks, as *unsinnig*, not just
*sinnlos*. One clear difference from game formalism however is
this: for Wittgenstein mathematics should not be conceived of as a
calculus separate from other uses of language. Rather he attempts to
show that parts of arithmetic, at least, can be seen as grounded in
non-mathematical uses of language. Frege by contrast, whilst arguing
that a proper account of arithmetic (and analysis) should show how its
generality enables one to give a uniform account of multifarious
different applications (cf. Dummett, 1991 Chapter 20) also argued
strongly for the view that mathematical utterances have a meaning
independent of, in conceptually prior to, their use in
applications.



Wittgenstein attempts no theory of mathematics in the
*Tractatus* beyond arithmetic, a rather narrow fragment of
arithmetic at that. The theory clearly shares the anti-platonism of
the game formalist. There are no numbers, arithmetic is to be
construed as a calculus in which one manipulates exponents or indices
of operators. What is an operator? Wittgenstein distinguishes operator
terms from function terms but commentators have struggled to explain
what the distinction comes to. It is clear that Wittgenstein held that
two occurrences of a function term \(f\) applied to different
strings \(t\) and \(u\) have different meanings, where by
'meaning' Wittgenstein means referent, something like
Frege's *Bedeutung*. Thus '\(f\)' in
"\(f(t)\)" does not refer to the same entity
as the outermost \(f\) in
"\(f(f(t)\)"; this is supposed to
be the basis of the solution to Russell's paradox (P3.333).
In particular 'the father of' in "the father of
John", means something different there compared to its outermost
occurrence in "the father of the father of John". There
can be no genuine iterated application of functions in other words, a
cure for Russell's paradox that many will find about as bad as
the disease.



*Operators*, however, are to be distinguished from functions in
at least that aspect: genuine iteration of operators--the
sentential operators of propositional logic are a prime
example--is possible without supposing a change of sense or
reference from one token to the other. What is their meaning or
referent then? Wittgenstein denies they have any referents, this is a
generalisation of his claim that the logical constants are not
representatives. Peter Hylton (1997: 96-98) argues that
Wittgenstein in the *Tractatus* has Russellian *propositional
functions* in mind when he talks of 'functions' and is
at pains to distinguish operators from these "substantial"
entities. Russellian propositional functions are not the same as
ordinary mathematical functions, the model for Frege's notion of
function. Rather they are structured entities, structurally related to
the propositions which are their values-- gappy states of affairs
might be one way to think of them. Operators, by contrast, do not
stand for any such entity, they are not parts nor any sort of
ingredients of propositions, they "leave no trace" in
them. 



Sentential operators are conceived as mapping not signs, nor
inscriptions, to other signs and ascriptions but rather propositions,
(in Wittgenstein's rather course-grained sense of the term) to
propositions. On Wittgenstein's account of proposition, repeated
application of an operation such as negation \(\ldots p,
{\sim}p, {\sim}{\sim}p\ldots\) may take one back to an earlier
point. Nonetheless Wittgenstein attempts to explicate arithmetic in
terms of sentential operators applied to non-mathematical language.
(One can see intimations here of Church's later rigorous working
out of a similar idea in his treatment of numbers in the lambda
calculus as functions which repeatedly apply input functions.) In
slogan form, numbers are exponents of operations (*ibid.*,
P6.021). Thus where \(\Omega\) is schematic for an operator and
\(\Omega p\) (or \(\Omega(p))\) for its application to a
proposition, then we can view the series



\[
p, \Omega p, \Omega \Omega p, \Omega \Omega \Omega p, \Omega \Omega \Omega \Omega p, \ldots
\]


as the starting point for a 'definition' of numbers, to be
had by rewriting it as



\[
\Omega^0 p, \Omega^{0+1}p,
\Omega^{0+1+1}p,
\Omega^{0+1+1+1}p,
\Omega^{0+1+1+1+1}p,\ldots.
\]


Here, then, we have infinitely many schematic rewrite rules. Indices
such as '\(^{0+1+1+1+1}\)' can be abbreviated by
numerals in the obvious fashion, with
'\(^{0+1+1+1+1}\)' being abbreviated '4'
and so on.



Wittgenstein's examples show (though he did not explicitly state
this) that addition of two number/exponents
\(\Omega^{n}p + \Omega^{m}p\) (likewise
\(\Omega^{n+m}p)\) is given by the
rule:



\[
\Omega^{n}p + \Omega^m p \Rightarrow \Omega^{n}(\Omega^m p)
\]


telling us we may replace the expression on the left in a formula by
the one on the right.



This is what underlies the 'correctness' of identities
such \(n + m = r\) except that for Wittgenstein
no such identity expresses a truth. On his account, the identity sign
disappears in a full analysis of language, wherein sameness and
distinction are shown by sameness and distinctness of names, no two of
which refer, in fully analysed language, to the same object (this view
gives grounds for interpreting mathematical utterances in the
*Tractatus* as *unsinnig*). Wittgenstein himself did not
trouble to show that abandonment of an identity sign would not cripple
the expressive power of language but others such as Hintikka (1956)
and Wehmeier (2004) have done so. What is left in the underlying
language, with identity excised, are substitution rules
(*Tractatus* P 6.23).



These have to be interpreted in a general, and schematic, fashion.
Thus when we plug in \({\sim}\) for \(\Omega\), we find that (Wittgenstein had no
intuitionist scruples here) a double application \({\sim}{\sim}p\) takes us
back to (has the same sense as) \(p\). But this does not ground
the truth of \(2 = 0\), since for many other operations
\(\Omega \Omega p\) is not equivalent to \(p\). On the other
hand



\[
\Omega \Omega(\Omega \Omega \Omega p) 
 \text{ always has the same sense as }
 \Omega \Omega \Omega(\Omega \Omega p)
\]


Wittgenstein implicitly assumes suitable rules for the interaction of
brackets with operators, in particular, generalised associativity. (In
fact, he uses a mix of brackets and the notation
\(\Omega 'p\) to express \(\Omega(p)\).)



Since an equation \(\Omega^{n}p=\Omega^{m}p\) is, in its underlying
logical form, not a universal generalisation \(\forall n,m(\Omega^{n}p
= \Omega^{m}p)\) but a purely schematic generalisation, there is no
form \(\exists n,m(\Omega^{n}p \ne \Omega^{m}p)\) with which we can
express inequality, even if we can make sense of
'\(\ne\)'. Nor can we express the inequality \(n \ne m\)
schematically as the holding of the inequivalence of \(\Omega^n p\)
from \(\Omega^m p\) for every choice for \(\Omega\). Otherwise \(2
\ne 0\) would fail since \({\sim}{\sim}p\) is equivalent to
\(p\). The *Tractarian* theory cannot handle inequalities.



So much for addition and the limitations of its account of that
operation. What of multiplication? Wittgenstein does define it, at
P6.241, by:




\[
\Omega^{n\times m}p \Rightarrow (\Omega^n)^m p
\]


but to grasp this as a general principle we need to know how to
interpret the notation \((\Omega^n)^m\). In more conventional
mathematics, one might simply define \((x^n)^m\) as
\(x^{n\times m}\) but clearly this (or rather the equivalent
for the interaction of exponents of operators) would introduce a
circularity into Wittgenstein's account. Alternatively one could
appeal to a recursive theory of exponents - \(a^{m\times 0}= a,
a^{m\times(n+1)} = a^m + a^{m\times n}\). Since the principle of
induction needed to show that the recursion is coherent features
nowhere in Wittgenstein's system, these rules would presumably
have to be taken as primitive. 



Overall, then, Wittgenstein in the *Tractatus* gives us no
account of mathematics in general other than for a fragment of
arithmetic, basically positive identities involving only addition. And
there he denies that the sentences express propositions with truth
values. Of course the book was written under extraordinarily difficult
circumstances. Perhaps his account could have been developed further
and more plausibly, despite the difficulties noticed above--but
for some scepticism on that front, see Landini, 2007. Certainly
Wittgenstein made no attempt to do so whilst engaged with F.P. Ramsey
and the Vienna Circle in the 1920s. If Wittgenstein's standpoint
was not capable of much further development, we have the choice of
either abandoning all of mathematics except, at most, a fragment of
the arithmetic of addition; or else of rejecting the
*Tractarian* account. One does not need to be slavishly
uncritical of contemporary mathematics to see what the reasonable
option is here. Admittedly, rejecting the *Tractatus* account
is also the option Wittgenstein himself seems to adopt at the end of
the book; here then we enter the issue of what the point is of taking
us through such a bizarre and unconvincing theory in order to throw it
away at the end. (For a more positive appraisal of
Wittgenstein's *Tractarian* position, see Floyd (2002).




Wittgenstein's later work on philosophy of mathematics, such as
the *Remarks on the Foundations of Mathematics* 1956/1978), for
a long time attracted even less approval than the *Tractarian*
account, though recently philosophers such as Juliet Floyd and Hilary
Putnam have come to its defence as an interesting and informed account
of mathematics (Floyd/Putnam, 2000). Its themes include the rejection
of the actual infinite (indeed the tendency in his writings is
strongly finitist); the denial that undecidable sentences are
meaningful; a rejection of Cantor's powerset proof; the idea
that proof discovery changes the very meaning of the terms involved;
and other very radical ideas. Among them we find a continued adherence
to formalist motifs:



> 
> In mathematics *everything* is algorithm and *nothing*
> is meaning; (*Philosophical Grammar*: 468).
> 



Another persistent theme in Wittgenstein's thought is that the
meaning of mathematics resides entirely in its utility in
non-mathematical applications. But there is no systematic theory of
how this applicability comes about, no proof of a conservative
extension theorem, for example, showing how application of
mathematical calculi to empirical premisses will never lead us to
derive an empirical conclusion which does not follow from those
premisses. And there is no resolution of the problem of the
metatheory. On the other hand, we should observe that these notes of
Wittgenstein on the philosophy of mathematics were not published by
him, but by others after his death. For an overall view of
Wittgenstein's philosophy of mathematics as a whole see
 Wittgenstein's philosophy of mathematics.


## 4. Formalism and the Positivists



Wittgenstein greatly influenced the Vienna Circle. The
'official' positivist theory of mathematics, as it were,
is not a formalist one. Mathematical theorems express truths, albeit
in a special way: true by virtue of meaning alone. The most
influential positivist has been Carnap, if one does not classify Quine
as a positivist (but Quine's views, in the 1930s at any rate,
were very close to Carnap's, indeed arguably Quine remained
truer to Carnap's radical empiricism than Carnap did). And one
can certainly discern strong elements of formalism in some of
Carnap's writings, for instance in *Logische Syntax der
Sprache* (1934 [1937]) and 'Empiricism, Semantics, and
Ontology' (1950 [1956]).



The former book was translated into English as *The Logical Syntax
of Language* in 1937. In it, Carnap argued that the correct method
in philosophy is to engage in conceptual analysis conceived of as
'logical syntax', roughly speaking syntax proper and proof
theory. To address philosophical differences, one proposes regimenting
the disputed positions in formal languages or 'frameworks'
which include a system of axioms and rules of proof; given these, some
sentences are 'determinate', are provable or refutable.
These are the analytic, and contradictory, sentences, relative to that
framework. How do we choose which system to adopt? Carnap's
*Principle of Tolerance* (1934 [1937], p. 52) allows us to
adopt any system we wish:



> 
> *In logic, there are no morals*. Everyone is at liberty to
> build up his own logic, i.e. his own form of language, as he wishes.
> [Italics in original]
> 



Carnaps extends this unbridled permissiveness to mathematics:



> 
> The tolerant attitude here suggested is, as far as special
> mathematical calculi are concerned, the attitude which is tacitly
> shared by the majority of mathematicians.
> 



Any such calculus can count as a piece of mathematics, even an
inconsistent one. By downplaying or outright discarding semantic
notions, we simply bypass traditional ontological disputes concerning
the nature of the entities mathematics is 'about'. The
only issue is the pragmatic utility, or otherwise, of any given
mathematical calculus.



A number of concerns arise here. How can Carnap distinguish between
empirical, scientific theories, and mathematical ones? Secondly, if
pragmatic utility is primarily a matter of empirical applications, how
does the Carnapian formalist know that a given calculus will
conservatively extend empirical theory, how can this be known without
appeal to meaningful mathematical results? Carnap writes:



> 
> The formalist view is right in holding that the construction of the
> system can be effected purely formally, that is to say without
> reference to the meaning of the symbols; ... But the task which
> is thus outlined is certainly not fulfilled by the construction of a
> logico-mathematical calculus alone. For this calculus does not contain
> ... those sentences which are concerned with the *application
> of mathematics* ... For instance, the sentence "In this
> room there are now two people present" cannot be derived from
> the sentence "Charles and Peter are in this room now and no one
> else" with the help of the logico-mathematical calculus alone,
> as it is usually constructed by the formalists; but it can be derived
> with the help of the logicist system, namely on the basis of
> Frege's definition of '2'. (Carnap 1934 [1937], p.
> 326)
> 



adding (italics are Carnap's) '*A structure of this
kind fulfils, simultaneously, the demands of both formalism and
logicism*.'



But what is to stop us freely stipulating, in accordance with the
principle of tolerance, 'bridge principles' for operators
which proof-theoretically behave like numerical operators--
'the number of \(\phi\)'s'--via their definitional
occurrence in formulations of arithmetic axioms and where the bridge
principles include the likes of:



\[\begin{align}
\text{the number of }\phi\text{'s} &= 0 \leftrightarrow \exists x(\phi x \amp{\sim}\exists y(\phi y \amp y \ne x)) \\
\text{the number of }\phi\text{'s} &= 1 \leftrightarrow{\sim}\exists x\phi x\, ?
\end{align}\]


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



Godel's incompleteness theorems pose very difficult
problems for Carnap in this and other regards. The first
incompleteness theorem tells us that in any \((\omega\)-) consistent
formal theory whose theorems are recursively enumerable and which
entails a certain (rather limited) amount of arithmetic, there will be
an arithmetical sentence such that neither it nor its negation is
provable. In Carnap's terminology, this seems to yield
non-determinate sentences, which is a problem for him if we are
convinced that nonetheless some of these sentences are true; and
indeed the key type of sentences used to prove the incompleteness
result--'Godel sentences'--are in fact true
in the standard model of arithmetic if these sentences are constructed
in an appropriate way (there are different ways of doing this) from a
theory itself true in that model.



Godel himself wrote, but did not publish, a searching critique of
Carnap's position (Godel, 1953-9). Godel focuses
not on his first incompleteness theorem but on the corollary he drew
in his second theorem: that, under a certain natural characterization
of the property of consistency, a characterization which can be given
mathematically via his arithmetization of syntax, no formal theory of
the type Godel considered could prove its own consistency. He
argued that Carnap, in order to make good his positivistic thesis that
mathematical theorems are devoid of content, needed to give a
consistency proof for mathematical calculi in order to show that they
do not have empirical content, an abundance thereof indeed, by dint of
entailing all empirical sentences. Warren Goldfarb notes, however,
(1995: 328) that this point fails to appreciate the deep holism of
Carnap's 1937 position where the distinction between analytic
and synthetic is relative to the system in question, the
'linguistic framework'. (This deep holism, of course, has
the counter-intuitive consequence that there is no
framework-transcendent distinction between mathematics and empirical
sciences.)



Carnap, in fact, understood the import of Godel's theorems
(Tennant, 2008); he knew of the results directly from Godel, who
indeed read drafts of the *Logical Syntax*. Despite this he
displayed what looks like remarkable insouciance with respect to its
implications for his position. He acknowledged the need, in
demonstrating consistency, to move to a stronger language (SS60c),
and freely helped himself to mathematical techniques which could in no
sense be classed as finitary (in SS14 he used, for example, rules
with infinitely many premisses, notably one which was later to be
called the \(\omega\)-rule). In this way he can deny, for arithmetic at
least, that there are any non-determinate sentences since every true
arithmetic sentence is provable using the \(\omega\)-rule (relative to a
fairly weak finitary logic, considerably weaker than classical
logic).



Carnap's relaxed attitude stems from his abandonment of the
search for epistemological foundations. If one hopes to secure our
knowledge of mathematics by appeal to a formalist interpretation then
the search for a consistency proof of the type Hilbert sought makes
sense and one will look for a vindication of mathematics as a whole
from within a limited fragment with respect to which our knowledge
seems hard to impugn. But Carnap, perhaps as a result of
Godel's deep theorems, seems to have abandoned that goal
and thought that the Principle of Tolerance absolved him of any such
need. One can stipulate what one likes, including stronger axiom
systems from which one can prove the consistency of a weaker theory.
This does not give any firmer grounds for believing or accepting the
weaker theory, but one does not need such grounds anyway.



Few nowadays look for Cartesian certitude in mathematics, so
Carnap's position here may seem reasonable. It is not so clear,
however, that he has answered the problem of applicability. Even if a
conservative extension result can only be given in a more powerful
system, we need the result to be a contentful truth, not just a string
of symbols we can derive from some system, if we are to have
reassurance that a particular calculus we are about to use in
designing bridges or computers will be pragmatically useful. And if
the Carnapian grants that the result is a contentful truth, we can ask
what, according to this formalistic position, constitutes that truth.
Carnap, to be sure, was motivated by a horror of becoming embroiled in
metaphysical disputation. But if his standpoint is emptied not only of
epistemological ambition but is so deflationistic as to say little
more than that metamathematical techniques can be applied to
formalisations of mathematical and scientific theories then it is also
emptied of all philosophical interest and ceases to make an
intervention in the debates in philosophy of mathematics.



Not that Carnap really is abandoning metaphysics: this erstwhile
opponent of metaphysics is really a brother metaphysician with a rival
theory of his own, as F. H. Bradley might have said. Thus the main
import of the later 'Empiricism, Semantics, and Ontology'
is the dismissal of ontological worries as pseudo-problems by dint of
the (highly contentious) distinction between 'internal'
questions, to be settled by the rules of the framework--of
mathematics, ordinary 'thing' talk or whatever--and
'external' questions such as 'which framework to
adopt?'. To these external questions there correspond, Carnap
claimed, no propositions with truth values; to no such question, for
example, is there a true answer of the form: "Yes, infinitely
many abstract numbers exist". Rather, they are to be answered by
*decisions*, decisions to adopt or not based on pragmatic
criteria regarding the efficiency, fruitfulness and utility of the
framework with respect to the aims of the discourse in question. But
what aims could these be? Predicting the flow of 'sense
data', taken for granted as forming the ultimate furniture of
the world? If so, we see that the vaunted ontological neutrality is a
sham and we have a radical form of empiricist anti-realist
metaphysics.



The vestige of formalism lies in this: Carnap takes
'correctness' to be determined by rules which govern a
system and not to consist, for example, in correspondence with a realm
of facts independent of the system of rules. And he thinks this
approach dissolves ontological worries and frees us from any
obligation to explain how finite, flesh-and-blood creatures like
ourselves could come to have detailed knowledge of this independent
realm of facts, this domain of configurations of abstract,
non-temporal, non-causal objects and properties, a realm exposed by
Carnap as a metaphysical illusion. One strong disanalogy with
formalism is that Carnap takes this line with *all* areas of
discourse, not just mathematics.


## 5. Nominalist Formalism



W. V. Quine famously rejected the positivist's doctrine of truth
in virtue of meaning and the quasi-logicist conception of mathematics
as a body of analytic truths (and, partly as a result, also rejected
his mentor Carnap's internal/external distinction). Quine, in
conjunction with Nelson Goodman, produced instead what amounts to a
formalist manifesto. His formalist phase does not seem to have lasted
very long: he later settled on a form of mathematical platonism,
downplaying, if not largely ignoring, his relatively youthful
flirtation with 'nominalism'. But while it lasted he and
Goodman greatly advanced the discussion of formalism by tackling
head-on the questions which other formalists shirked or ignored.



Goodman and Quine's 'Steps towards a constructive
nominalism' (1947) sets forth an uncompromising game
formalism:



> 
> The gains which seem to have accrued to natural science from the use
> of mathematical formulas do not imply that these formulas are true
> statements. No one, not even the hardiest pragmatist, is likely to
> regard the beads of an abacus as true; and our position is that the
> formulas of platonistic mathematics are, like the beads of an abacus,
> convenient computational aids which need involve no question of truth.
> (p. 122).
> 



They regard



> 
> the sentences of mathematics merely as strings of marks without
> meaning
> 



so that



> 
> such intelligibility as mathematics possesses derives from the
> syntactical or metamathematical rules governing those marks. (p. 111)
> 



Commendably, Goodman and Quine do not shy away from the metatheory
problem, the difficulty that syntax and metamathematics itself seems
as ontologically rich and committed to abstract objects as arithmetic.
On the contrary, they face it squarely and attempt to make do entirely
with an ontology of concrete objects, finitely many such objects in
fact. (However they do assume fairly powerful mereological principles,
essentially universal composition: they assume that any fusion of
objects, however scattered or diffuse, is also an object in good
standing.)



With much ingenuity they try to develop a syntax which "will
treat mathematical expressions as concrete objects"
(*ibid.*)--as actual strings of physical marks--and
give concrete surrogates for notions such as 'formula',
'axiom' and 'proof' as platonistically
defined. However they do not address the issue of the application of
mathematics, construed in this concrete, formalistic fashion.



In addition to the applicability problem, there are two further
crucial problems for formalism as developed by Goodman and Quine.
Firstly it is not clear that they are entitled to the general claims
they make about syntax, construed as a theory about certain concrete
marks and fusions of marks. Thus when arguing that their definition of
formula in terms of 'quasi-formula' gives us the results
we want, they say:



> 
> By requiring also the next more complex alternative denials in
> \(x\) to be alternative denials of quasi-formulas, the definition
> guarantees these also will be formulas in the intuitively intended
> sense; and so on, to \(x\) itself. (p. 116)
> 



('Alternative denial' is the Sheffer stroke operation
\(P|Q\) which is true if and only if one component is
false.) The problem lies with the 'and so on'. Goodman and
Quine are trying to work their way up through an arbitrary formula
showing that their definition will ensure that each larger component
is a formula. It is not clear how we can have a guarantee of this for
arbitrary \(x\), without something like induction over formula
complexity; but this is not available as formulas are not generated in
the usual inductive set-theoretic fashion. Similar remarks apply to
the demonstration that proofs, as defined nominalistically on p. 120,
have the internal order of precedence among immediate sub-premisses
and conclusions that we intuitively expect. The demonstration proceeds
with a generalisation over all numbers \(k\) which number the
axioms in a concrete proof and then performs a sequence of selections
on them. This seems to presuppose the truth of generalisations over
all numbers and indeed countable choice, resources unavailable to a
strict nominalist.



Secondly, what can Goodman and Quine say about a sentence such as



\[
2^{2^{2^{2^{ 2^2} }}}+1 \text{ is prime}\,?
\]


(That is--with '2^\(n\)' representing '2 to
the power \(n\)'--'[2^(2^(2^(2^(2^2))))]\(+1\) is
prime'; cf. Tennant, 1997 p. 152.) They cannot deny the sentence
exists, for there is the token before our very eyes. But there are
strong grounds for thinking that no concrete proof or disproof will
exist, for the only methods available may use up more time, space and
material than any human could have at her disposal, perhaps than
actually exists. There are countless sentences with this property:
concrete tokens of them exist but no concrete proof or refutation
actually exists, none that a human could manipulate as a meaningful
utterance anyway. (Cf. Boolos, 1987.) Formalists of Goodman and
Quine's persuasion seem forced to the conclusion that sentences
like the above, sentences which are decidable in the usual formal
sense, are neither true nor false, since neither (concretely) provable
nor refutable. To embrace this view, however, would be to butcher
mathematics as currently practiced; such a consequence should rather
be viewed as a *reductio ad absurdum* of their position.


## 6. Term Formalism: Curry



The most substantive attempt at a non-Hilbertian formalist philosophy
of mathematics is Haskell Curry's book *Outline of a
Formalist Philosophy of Mathematics* (Curry, 1951). Curry is no
game formalist, his position is closer to term formalism, of the two
views we started out from. Curry's philosophy of mathematics,
however, is, or attempts to be, a highly anti-metaphysical one, at
least to the extent that he thinks he can remain neutral on the issue
of the ontological commitments of mathematics.



> 
> Mathematics can be conceived as a science in such a way as to be
> independent of any except the most rudimentary philosophical
> hypotheses. (p. 3)
> 



Hence he is not motivated by an anti-platonist horror of abstract
objects. His neutrality, indeed, is somewhat compromised by the fact
that Curry is perfectly happy to commit to an infinite ontology of
presumably abstract expression types. Officially he evinces
disinterest in what the primitives--he misleadingly calls them
'tokens'--of his formal systems are:



> 
> We can take for those tokens any objects we please, and similarly we
> can take for operators any ways of combining these objects which have
> the requisite formal properties. (p. 28)
> 



But since for many systems there are infinitely many primitive
'tokens', they cannot all be identified with concrete
marks which mathematicians have actually produced.



Like the term formalist, Curry takes mathematics, properly
reconstructed after philosophical reflection, to have an essentially
syntactic subject matter, namely formal systems. Unlike Frege's
adversaries, though, Curry, writing after the development of the
discipline of metamathematics, is able to give a far more rigorous
(albeit in his case somewhat eccentric) account of what a formal
system is.



No restrictions are placed on what form the axioms, rules and
therefore theorems of a formal system are to be. Truth for elementary
propositions of a formal system consists simply in their provability
in the system. One of his formal systems (Example 7, p. 23) has only
one predicate "a unary predicate expressed by Godel by the
words 'ist beweisbar'" (p. 23), i.e., a provability
predicate. The elementary truths of this system can be interpreted as
claims about provability in the underlying system. Any formal system
of the usual sort could be 'reduced' into one in which
there is only one provability predicate and truth (= provability) in
the reducing system of the elementary proposition \(\vdash \langle
\phi \rangle\) holds only when \(\phi\) is provable in the reduced
system (pp. 34-35). Curry allows that one can form compounds
from elementary propositions by means of the usual logical operators
in order to express complex propositions in the language of proof
theory (Chapter IX).



The upshot is that mathematics in general becomes metamathematics, a
contentful theory--Curry's sentences express propositions
with truth values--setting out the truths about what is provable
from what in underlying formal systems whose interpretation, or rather
interpretations, are not taken to be mathematically important. This
standpoint, however, threatens to collapse into structuralism, into
viewing mathematical utterances as schemata implicitly generalising
over a range of (in general) abstract structures which satisfy the
schemata. As to the problem of the metatheory, Curry does not seek to
answer this; there is no real attempt to avoid commitment to a rich
ontology of objects, except that, by considering only standard formal
systems, one could make do with a countable ontology which can play
the role of linguistic expressions. Only, however, at the cost of
grossly distorting mathematical practice. Set theorists, topologists,
analysts *et al*. entertain conjectures, and try to prove things
'about' sets, topological spaces, functions on the complex
numbers and so on. In their philosophical moments they may wonder just
what the concepts they are wrestling with are 'about', but
they do not in general entertain conjectures or try to prove things
'about' expression strings, except where it is of
instrumental value to them in proving things about sets, spaces, the
complex plane and so on (cf. Resnik pp. 70-71).


## 7. The Curry-Howard Correspondence



Haskell Curry was also to play an important role in
developments linking logic to computer science which some argue can
lend support to formalism in mathematics. His work on combinatory
logic along with work of W.A. Howard's led to the
'Curry-Howard correspondence' ('Curry-Howard'
henceforth written 'CH') or 'CH isomorphism'
linking logic, proof theory and computer science.



Curry intended to provide a general theory of functionality as part of
a foundation for logic, "pre-logic", as Curry called it.
See in particular (Curry 1934) and, with Robert Feys, (Curry and Feys
1958). At around the same time as Curry's first publications in
the area, Alonzo Church developed his untyped \(\lambda\)-calculus, also
designed to provide a foundation for logic, indeed mathematics more
generally, and also taking functions, very generally applicable, as
fundamental. In these functional calculi, (for a comprehensive account
see Barendregt (1984), also the entry on
 the lambda calculus)
 a concatenation \(fg\) is used to represent the application of
function \(f\) to argument \(g\) yielding an output value, where
both argument and value can themselves be functions and where
self-application is allowed. Whereas Curry's system is
variable-free, variable binding in Church's occurs by means of
the \(\lambda\) term, the variable \(x\) if and where it occurs in
\(N\) being bound in \(\lambda x.N\). The fundamental
operation in \(\lambda\)-calculus is \(\beta\)-reduction, the transformation
which takes us from \((\lambda x.N)M\) to
\(N[x := M].\) Here \(N[x := M]\) is
the result of substituting \(M\) for all free occurrences of
\(x\) in
 \(N\).[1]
 So, for example, \((\lambda x.xx)f\) \(\beta\)-reduces
to \(ff\). We can write this as: 



\[
(\lambda x.xx)f \rhd ff
\]


Thus these calculi achieve what Wittgenstein in the *Tractatus*
(see above) seems to be gesturing at in his operations/function
distinction, for in Church and Curry we have a fully developed theory
of 'operations', that is functions which can take
functions as arguments and values. Of course self-application, as in
the infinite looping \(\beta\)-reduction:



\[\begin{align}
(\lambda x.xx)(\lambda x.xx) &\rhd (\lambda x.xx)(\lambda x.xx) \\
 &\rhd (\lambda x.xx)(\lambda x.xx) \\
 &\vdots
\end{align}\]


raises worries that paradox may emerge. Church thought that eschewing
the use of free variables and restricting excluded middle (Church,
1932: 346-7) blocked paradox but Kleene and Rosser showed
(1935), using a strategy based on Richard's paradox, that the
system was trivial: every formula could be derived using the rules.
Church fixed this up to produce a consistent untyped \(\lambda\)
calculus, but the important step with regard to the CH correspondence
was the development of typed \(\lambda\) calculi.



Now 'type' is a very overworked word. Tokens of that type,
to use one of its meanings (roughly as abstract syntactic object),
have sometimes been used to stand for properties, including
higher-order properties, as in Russell's various type theories.
Church was to continue this tradition in his simple typed version of
\(\lambda\)-calculus (Church, 1940). In this usage, types, such as the
property of caninity or of being a shape, have instances such as Fido,
in the first case, or the lower-order property of being square in the
second case. Thus instances of types in this sense need not be
abstract entities. Another usage is syntactic, as when the basic
expressions of a language are divided into various disjoint categories
('types') and formation rules for generating well-formed
expressions are set down making use of the type distinctions. In this
usage 'type' is an expression in the syntactic metatheory
of the language of some object theory. In some cases the syntactic
theory is in fact part of the object theory under discussion.
Independently of this, some presentations of syntactic type theories
'push down' the syntactic metatheory into the object
theory under discussion by stipulating that the well-formed
expressions of the object language contain syntactic proper parts,
tags as it were, which are correlated with the metatheoretic type of
the expression and have no semantic role. In Howard (1969), for
example, the well-formed formula of the conditional fragment of
propositional logic are used as type symbols superscripting terms of
the type theory. 



The expression <<\(N: \tau\)>> which is usually read
along the lines of: 



\[
\text{term } N \text{ is of type } \tau
\]


can be, therefore, read in a number of ways, for example: 



I: Non-syntactic: the entity referred to by \(N\) is an instance of
the class/set/property referred to by \(\tau\), an instance which need
not be syntactic nor, more generally, abstract. 



II: Meta-syntactic: the expression referred to by \(N\) is an
instance of the syntactic category \(\tau\). The syntactic theory in
which the ascription of term to type occurs is 'meta',
relative to a background intellectual context, in that it is not there
presented as (under its intended interpretation) part of a more
general, non-syntactic theory expressed in a language for which it
itself provides the syntax. 



III: Syntactic: the expression \(N\) is an instance of the
syntactic category \(\tau\) and the type theory \(is\) a syntactic
theory for the language which it itself belongs to. 



Whilst some textbooks on type theory can seem, to the logician, to be
rather hazy on exactly which, if any, of the above interpretations of
'type' is at issue this is certainly not generally the
case with the pioneers of these theories. Howard, for example, writes
in his classic paper 'The Formula-as-Types Notion of
Construction':



> 
> The title has a second defect; namely, a *type* should be
> regarded as a [sic] abstract object whereas a *formula* is the
> name of a type. (1969: 479)
> 



and distinguishes between types and type symbols (480).



There are non-syntactic models of various type theories, though these
were developed later, for example (Scott, 1970), and make fairly
strong set-theoretic cardinality assumptions, such as the existence of
inaccessible cardinals. Much more relevant for the formalist are
'term models', syntactic models similar to the
interpretations of languages which make use of their own symbols as
the members of the domain, interpretations to be found in Henkin
completeness proofs; and especially 'syntactic semantic'
approaches, as in some interpretations of Per Martin-Lof's
intuitionist type theory (see the entry on
 intuitionistic type theory),
 or the 'proof-theoretic semantics' of Peter
Schroeder-Heister
 (see the entry on proof-theoretic semantics)
 which eschew attempts to give meanings to types and terms by
reference to 'external' truth-conditions: mathematical
language is not to be taken as a representation of some independent
reality. 



In such a context, the distinction between meta-syntactic and
syntactic readings of 'type' is not very important. Even
if 'type' is taken to be a purely metatheoretic notion,
the axioms and rules of inference of type theory enable us to prove
meta-theorems about types. The situation can be compared to the
relation between sequent calculus and natural deduction, with the
turnstile \(\vdash\) of the former interpretable as a relation of
derivability in some underlying natural deduction system, and the
sequent calculus providing higher-order theorems about object language
derivability. Thus whether or not one thinks of types as
meta-theoretic notions, the calculi of type theory are such that one
can prove theorems to the effect that term \(N\) is of type \(\tau\).




Now Curry's work in (1934) and more fully with Feys in (1958)
had shown a certain correspondence between provable formulae in the
theory of the conditional and the types of basic combinators in type
theory. In particular, with \(\rightarrow\) as the conditional and
\(\alpha \Rightarrow \beta\) representing function types, that is
types (construed non-syntactically) whose inputs are type \(\alpha\)
functions and outputs type \(\beta\) functions, we have, (here
\(\vdash\_{T\rightarrow}\) means provability in the positive
(non-relevantist) theory of the conditional and \(\vdash\_{CL}\) means
provability in a suitable combinatory logic): 




\[
\vdash\_{T\rightarrow} \mathrm{A} \rightarrow \mathrm{B} \text{ iff for some }
N, \vdash\_{CL} N : \alpha \Rightarrow \beta
\]


where \(N\) is a term built from basic combinators and \(\alpha\) is
structurally isomorphic to A (likewise \(\beta\) to B). That is, one
can generate A \(\Rightarrow\) B from A \(\rightarrow\) B by replacing
each occurrence of \(\rightarrow\) by \(\Rightarrow\), given a uniform
substitution (perhaps the trivial identity one) of sentential letters
in the formulae of the propositional language by names for basic
types. 



Curry and Feys (1958) extended the correspondence idea to one between
type theory and Gentzen's sequent calculus. In the paper already
cited, circulated in 1969, but only published in a volume in a
Festschrift for Curry in 1980, W.A. Howard (1969) deepened the CH
correspondence by demonstrating a correspondence between
intuitionistic sequent form natural deduction and type theory in
\(\lambda\)-calculus format, generalising to encompass intuitionist
arithmetic- 'Heyting arithmetic' (HA)- (thus requiring an
extension from all of propositional to predicate logic), all as part
of a project of investigation of the constructivist notion of a
construction. Howard deepened the results by making clear not only a
correspondence between provable formulae in the sequent calculus and
type ascriptions, but also between the terms in the type ascriptions
and proofs of the corresponding
 formulae.[2]
 For example, (to the horror of relevantists) A \(\rightarrow\) (B
\(\rightarrow\) A) is provable in T\(\_{\rightarrow}\). The
corresponding type is \(\alpha \Rightarrow(\beta \Rightarrow
\alpha)\), the type of the basic operator K, whose action is 



\[
NM \rhd N
\]


and whose \(\lambda\) representation is
\((\lambda x.(\lambda y.x))\) (usually abbreviated
\(\lambda xy.x)\), as can be seen by the \(\beta\)-reduction
 chain:[3]



\[
(\lambda x.(\lambda y.x)N)M \rhd (\lambda y.N)M \rhd N.
\]


The simplest proof of A \(\rightarrow\) (B \(\rightarrow\) A) in T\(\_{\rightarrow}\) is:




\[
\dfrac{\dfrac{\dfrac{}{A}\scriptsize{1}}{B \to A}}{A \to (B \to A)}\scriptsize{1}
\]

Figure 1




in which the second step, to the intermediate conclusion B \(\rightarrow\) A,
is an instance of \(\rightarrow\)I(ntroduction) with vacuous discharge of the
unassumed antecedent B (in a sequent calculus version, the rule of
thinning, adding extra assumptions in the sequent antecedent, would be
used). The type theoretic proof in type theory
 TT[4]
 of the construction of a term "inhabiting" the type \(\alpha
 \Rightarrow(\beta \Rightarrow \alpha)\) takes the form: 



\[
\dfrac{\dfrac{\dfrac{}{x:\alpha}\scriptsize{x}}{\lambda y.x:\beta \Rightarrow \alpha}}{\lambda xy.x:\alpha \Rightarrow (\beta \Rightarrow \alpha)}\scriptsize{x}
\]

Figure 2




Here \(\lambda\) abstraction, the introduction of \(\lambda\) terms,
corresponds to \(\rightarrow\)I, so the \(\lambda\) term
\(\lambda xy.x\) which is shown to have type \(\alpha \Rightarrow 
(\beta \Rightarrow \alpha)\) 'codes' two steps of the type
correlate of \(\rightarrow\)I, namely the rule \(\Rightarrow\)I introducing function
types, and we can recover the above proof of the propositional
theorem. Moreover given the tight connection between type theoretical
calculi and programs in certain types of programming languages, we can
also see the TT proof as a program of steps in the construction of a
certain type of computational object.



In natural deduction systems, normalisation is the procedure by which
redundant inferential loops are eliminated. In certain logics (such as
intuitionistic logic), a normalisation metatheorem holds and tells us
that any proof can be stripped of its redundancy and reduced to a
normal form. A further level of correspondence, brought out by Howard,
links normalisation with the 'evaluation' of programs in
which complex terms are reduced to their simplest forms (this is not
always possible in the more expressively powerful type systems). 



It seems to be Kreisel who introduced the slogan 'formulae as
types', with Martin-Lof responsible for the more widespread
'propositions as types' slogan (See again, Wadler, 2015).
In the philosophical context, 'proposition' is often used
to mean something like the meaning of a sentence, i.e. of a formula of
a certain sort. Using this terminology, a widespread intuitionist
position is that that the proposition expressed by a formula is the
set (or species, for the intuitionist) of all proofs of the formula.
Given that different provable formulae will correspond to different
types, the CH correspondence allows us to rephrase this
'syntactico-semantics' position as: the proposition
expressed by a formula of HA is the type of its proofs, where
'type' is not a straightforward synonym for
'set' or 'species' but the notion from
\(\lambda\)-calculus. As noted, this calculus is a formal system with
rich interconnections with programming and computer science and
readings in which the instances of types are purely syntactic, for
example proof-theoretic entities. Hence the meaning of a formula, the
proposition expressed, on such readings, does not represent a reality
distinct from the linguistic system in which the formula occurs. 



The connection with intuitionism, then, is clear: but what is the
relevance of the CH correspondence to formalism? There are, in the
first place, clear overlaps between some forms of intuitionism and
certain formalist positions. Not the philosophical intuitionism of the
founding father Brouwer, of course, with its ontology of mathematical
objects as mental constructions and an epistemology in which
mathematical knowledge is based on internal reflection on the
succession of ideas; this is a mathematical metaphysics far removed
from formalism. But many constructivists have embraced, without
accepting his metaphysics, the Brouwerian identification, or close
linkage, of mathematical correctness (truth, if one is prepared to
speak of mathematical truth) with provability. This sort of
identification is more than congenial to a certain brand of formalism,
one which rejects the idea that mathematical theses represent a
mind-independent reality and which also divides the sheep from the
goats on the basis of those which are provable, in some formal system,
versus those which are disprovable.



But there are also substantial differences between the intuitionist
and the formalist. For one thing, not only Brouwer but also many later
constructivists refuse to identify provability with provability in
some formal system. For another, formalists have generally felt free
to help themselves to classical logic, and have emphasised the *free
creativity* of the mathematician: she should be free to generate
whatever mathematical theories she wishes, subject only to withdrawing
them if they turn out to be inconsistent (in the chosen background
logic). 



On the first point the formalist will, of course, be a formalist! She
will link correctness, at least at the most fundamental level, to
formal proof. Here, then, the CH correspondence, or better
correspondences, are surely very attractive to the formalist. The
linkage between propositions and computations, algorithmic reductions
of terms coding proofs to irreducible normal forms in particular, fits
very snugly with those versions of formalism which take mathematics to
be, at heart, shuffling of symbols with no external reference.



On the second point, further work on the CH correspondences
generalised the results from intuitionistic logic to a wide variety of
other logics, in particular to classical logic (Griffin, 1990), as
well as to other logical frameworks such as modal logic and linear
logic. There is no burdensome logical restriction imposed by
'Formulae-as-Types' then, no need for the formalist to
fight with one hand tied behind her back.



What of the free creativity the formalist cherishes? Constructivist
type theory has, of course, been extended well beyond Heyting
arithmetic, particularly ambitious extensions are to be found in the
univalent foundations project based on homotopy type theory (Awodey,
2014). This, then, is an avenue a formalism based on Formulae-as-Types
might pursue. But for a formalist who wishes to be non-revisionist
about non-constructivist mathematics the prospects are perhaps less
clear. It is not enough just to add in the extra axioms or inference
rules which yield the particular theory, in a standard framework, e.g.
of a first-order or higher-order language. For one needs to do the
further work needed to show that an extension of the CH correspondence
obtains in this system. 



Moreover there is also an issue with primeness in the sense of the
proof-theoretic property (which intuitionist logic satisfies) that
when \(\vdash\) A \(\vee\) B then either \(\vdash\) A or \(\vdash\) B. Classical
theories typically do not have this property and this will pose
problems for generalising the CH correspondence and for justifying
excluded middle (assuming the formalist does not simply take all
non-trivial calculi as legitimate without need of justification). If
correctness of a mathematical claim, relative to a particular
framework, is identified with provability and if a disjunction can be
correct where neither disjunct is provable then the formalist would
seem to require some fancy footwork--supervaluationalism is not
obviously appropriate here--to justify using classical logic.




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



Finally it should be noted that CH formalism, if we can call it such,
will be unacceptable to the formalist who is motivated by
anti-platonist concerns and wishes to exclude abstract objects from
all mathematics, including metamathematics. For the meanings of
concrete utterances of mathematical formulae, according to CH
formalism, are sets/species/types of proofs, and the latter are
abstract objects, infinitely many of them, of arbitrarily long finite
length. The problem of the metatheory, in other words, has not been
met. The anti-platonist cannot straightforwardly lift the ideas of
syntactico-semantics and apply the CH correspondence in support of
anti-platonism; a great deal more philosophical work is required.


## 8. Contemporary Formalism



Later developments have been primarily in the 'Hilbertian'
wing of the formalist movement. P. J. Cohen's work on the
generalised continuum hypothesis showed, in tandem with
Godel's relative consistency proof, that it and countless
related set-theoretic propositions concerning the relations of the
cardinality of a set to that of its powerset are undecidable by
current axioms. With no obvious, non-ad hoc, ways to extend the axioms
to decide these questions, this led some mathematicians, such as Cohen
himself (Cohen, 1971) and Abraham Robinson, (Robinson, 1965; 1969) to
despair of a realistic interpretation of higher set theory. They
therefore treat branches of mathematics in which no plausible axiom
set will decide the key questions as 'ideal' parts of
mathematics, lacking the content to be found in other areas.



As to game formalism, although philosophers may accuse mathematicians
of a tendency to lapse into this seemingly discredited position, very
few philosophers advance views resembling the game formalists. Gabbay
(2010) and Azzouni (2004; 2005; 2006; 2009) have flown under the
formalist flag. Gabbay's formalism (which he develops only with
respect to arithmetic in the paper cited) occupies 'lush middle
ground between traditional formalism, fictionalism, logicism and
realism' (Gabbay, 2010: 219). Moreover he writes 'Contrary
to traditional (game) formalism, my proposal shall not involve an
attempt to provide formal derivations of each and every arithmetic
truth' (Gabbay, 2010: 221).



Azzouni describes his 'version of formalism' (Azzouni,
2004: 105) as one in which ordinary mathematical proofs
'indicate' formal derivations. The indication relation is
left rather open: Azzouni does not claim that the indicated
derivations all belong to a single formal system; rather ordinary
proofs can indicate derivations from a 'family' of formal
derivations. However, the indicated derivations, Azzouni says need not
exist; certainly concrete tokens of them need not exist in the same
period as the informal proofs which indicate them. Proofs in ancient
Greek geometry indicate 21st century or later derivations. They may
never, in fact, exist--they may be too long ever to be written
down (Azzouni, 2006: 154) though these non-existent proofs are
supposed to explain the consensus among mathematicians about which
informal proofs are correct! In later work Azzouni seems to retreat
from this (or any) brand of formalism:



> 
> I kept falling (against my will) into a view that mathematicians had
> to be engaged in *something* like sophisticated syntactic
> pattern-recognition while perusing informal mathematical proofs, so
> that they would be sensitive (without realizing it) to a background of
> *nonexistent* formal derivations. (Azzouni, 2009: 25)
> 



moving to an 'inference package' view of mathematical
reasoning which does not seem formalist: see again Azzouni (2009).




There are, however, another group of contemporary philosophers of
mathematics whose views seem close to formalism, namely (some among
the) fictionalists. Now the term 'fictionalist' can
mislead as not all fictionalists assimilate mathematics to fiction.
And even if one did, the question would arise: 'What
philosophical account of fiction, and discourse about fiction, does
one adopt?' Many philosophers resile from a realist ontology of
fictional characters just as many reject a
realist--platonistic--ontology for mathematics. A very
simplistic anti-realism about fictionalism might analyse such
statements as 'Oliver Twist was born in London' as true
(or correct) just in case that sentence, or a synonym, occurs in
Dickens's novel (Field, 1989: 3). Even if this worked for
fiction, which it pretty clearly does
 not,[5]
 a parallel approach for mathematics is patently absurd. If a
mathematical thesis is proclaimed a theorem, with or without proof, in
a journal, be it however respectable then, even if this claim is never
challenged and the assertion accepted by the mathematical community
this by no means entails that the thesis is true (or correct, if one
dislikes applying truth predicates to mathematical sentences). Given
the number of "theorems" whose purported proofs were later
discovered to be incorrect, we can be pretty sure some falsehoods will
wrongly remain accepted for all time as proven. Moreover there will be
no end of mathematical claims, some true, some false, which never make
it into the mathematical literature at all, never being considered by
actual mathematicians.



Now the Oliver Twist example is owing to Hartry Field, the founder of
the school of fictionalism, if we may term it such. But he qualifies
his position as follows:



> 
> most of us believe that Oliver Twist lived in London only in the sense
> that we believe that the novel says that or *has as a
> consequence* that Oliver Twist lived in London (1989, 3) [emphasis
> mine].
> 



Whether or not this will work for fiction (What if the work is
inconsistent: must one use a relevant consequence relation? How can it
handle comparisons among different works as in the Tolstoy/Dostoyevsky
example above?) this is an interesting position on mathematics with
definite formalistic overtones. The mathematician can put forward any
(consistent) theory she likes. The truths of the theory are then just
the consequences of the theory, there is no need to think that the
theory represents an external reality. Fictionalism along these lines
has been investigated by Mary Leng (2010). But a key question is: how
does one read 'consequence'? For a formalist, this has to
be consequence as derivability. But Leng rejects such a reading: her
fictionalism is one in which logical consequence is interpreted not
syntactically but modally, with the necessity in question taken as
primitive. Hence, this subspecies of fictionalism cannot be classed as
formalist.



Weir, by contrast, explicitly embraces formalism (1991; 1993; 2010;
2016), moreover formalism in the game formalism tradition. His
position, if situated with respect to fictionalism, can be seen as one
in which 'consequence' is read, in the formalist
tradition, syntactically, in terms of formal derivability. As a first
approximation, the position is that a mathematical sentence is true if
there exists a *concrete* derivation of a token of it, false if
there exists a concrete derivation of a token of its negation. Since
truth and falsity conditions make no appeal to abstract proofs, this
type of formalism is firmly anti-platonist. 



This bluntly concretist formalism would seem to face insuperable
problems: for example in the shape of the 'concretely
undecidable', those short theses with unfeasibly long proofs or
disproofs mentioned above in connection with Goodman and Quine's
nominalism. Weir's attempt to address such problems takes as its
basis a fairly common 'post-Fregean' or
'neo-Fregean' perspective on language. Frege, at least
early in his career, held that the truth value of a sentence is
determined by two factors, the *Sinn*, the sense, literal meaning
or informational content, of the sentence; and the way the world is.
Indexicality and wider context relativity of sentences he initially
supposed could be met by assuming speakers who uttered and grasped
such sentences took them as elliptical for more complete utterances
whose sense fixed, in combination with the world, a unique truth
value.



Later work (including Frege's own) revealed the inadequacy of
this picture, revealed that some indexicality, for instance, is
'essential' in John Perry's phrase, to the thought
expressed. I can utter 'it's hot now' truly without
knowing where, when or even who (if sufficiently disoriented or out my
head) I am. Those who are not utterly sceptical, as radical
contextualists are, of systematic theories of meaning, will amend
Frege's view to a tripartite one. A sentence's truth
value, in a particular context, is determined by its informational
content, by contextual circumstances related to, 'fitted
to', the utterance by aspects of the speakers' practice,
and finally by the mind and language-independent world. The contextual
circumstances need not figure in the sense or informational content of
the utterance; thus a specification of them may include dates and
locations, though this is not part of the sense of 'it's
hot now' (compare Kaplan's *character* versus
*content* distinction, especially the second sense of
'content' discussed in his 1989 (fn 28: 503)). 



This picture in turn suggests the idea that the contextual
circumstances which 'make true', in conjunction with
independent reality, an utterance, may make it true in a non-realist
fashion. In Weir's version of game formalism, the fundamental
idea is that what makes true (or false) \(\text{'}\sin^2\theta
+\cos^2\theta = 1\text{'}\) in a particular system is the existence of
a concrete proof or refutation, though the statement that such a
concrete proof exists is no part of the literal meaning or sense of
the claim. 



The advantage of this type of formalism is that it not only affirms
the meaningfulness, the possession of sense, of mathematical
utterances; in sharp contrast with traditional game formalism, it
holds that such utterances have truth values, where proofs or
refutations exist. The problem of the metatheory is met if one can
give, after the fashion of Goodman and Quine 1947, a non-mathematical
account of concrete proof. Of course, as noted above, severe problems
remain. The problem of applicability has to be met, by providing
conservative extension proofs for example. And of course the theory is
threatened not only by Godel--type incompleteness and the
undecidable--but intuitively truth-valued- sentences it
throws up; even more devastating indeterminacy looms in the form of
'concretely undecidable' sentences such as Tennant's
primality claim of section 5. 



One strategy for dealing with these problems is to combine formalism
with strict finitism (for one brand of which see Yessinin-Volpin
(1961; 1970) and for criticism Dummett (1975): only feasibly long
"digestible" tokens--of formulae and of proofs--exist.
Formulae with no feasible proofs or disproofs simply lack truth value.
Since we know from Godelian speed-up considerations that for many a
provable sentence the shortest derivations of it or its negation are
vastly longer than the sentence itself, this finitist/formalism
position threatens to cause havoc across large areas of perfectly
ordinary mathematics. 



Weir argues (2010; 2016) that finitist formalism is not only extremely
radical, it is incoherent. The reason is the all-pervasive role of
abbreviation which generates complex tokens where most of their
subparts will never exist, for example abbreviations which create
numerals naming (speaking with the platonist) arbitrarily high
numbers. As a result, strict finitist specifications cannot be given
in the usual inductive fashion, as the intersection of all inductive
sets containing the base set and closed under the complexity-forming
operations. And this in turn means we cannot prove even very simple
facts about wffs and proof.



Idealisation, then, is essential in metamathematics, including the
idealised notions of truth and proof found in metamathematics. If the
formalist is entitled to assert there are infinitely many primes,
whilst denying abstract objects exist, there seems no reason why she
cannot affirm that there are infinitely many formulae or infinitely
many proofs, whilst also denying that abstract objects exist. Weir
(2016: 38-39) therefore argues that the formalist can answer the
problem of concrete undecidables so long as there are *concrete*
proofs (or proof sketches) of results to the effect that formal truth,
for a given language or sub-language, coincides with formal
provability, and that there is no reason to restrict idealisation for
finitary languages. 



In conclusion, the formalist who espouses the meaningfulness and
indeed truth of standard mathematical theories, including proof theory
for example, has more resources to meet such claims than classic game
formalism. The question is whether these are enough to salvage a
position which it is fair to say most philosophers of mathematics
still think hopeless. There is, on the other hand, certainly no
universal consensus that formalism is dead and buried and signs of
strong sympathy for formalism among some mathematicians and computer
scientists.