###### Abstract

We re-examine the old question to what extent mathematics may be compared to a
game. Under the spell of Wittgenstein, we propose that the more refined object
of comparison is a “motley of language games”, the nature of which was
(implicitly) clarified by Hilbert: via different language games,
axiomatization lies at the basis of both the rigour and the applicability of
mathematics. In the “formalist” game, mathematics resembles chess via a clear
conceptual dictionary. Accepting this resemblance: like positions in chess,
mathematical sentences cannot be true or false; true statements in mathematics
are _about_ sentences, namely _that they are theorems_ (if they are). In
principle, the _certainty_ of mathematics resides in proofs, but to this end,
in practice these must be “surveyable”. Hilbert and Wittgenstein proposed
almost oppositie criteria for surveyability; we try to overcome their
difference by invoking computer-verified proofs. The “applied” language game
is based on Hilbert’s axiomatization program for physics (and other scientific
disciplines), refined by Wittgenstein’s idea that theorems are yardsticks to
which empirical phenomena may be compared, and further improved by invoking
elements of van Fraassen’s constructive empiricism. From this perspective, in
an appendix we also briefly review the varying roles and structures of axioms,
definitions, and proofs in mathematics. Our view is not meant as a philosophy
of mathematics by itself, but as a coat rack analogous to category theory,
onto which various (traditional and new) philosophies of mathematics (such as
formalism, intuitionism, structuralism, deductivism, and the philosophy of
mathematical practice) may be attached and may even peacefully support each
other.

Is mathematics a game?

Klaas Landsman and Kirti Singh

Institute for Mathematics, Astrophysics, and Particle Physics
Radboud Center for Natural Philosophy
Radboud University, Nijmegen, The Netherlands
<EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Contents

  1. 1 Introduction
  2. 2 Wittgenstein
  3. 3 Mathematics as a motley of language games
  4. 4 Hilbert
  5. 5 Applied mathematics
  6. 6 Truth
  7. 7 Certainty
  8. A Formalism, axioms, definitions, and proofs

##  1 Introduction

The aim of this paper is to re-examine the old question to what extent
mathematics may be compared with a game (like chess).111The review by Epple
(1994) provides a nice historical and philosophical introduction to this
question; see also Detlefsen (2005). For the analogy with chess especially in
Frege and Wittgenstein see also Kienzler (1997), Mühlhölzer (2008, 2010),
Stenlund (2015), Max (2020), and Lawrence (2023). Wittgenstein’s reflections
on Frege in general (which fed much of his thought throughout his career) are
reviewed by many authors. Apart form the above literature, see e.g. Reck
(2002), Baker & Hacker (2009ac), Kienzler (2012), Dehnel (2020), Potter
(2020), Schroeder (2021), etc.  Whatever the answer, it is a good point of
entry into the philosophy of mathematics, and even into its history; for
example, it is no accident that (barring some “playful” earlier allusions) the
idea of such a comparison originated in the late 19th century, since that was
the time of the “modernist transformation” in which mathematics gradually lost
its connection with physical reality, intuition, and visualizability; these
were replaced by abstraction and rigorous proof.222See, for example, Mehrtens
(1990), Heintz (2000), Gray (2008), and Maddy (2008). Serious analysis started
with Frege’s criticisms of Thomae (1898), Heine (1872), and Illigens
(1893).333See Frege (1903), §§86–137, translated in Geach and Black (1960).
This degenerated into a hilarious public polemic between Thomae (1906ab, 1908)
and Frege (1906, 1908ab). Frege (1899) is also relevant (and very funny).
Frege fiercely rejected the analogy between mathematics and chess; for he was
primarily unable to comprehend how a mere game could describe any
“thought”.444 This was the central ingredient of his philosophy of both
mathematics and language. The word ‘thought’ (_Gedanke_) is ambiguous in both
English and German. Frege (1892) clarifies that for him, thoughts do not refer
to the subjective act of thinking, but to the objective content thereof, which
can be shared by many people. All of his more specific reasons for
disapproving of the analogy essentially reduce to this inability, such as:

  * •

Mathematics is _meaningful_ , since it refers to real objects. Games are
meaningless.

  * •

Thus the rules of mathematics originate in reality, whereas for games they are
arbitrary.

  * •

Grounded in reality, there is _truth_ in mathematical theorems, which games
lack.

  * •

The applicability of mathematics would be incomprensible if it were merely a
game.

  * •

Mathematics also incorporates _the theory of the game_ instead of only _being
the game_.555The argument seems to be that this is something a game could not
accomplish, whence the analogy breaks down.

Such arguments go to the heart of the philosophy of mathematics. Seeing
mathematics as a game provides a yardstick for answering the main
(traditional) questions in this area, such as:666This list is mostly
paraphrased from Linnebo (2017), §1.1, except for the last question. See also
Tait (2001).

  1. 1.

What is mathematics _about_?777The question what it means that mathematics (or
rather some language game) is _about_ certain objects external to it is a
difficult one. A useful answer has been given by Mühlhölzer (2012), p. 114:
‘An object is _given in advance_ iff the criteria of identity for the object
which the language game is about are _not_ completely stated or presented by
the language game itself; and it is _not given in advance_ iff the criteria of
identity for the object are completely stated or presented in the language
game – [so] that this identity is given by the language game alone and by
nothing else.’ Do “mathematical objects” exist? If so, are they “ _abstract_
”?888Abstraction may be taken to mean: not located in space and time and not
participating in causal relationships.

  2. 2.

Is mathematics _a priori_ , in the sense that it does not rely on experience
or experimentation?

  3. 3.

What is the nature of mathematical truth? How can we know it? Is what we know
certain?

  4. 4.

Are mathematical truths _necessary_ , in the sense that _they could not have
been otherwise_?

  5. 5.

How is applied mathematics/mathematical physics possible (and why is it so
powerful)?

Platonists like Frege answer these question from the belief that mathematics
is _referential_ in a very specific way, namely that mathematical language
refers to _abstract_ mathematical objects that “exist” outside this language
and independently of those using it. In particular: _there are real
mathematical objects, which mathematicians refer to and describe._ Thus
mathematicians _discover_ (rather than _invent_) mathematical truths, which
consist of correct descriptions of these mathematical objects and their
properties. Being discoverers, mathematicians are supposed to be similar to
physicists who unearth properties of nature, with the flattering difference
that what mathematicians discover is both _a priori_ and necessary. As Hardy
put it:999And similarly: ‘It seems to me that no philosophy can possibly be
sympathetic to a mathematician which does not admit, in one manner or another,
the immutable and unconditional validity of mathematical truth. Mathematical
theorems are true or false; their truth or falsity is absolute and independent
of our knowledge of them. In some sense, mathematical truth is part of
objective reality. “Any number is the sum of 4 squares”; “any number is the
sum of 3 squares”; “any even number is the sum of 2 primes”. These are not
convenient working hypotheses, or half-truths about the Absolute, or
collections of marks on paper, or classes of noises summarising reactions of
laryngeal glands. They are, in one sense or another, however elusive and
sophisticated that sense may be, theorems concerning reality, of which the
first is true, the second is false, and the third is either true or false,
though which we do not know. They are not creations of our minds; Lagrange
discovered the first in 1774; when he discovered it he discovered some- thing;
and to that something Lagrange, and the year 1774, are equally indifferent.’
(Hardy, 1929, p. 4). Yet Hardy (1940) takes the analogy with chess quite
seriously, but rejects it: not on grounds of ontology or epistemology, but
_triviality_ : ‘A chess problem is genuine mathematics, but it is in some way
“trivial” mathematics. However ingenious and intricate, however original and
surprising the moves, there is something essential lacking. Chess problems are
_unimportant_.’ (p. 88)

> I have often used the adjective ‘real’, and as we use it commonly in
> conversation. I have spoken of ‘real mathematics’ and ‘real mathematicians’,
> as I might have spoken of ‘real poetry’ or ‘real poets’, and I shall
> continue to do so. But I shall also use the word ‘reality’, and with two
> different connotations.
>
> In the first place, I shall speak of ‘physical reality’, and here again I
> shall be using the word in the ordinary sense. By physical reality I mean
> the material world, the world of day and night, earthquakes and eclipses,
> the world which physical science tries to describe.
>
> I hardly suppose that, up to this point, any reader is likely to find
> trouble with my language, but now I am near to more difficult ground. For
> me, and I suppose for most mathematicians, there is another reality, which I
> will call ‘mathematical reality’; and there is no sort of agreement about
> the nature of mathematical reality among either mathematicians or
> philosophers. Some hold that it is ‘mental’ and that in some sense we
> construct it, others that it is outside and independent of us. A man who
> could give a convincing account of mathematical reality would have solved
> very many of the most difficult problems of metaphysics. If he could include
> physical reality in his account, he would have solved them all.
>
> I should not wish to argue any of these questions here even if I were
> competent to do so, but I will state my own position dogmatically in order
> to avoid minor misapprehensions. I believe that mathematical reality lies
> outside us, that our function is to discover or _observe_ it, and that the
> theorems which we prove, and which we describe grandiloquently — as our
> ‘creations’, are simply our notes of our observations. This view has been
> held, in one form or another, by many philosophers of high reputation from
> Plato onwards, and I shall use the language which is natural to a man who
> holds it. A reader who does not like the philosophy can alter the language:
> it will make very little difference to my conclusions. (Hardy, 1940, pp.
> 63–64)

See Balaguer (1998), Panza & Sereni (2013), Landry (2023), and Linnebø (2023)
for studies of platonism. Our goal is not to dispute this ideology in any
detail, if only because the above summary is very superficial. But here is the
main point of criticism: its ontology is obscure, and because of that it is
also difficult to explain how we can know anything about it (as Aristotle
already demurred against Plato).101010See especially his _Metaphysics_ , books
M and N ($\mu$ and $\nu$), available in many editions and translations.  Its
explanation of the applicability of mathematics is also at best shadowy.

More recently, Benacerraf (1973) argued that one cannot get both the ontology
and the epistemology of mathematics right,111111See also Nutting (undated) and
references therein for an excellent review of “Benacerraf’s problem”. His
argument is based on the same assumption as Frege’s, namely that mathematical
reference is similar to ordinary linguistic reference, and hence is vulnerable
to a late Wittgensteinian critique (Barco, 2018). We only use it rhetorically.
so that some philosophies of mathematics bend towards a sound ontology at the
expense of the epistemology, whilst others do the opposite. Platonism seems to
be the unique philosophy of mathematics that manages to get neither of the two
right. Hence despite its apparent popularity, even among top mathematicians of
the 20th century like Gödel (Kennedy, 2020), Connes (2001), and Penrose
(2004), platonism is, in our view, best seen as a foil (like solipsism), to
which a philosophy of mathematics based on games provides a useful
antidote.121212The broader notions would be “constructivist” or “anti-realist”
philosophies of mathematics. In the more general setting of realist versus
non-realist philosophies of mathematics, Shapiro (1997) and Rush (2022) argue
that this contrast is less than it seems, once these positions have been made
sufficiently precise.

In analyzing this cure for platonism, Wittgenstein’s late philosophy is an
important source, especially since it partly originated in his thoughts and
reflections on the foundations of mathematics, whilst his philosophy of
language constantly challenged referential (“Augsutinian”) theories, see
below. Our other protagonist is Hilbert, whose formalism—broadly taken and
_correctly interpreted_ —provides the key to modern mathematics and thence to
our analysis.131313Please note that this paper is _not_ supposed to be a study
of their respective philosophies of mathematics as such. Adding a crucial
ingredient of the constructive empiricism of van Fraassen, we arrive at a
synthesis to the effect that, in Wittgensteinian parlance, our answer to the
question in the title is that mathematics may be seen as —or at least may be
favourably compared with—a so-called _motley of language games_.

The motley we have in mind is like a branched tree. At the multiple roots
(some would say: at the top) one finds foundational theories or overall
frameworks like ZFC set theory.141414Their axioms are what Feferman (1999)
calls _foundational_. The others are _structural_. See also Schlimm (2013).
Above (or below) these roots, interwoven individual areas of mathematics spurt
out. Within most of these areas (or even within the foundational theories at
the top), one has two absolutely crucial language games: the first concerns
formalized proofs, whereas the second governs applications.151515From a
Kantian point of view, we might say that the _a priori_ side of mathematics is
one language game, and the _a posteriori_ side is another. These may also
alternate, as in the case where some area of mathematics originates in
experience (as all the traditional areas do), then becomes “purified” via
formalization, and subsequently may be applied again, perhaps in a different
context. Kant’s analytic/synthetic distinction is less relevant for modern
mathematics, except perhaps in propositional logic, where the tautologies
could be called “analytic”, and, throughout mathematics, in the context of
_explicit_ definitions, see §A.3. But invoking the latter distinction would
add little to our discussion.

This picture (or tool of comparison) is not meant as a philosophy of
mathematics by itself, but rather as a coat rack onto which various
(traditional and new) philosophies of mathematics may be attached and may even
support each other. Among many, the basic ones we have in mind are:

  * •

_Formalism_ as the starting point of any game-theoretic picture of
mathematics, cf. §A.1.

  * •

_Intuitionism_ as one of the possible language games giving a foundational
framework. Even if the latter is classical (as in ZFC), intuitionism may
provide the logic of the metamathematics used to analyze the framework (and
similarly for _finitism_ , as Hilbert tried).

  * •

_Structuralism_ entering pervasively in the proposed picture, starting with
Hilbert’s notion of implicit definitions, and moving to the structural
empiricism as just explained.

  * •

_Deductivism_ governing our concept of mathematical truth.

  * •

_Philosophy of mathematical practice_ (Mancosu, 2008; Hamami & Morris, 2020),
which is also favourably disposed towards the later Wittgenstein (Pérez-
Escobar, 2022).

Rather than contradicting each other, such philosophies in fact complement and
reinforce each other, as Hilbert himself clearly saw (at least for the first
two, whilst embodying the third).161616Even platonism is not excluded by our
proposal, although it would be unnatural, given that it was a foil.  This
unifying role of the motley of language games may be comparable to the
fruitful way category theory (or more generally topos theory) is a coat rack
for numerous mathematics disciplines; cf. Krömer (2007) and Marquis (2009) for
philosophically informed histories of this area.

To summarize this picture, let us answer the five traditional questions in the
philosophy of mathematics listed in the beginning of this Introduction, from
the point of view developed here:

  1. 1.

There are no mathematical objects in the platonic sense. Whatever its origins,
once mathematics has become mathematics, it is a dynamical collection of
families of rules. In Hilbert-style proof theory certain families of rules
become the object of study, once again via specific (new) rules; in that case,
the rules themselves become the object of (meta)mathematics.

  2. 2.

Historically, all of mathematics originated in experience. Some areas
subsequently took a life of their own, like a game, but even these remain a
_human practice_. Others areas always remained close to experience, and many
alternate between “pure” and “applied” (as a consequence of which this
distinction seems to make little sense). In applied mathematics and
mathematical physics, it is not so much the _mathematics itself_ that relies
on experimentation but the _match_ between the rules inherent in it and the
intended application.

  3. 3.

Theorem as such are not “true”. What may be true is the claim that some
sentence is a theorem. This kind of truth is intersubjective: we should all
agree about it. By definition, we can find truth through proof (which in turn
is an example of strict rule following). Like Hilbert and Wittgenstein, we
claim that the certainty of some proof is determined by its surveyability, but
we redefine this using the more recent notion of proof verification.

  4. 4.

Such truths are necessary, given the set of rules in which they originate. But
this necessity is admittedly remote from the necessity that mathematics has in
the eyes of a platonist.

  5. 5.

Applied mathematics and mathematical physics—indeed all of
mathematics—originate in attempts to axiomatize sufficiently mature theories
of physics, space, and quantity, etc. Their power comes from the fact that
this is a progressive practice, in which the axiomatizations improve upon
checks against “reality”, represented by data models which in turn improve.

The plan of the remainder of this paper is as follows. We start with a summary
of what we learnt from Wittgenstein (§2), and then walk straight into our
“motley” (§3). This is backed up by a study of what we take to be Hilbert’s
main views on the foundations of mathematics in so far as these are relevant
to our program (§4).171717What is called “Hilbert’s program”, i.e. his
attempts to give a finitist proof of the consistency of classical mathematics,
plays no role here; its goal was never achieved or perhaps even well defined.
See footnote 50 for references. Combining Hilbert, Wittgenstein, and van
Fraassen (as we see them), we move on to the associated concepts of applied
mathematics (§5), and finish with an analysis of truth (§6) and certainty (§7)
in this light. These are the main sections of the paper.

The appendix of the paper, which may be skipped on a first reading, is a crash
course in the philosophy of axioms (§A.2), definitions (§A.3), and proofs
(§A.4), in so far as this is relevant to our approach. These topics are
preceded by a short survey of formalism (§A.1), since this concept is often
misunderstood (especially by philosophers and notably in connection with
Hilbert).

Throughout this paper, readers who are cognizant of the (philosophical)
literature on Hilbert and Wittgenstein may feel that they are on familiar
territory, but even those readers might find some new analysis or context.
Serious examples from physics must be left to a successor paper.

##  2 Wittgenstein

The main sources of Wittgenstein’s philosophy of mathematics are _Wittgenstein
und der Wiener Kreis: Gespräche_ (Wittgenstein, 1984b), which records
conversations between 1929–1931; _Philosophische Grammatik, Teil III:
Grundlagen der Mathematik_ , compiled from 1930–1933 (Wittgenstein1984c);
_Bemerkungen über die Grundlagen der Mathematik_ (BGM) from 1937–1944
(Wittgenstein, 1984d); and _Lectures on the Foundations of Mathematics,
Cambridge 1939_ (LFM; Diamond, 1975).181818Weiberg & Majetschak (2022) define
Wittgenstein’s middle period as 1929–1936, so that BGM, LFM, and the
_Philosophische Untersuchungen_ (PU), written between 1936–1946 (Wittgenstein,
1984a), fall in his late period. Except for the _Lectures_ and the _Wiener
Kreis_ volume, the original notebooks and typescripts from which these
published works were assembled may be found in the _Wittgenstein’s Nachlass:
The Bergen Electronic Edition_ , available online in Open Access at
http://www.wittgensteinsource.org/ or
https://wab.uib.no/transform/wab.php?modus=opsjoner. English translations may
be found online in the _The Collected Works of Ludwig Wittgenstein. Electronic
Edition_ , at https://www.nlx.com/collections/121. Pichler et al. (2011) is a
useful survey of major Wittgenstein editions until 2011. Wittgenstein did not
prepare BGM for publication himself; the selection and polishing process that
led to the PU therefore did not take place, which makes the quality uneven
(Mühlhölzer, 2010; Hawkins & Potter, 2022). Recent secondary literature
includes e.g. Mühlhölzer (2006, 2008, 2010, 2012), Floyd (2021), and Schroeder
(2021), with useful summaries by Gerrard (1996), Potter (2011), Rodych (2018),
and Bangu (undated). For our purposes, Wittgenstein’s reply to Hardy (above)
seems a good way to start:191919 We side with Gerrard (1991) that a good way
to understand at least the start of Wittgenstein’s philosophy of mathematics
is to put his critique of the Augustinian conception of language at the
beginning of the _Philosophical Investigation_ (cf. Baker & Hacker, 2009a,
Essay I) next to his critique of what Gerrard calls the Hardyian Picture of
mathematics: ‘Wittgenstein, in fact, had two chief post-Tractatus accounts of
mathematics. I have labelled these the calculus conception and the language-
game conception. The calculus conception dominated Wittgenstein’s thought from
1929 through the early 1930s, although in some areas (such as contradiction)
its influence lasted longer. In the middle 1930s, his views began to change to
the language-game conception, and by the early 1940s, the view of mathematical
language as a nexus of language-games had completely overturned the calculus
view. In the transitional (calculus) period Wittgenstein saw mathematics as a
closed, self-contained system. The rules (construed extremely narrowly) alone
determine meaning, and thus become the final and only court of appeal. In the
more mature (language-game) period, meaning and truth can be accounted for
only in the context of a practice, and mathematics is examined by seeing what
special role it plays in our lives and its special relationship to other
language-games. But not everything changed; throughout all the stages of
Wittgenstein’s work on the philosophy of mathematics, he remained opposed to
and tried to undermine what he considered to be a misleading picture of the
nature of mathematics. According to this opposing picture, mathematics is
somehow transcendental: a mathematical proposition has truth and meaning
regardless of human rules or use. According to this picture there is an
underlying mathematical reality which is independent of our mathematical
practice and language and which adjudicates the correctness of that practice
and language. This plays a similar (negative) role for Wittgenstein’s
philosophy of mathematics as does the Augustinian Picture for his later
philosophy of language. For reasons given in the next section, I call this the
“Hardyian Picture” after the mathematician G. H. Hardy. The Hardyian Picture
helps to give a structure to Wittgenstein’s work on the philosophy of
mathematics and to unite seemingly disparate discussions. Regardless of what
else he was doing, Wittgenstein always kept this picture in mind and tried to
distance himself from its temptations and confusions.’ (Gerrard, 1991, p.
127).

> Consider [Hardy (1929)] and his remark that “to mathematical propositions
> there corresponds—in some sense, however sophisticated–a reality”. (The fact
> that he said it does not matter; what is important is that it is a thing
> which lots of people would like to say). Taken literally, this seems to mean
> nothing at all— _what_ reality? I don’t know what this means.— But it is
> obvious what Hardy compares mathematical propositions with: namely physics.
> Suppose we said first, “Mathematical propositions can be true or false.” The
> only clear thing about this would be that we affirm some mathematical
> propositions and deny others. If we then translate the words “It is true …”
> by “A reality corresponds to …”—then to say a reality corresponds to them
> would say only that we affirm some mathematical propositions and deny
> others. We also affirm and deny propositions about physical objects.—But
> this is plainly not Hardy’s point. If this is all that is meant by saying
> that a reality corresponds to mathematical propositions, it would come to
> saying nothing at all, a mere truism: if we leave out the question of _how_
> it corresponds, or in what sense it corresponds.
>
> We have here a thing which constantly happens. The words in our language
> have all sorts of uses; some very ordinary uses which come into one’s mind
> immediately, and then again they have uses that are more and more remote.
> For instance, if I say the word ‘picture’, you would think first and
> foremost of something drawn and painted and, say, hung up on the wall. You
> would not think of Mercator’s projection of the globe; still less of the
> sense in which a man’s handwriting is a picture of his character. A word has
> one or more nuclei of uses which come into every body’s mind first; so that
> if one says so-and-so is also a picture—a map or _Darstellung_ in
> mathematics—in this lies a comparison, as it were, “Look at this as a
> continuation of that.” So if you forget where the expression “a reality
> corresponds to” is really at home— What is “reality”? We think of “reality”
> of something we can _point_ to. It is _this, that_. Professor Hardy is
> comparing mathematical propositions to propositions of physics. This
> comparison is extremely misleading. (LFM = Diamond, 1975, pp. 239–240)

Indeed, apart from Plato, Frege, and Russell, Hardy was one of Wittgenstein’s
favorite targets.202020Wittgenstein’s opposition to Frege and Hardy aligns
with his objections to Plato (Kienzler, 1997, 2013). He commented on the
dialogues _Sophist, Theaetetus, Charmides, Philebus_ , and _Cratylus_ (and
undoubtedly read others). The first four are, roughly speaking, searches for
the definition of sophistry, knowledge, temperance, and pleasure, whilst the
last is a quest for the nature of names and signs, asking questions about
conventions and reference very similar to those analyzed about 2300 years
later by Frege, Russell, and Wittgenstein. In Plato (as well as Aristotle),
defining is seen as a search for essence, although ironically, almost all such
attempts fail. Wittgenstein’s concept of a _family resemblance_ (see also
§A.3) may even be seen as his answer to the Socratic quest for definitions
through essence. Wittgenstein is often negative about Plato (‘Wenn man die
Sokratischen Dialoge liest, so hat man das Gefühl: welche fürchterliche
Zeitvergeudung! Wozu diese Argumente die nichts beweisen & nichts klären.’
Ms-120,25r[2]), particularly about his efforts to _transcend_ language; what
Plato sought in the lofty realm of forms, Wittgenstein simply found on earth
in the use of language, obviating the need for Plato’s ethereal reifications
(Kuusela, 2019b). Nonetheless, it seems that Wittgenstein never _denied_ the
existence of some external reality, even a mathematical one (Gerrard, 1991;
Dawson, 2014). His point was that the correctness of a mathematical
proposition (or some other linguistic utterance) is not established by
comparing it with such a reality (if it exists), but with some linguistic
practice (such as a proof). But in the direction of philosophical therapy he
could have argued that a lack of real mathematical objects also relieves us of
the obligation to explain what these exactly would be and how we could
possibly access them.

On the positive side, in his middle period (notably in the _Blue and Brown
Books_ from 1933–1935) Wittgenstein introduced _language games_ as a central
tool of his analysis of language, which he generalized later on in the
_Philosophical Investigations_. Although he refrains from giving a
definition—and would regard any such definition as misguided, since games,
languages, and language games are among his main examples of a _family
resemblance_ , cf. §A.3 below—we try:

  1. 1.

A language game is a _practice_ where certain words and symbols are _used_
(operationally);212121Floyd (2015) makes the following point: ‘In English we
lack the German word “ _Praxis_ ” (…). “Practice” is a poor substitute,
because it carries a connotation of something contingent and coordinative, a
convention, or a mere matter of “what we choose to do”—which is of course
vague until spelled out, and impossible to describe without theorizing in some
way or other, for every description explains to some extent, and every
explanation describes to some extent. (…) But in German _Praxis_ forms part of
what any theory must itself manage to theorize and incorporate, critically.’

  2. 2.

This _use_ is determined by specific _rules_ (forming the _grammar_ of the
language game);222222Kuusela (2019a), Chapter 6, holds that language games
_need_ not be based on rules; but those in mathematics are.

  3. 3.

The _meaning_ of (most) words and symbols is in turn determined by their
_use_.

This inverts the traditional (‘Augustinian’) view in which the meaning of
words is determined by the external objects that they are supposed to refer to
(PI, §1). The example in §2 of the PI then shows that such a reference, even
if it exists, is insufficient to determine the use of words.232323In §2 of the
_Philosophical Investigations_ , an assistant $A$ has to pass building-stones
to a builder $B$ on a construction site. For this purpose they use a language
consisting of the four words ‘block’, ‘pillar’, ‘slab’, and ‘beam. After $B$
calls them out, $A$ brings the stones, ‘as he has learnt’. Here the use lies
in the _action_ involved. Wittgenstein’s examples also suggest that there are
natural as well as invented language games, and that language games may be
open-ended or “alive”: e.g., new words, symbols, as well as rules may be
added, and old ones changed. Furthermore, the _existence_ of a rule does not
automatically _impose_ it: both the rules and the habit of following them form
part of the language game in question and hence are matters of (in our case:
mathematical) “practice”. This may require strict adherence to the rule, or
some leeway, see §7 for the case of proofs as an example of rule
following.242424 In §185 of the PI the interlocutor asks a pupil to continue a
series $2,4,6,8,\ldots$. Surprisingly, as soon as the pupil has reached 1000,
(s)he continues with $1000,1004,1008,1012$, claiming this is correct. The
literature on this is enormous: see e.g. Baker & Hacker (2009c), pp. 116–134
in general, and Mühlhölzer (2010), §§I.5, I.6, for mathematics.

We will use language games in mathematics as indicated by Wittgenstein for
languages:252525See Kuusela (2019a), Chapter 5, for a detailed exegesis of
these paragraphs, which we endorse.

> §130. Unsere klaren und einfachen Sprachspiele sind nicht Vorstudien zu
> einer künftigen Reglementierung der Sprache, gleichsam erste Annäherungen,
> ohne Berücksichtigung der Reibung und des Luftwiderstands. Vielmehr stehen
> die Sprachspiele da als _Vergleichsobjekte_ , die durch Ähnlichkeit und
> Unähnlichkeit ein Licht in die Verhältnisse unsrer Sprache werfen
> sollen.262626‘Our clear and simple language-games are not preliminary
> studies for a future regimentation of language a as it were, first
> approximations, ignoring friction and air resistance. Rather, the language
> games stand there as _objects of comparison_ which, through similarities and
> dissimilarities, are meant to throw light on features of our language.’
> (_Philosophische Untersuchungen_)
>
> §131. Nur so nämlich können wir der Ungerechtigkeit, oder Leere unserer
> Behauptungen entgehen, indem wir das Vorbild als das, was es ist, als
> Vergleichsobjekt sozusagen als Maßstab hinstellen; und nicht als Vorurteil,
> dem die Wirklichkeit entsprechen _müsse_. (Der Dogmatismus, in den wir beim
> Philosophieren so leicht verfallen.)272727‘For we can avoid unfairness or
> vacuity in our assertions only by presenting the model as what it is, as an
> object of comparison a as a sort of yardstick; not as a preconception to
> which reality _must_ correspond. (The dogmatism into which we fall so easily
> in doing philosophy.)’ (_idem_)

Thus language games—or at least the ‘clear and simple’ ones—are tools of
_examination_ rather than tools of _description_ (or even of
_representation_): they provide benchmarks or yardsticks with which some
linguistic practice can be _compared_. The result of such a comparison could
be a perfect match, but it is equally interesting to find out where the
differences lie (Kuusela, 2019a).282828 What Hempel wrote about Carnap’s
notion of _explication_ also applies to such comparisons, namely that the
situation is not black and white: ‘Thus understood, an explication cannot be
qualified simply as true or false; but it may be adjudged more or less
adequate according to the extent to which it attains its objectives.’ (Hempel
1952, p. 12)

Throughout his career, Wittgenstein claimed that philosophical problems
originate in misunderstandings of the logic of language. In the _Philosophical
Investigations_ , such misunderstandings are traced to the confusion that
arises if different language games are mixed up, in the sense that rules
determining the use (and hence the meaning) of some linguistic structure (like
a word, or sentence, or grammatical form) in one language game are mistakenly
used in a different one:

> seltsam erscheint der Satz nur, wenn man sich zu ihm ein anderes Sprachspiel
> vorstellt als das, worin wir ihn tatsächlich verwenden. (_Philosophische
> Untersuchungen_ , §195).292929 ‘the sentence seems odd only only when one
> imagines it to belong to a different language-game from the one in which we
> actually use it.’

Phrased during his work on the _Philosophical Investigations_ , Wittgenstein’s
comments on Hardy (quoted above) follow this pattern. More briefly, the
following pair of sentences misled e.g. Frege:

“The Earth is very old” “17 is a prime number”.

These have the same structure, which suggests that the singular terms “Earth”
and “17” both refer to objects that really exist. But this is based on a
confusion between different language games.303030See e.g. Baker & Hacker
(2009a), especially §I.4, and Mühlhölzer, 2010, especially §I.4.

Another mistake is to confuse properties of some language game used as a tool
of examination (i.e. comparison) with properties of the things that are
examined (Kuusela, 2019a, §4.6). This is a key mistake of the platonists: the
language of mathematics (or, for Wittgenstein, of logic) is _abstract_ , in
the sense of acausal, aspatial, and atemproal. The things for which
mathematics is a tool of examination, such as (actual) circles or triangles or
numbers, are not: any of those is present somewhere in space and time, and may
be causally effective. And yet we can apply mathematics to them: in §5 we
describe in detail how this is done without postulating a platonic realm.

##  3 Mathematics as a motley of language games

Although Wittgenstein is sometimes taken to have proposed that a (natural)
language is a “motley of language games”, such a claim is difficult to find in
his published works or in his _Nachlass_ , and indeed would be at odds with
§§130–131 of the PU just quoted. What he did say was:

> Die Mathematik ist ein buntes _Gemisch_ von Beweistechniken. — Und darauf
> beruht ihre mannigfache Anwendbarkeit und ihre
> Wichtigkeit.313131‘Mathematics is a motley of techniques of proof. — And
> upon this is based its manifold applicability and its importance.’ A motley
> is the traditional costume of the court jester or fool, which seems well in
> the spirit of Wittgenstein! See, however, Mühlhölzer (2005), p. 66, footnote
> 15, for a critique of this translation. (BGM, §III.46)

But this limited view of mathematics seems insufficient: apart from proofs, it
involves a lot more:

  1. 1.

_A long history:_ from numerical tables in Mesopotamia almost 4000 years ago
to the rigorous concept of a function in the 19th and 20th centuries; from
quantitative methods of surveying to Riemannian geometry; from counting to
class field theory, _et cetera_. It has thereby led to:

  2. 2.

A number of different _formal foundations of mathematics_ , like ZF or ZFC or
BNG set theory, intuitionistic set theory, $\lambda$-calculus, topos theory,
homotopy type theory, _et cetera_.

  3. 3.

Associated _notions of proof_ ranging from the informal reasoning of ancient
Babylonian and Chinese mathematicians to the pseudo-axiomatic setting of
Euclid (which lacked explicit rules of deduction) to the advanced logical
apparatus of Frege, Russell, Hilbert, and Gödel. But even the logic differs
not only between the formal foundational systems just mentioned (and others),
but also includes considerable diversity in what is being tolerated within
each of them, from informal rigour to bending the rules. See also
Wittgenstein’s quote above.

  4. 4.

Within each of these foundational systems: a wide collection of mathematical
theories (also called areas, branches, disciplines, or fields),323232See the
MSC at https://mathscinet.ams.org/mathscinet/msc/msc2020.html, or the
‘Branches of Mathematics’ listed in Gowers (2008). To get an idea, under the
letter $a$ the latter sums up: ‘algebraic numbers; analytic number theory;
algebraic geometry; arithmetic geometry; algebraic topology’.  each with its
own (sub) community, goals, and standards of proof. These areas typically also
overlap (e.g. Lie groups combine group theory and differential geometry;
functional analysis combines linear algebra and topology, etc.). Following
Hilbert (see below) we find it hard to maintain the traditional distinction
between “pure” and “applied” mathematics (although many mathematics
departments do).

  5. 5.

The meta-theory of the axiomatized theories (i.e. Frege’s “theory of the
game”), including _both_ formal aspects like proof theory _and_ informal
aspects like “the strategy of the game”.333333Developing the formals aspect of
thist, i.e., metamathematics, was of course Hilbert’s achievement.

It would therefore be better to answer our title question ‘ _Is mathematics a
game?_ ’ by:

_Mathematics is a motley of language games of a very specific (formal) kind._

This remains well within the spirit of Wittgenstein,343434See e.g. Baker &
Hacker (2009c), Essay I: Two fruits upon one tree, and Mühlhölzer (2010),
§§I.6, II.8. especially if, encouraged by §§130–131 of the PU just quoted, it
is taken as a proposal to merely _compare_ mathematics with a motley of
language games, seen as a yardstick which in part describes mathematics well,
and in part may not (although the latter is hard to find). But the point, of
course, is to flesh this out by explaining what is meant by “mathematics” and
showing what actually makes it a “motley of language games”.

To start, let us take some piece of formalized mathematics, such as ZFC set
theory at the top level,353535Omitting the axiom of choice would be like
omitting e.g. castling from the rules of chess. or some formalized theory
within it (or taken separately), such as Peano arithmetic or (Hilbert-style)
Euclidean geometry, including its (logical) rules of inference (e.g. first-
order logic). We then favour the specific analogy between chess and
mathematics proposed by Weyl (1926):363636This works both at a purely logical
level and in interpretations of syntactic theories in set theory (cf. footnote
56). What is admittedly missing here is a translation of the goal of _winning_
in chess: there seems to be no analogue of checkmate in mathematics (although
there is an emotional analogue of resigning, i.e., “giving up”.). Indeed, in
the latter the goal is to establish the counterpart not of a winning position
but of an arbitrary legal position (i.e. a theorem). Perhaps the shared aspect
of beauty in both games of chess and proofs somewhat compensates for this
discrepancy.

  * •

The _axioms_ of some theory are analogous to the starting position of a game
of chess;

  * •

The _deduction rules_ (à la Natural Deduction) are analogous to the possible
moves;373737 In a Hilbert-style calculus (Hilbert & Ackermann, 1928), most
deduction rules are seen as axioms, _modus ponens_ being the only deduction
rule. What we here have in mind is the opposite: the axioms are supposed to
describe some specific mathematical theory (such as set theory, or arithmetic,
or Euclidean geometry) whereas all deduction rules are logical in character
and, with a few exceptions, are universal for all fields of mathematics. See
e.g. von Plato (2017).

  * •

A _sentence_ (as defined in logic) is analogous to _some_ position on a chess
board;

  * •

A _theorem_ is like a _legal_ position in a correctly played chess game;

  * •

A _proof_ is like a game leading to that position, played according to the
rules;

  * •

A _definition_ resembles the idea that chess pieces are defined by the rules
of chess.

The last point was not mentioned by Weyl (1926) and should be attributed to
Wittgenstein:

> Es ist übrigens sehr wichtig, daß ich den Holzklötzchen auch nicht ansehen
> kann, ob sie Bauer, Läufer, Turm, etc. sind. Ich kann nicht sagen: Das ist
> ein Bauer und für diese Figur gelten die und die Spielregeln. Sondern die
> Spielregeln bestimmen erst diese Figur: Der Bauer ist die Summe der Regeln,
> nach welchen er bewegt wird (auch das Feld ist eine Figur), so wie in der
> Sprache die Regeln der Syntax das Logische im Wort bestimmen.
> (Wittgenstein, 1984b, p. 134).383838‘It is, incidentally, very important
> that by merely looking at the little pieces of wood I cannot see whether
> they are pawns, bishops, castles, etc. I cannot say, “This is a pawn and
> such-and-such rules hold for this piece.” Rather, it is only the rules of
> the game that define this piece. A pawn is the sum of the rules according to
> which it moves (a square is a piece too), just as in language the rules of
> syntax define the logical element of a word.’

This was part of Wittgenstein’s general criticism of Frege’s arguments against
the chess analogy:

> _Frege_ hat sich mit Recht gegen die Auffassung gewendet, daß die Zahlen der
> Arithmetik die Zeichen sind. Das Zeichen “0” hat doch nicht die Eiegnschaft,
> zu dem Zeichen “1” das Zeichen “1” zu ergeben. In dieser Kritik hatte Frege
> recht. Nur hat er nich das andere gesehen, was am Formalismus berechtigt
> ist, daß die Symbole der Mathematik nicht die Zeichen sind, aber doch keine
> Bedeutung haben. Für Frege stand die Alternative so: Entweder wir haben es
> mit den Tintenstrichen auf dem Papier zu tun, oder diese Tintenstriche sind
> Zeichen _von etwas_ , und das, was sie vertreten, ist ihre Bedeutung. Daß
> diese Alternative nicht richtig ist, zeigt gerade das Schachspiel: Hier
> haben wir es nicht mit den Holzfiguren zu tun, und dennoch vertreten die
> Figuren nichts, sie haben in Freges Sinn keine Bedeutung. Es gibt eben noch
> etwas drittes: die Zeichen können verwendet werden wie im Spiel. Wenn man
> hier (beim Schachspiel) von ‘Bedeutung’ reden wollte, so wäre es am
> natürlichsten zu sagen: Die Bedeutung des Schachspiels ist das, was alle
> Schachspiele gemeinsam haben.393939 ‘ _Frege_ was right in objecting to the
> conception that the numbers of arithmetic are signs. The sign “0” , after
> all, does not have the property of yielding the sign “l” when it is added to
> the sign “I” . Frege was right in this criticism. Only he did not see the
> other, justified side of formalism, that the symbols of mathematics,
> although they are not signs, lack a meaning. For Frege the alternative was
> this: either we deal with strokes of ink on paper or these strokes of ink
> are signs of something and their meaning is what they go proxy for. The game
> of chess itself shows that these alternatives are wrongly conceived—although
> it is not the wooden chessmen we are dealing with, these figures do not go
> proxy for anything, they have no meaning in Frege’s sense. There is still a
> third possibility: the signs can be used the way they are in the game. If
> here (in chess) you wanted to talk of “meaning” , the most natural thing to
> say would be that the meaning of chess is what all games of chess have in
> common.’
>  (Wittgenstein, 1984b, p. 105)
>
> Frege ridiculed the formalist conception of mathematics by saying that the
> formalists confused the unimportant thing, the sign, with the important, the
> meaning. Surely, one wishes to say, mathematics does not treat of dashes on
> a bit of paper. Frege’s idea could be expressed thus: the propositions of
> mathematics, if they were just complexes of dashes, would be dead and
> utterly uninteresting, whereas they obviously have a kind of life. And the
> same, of course, could be said of any proposition: Without a sense, or
> without the thought, a proposition would be an utterly dead and trivial
> thing. And further it seems clear that no adding of inorganic signs can make
> the proposition live. And the conclusion which one draws from this is that
> what must be added to the dead signs in order to make a live proposition is
> something immaterial, with properties different from all mere signs.
>
> But if we had to name anything which is the life of the sign, we should have
> to say that it was its _use_. (_Blue Book_ , p. 4).

In other words, Frege (allegedly) just saw two possibilities: either symbols
refer to something in reality, in which case the game is meaningful (which, in
his view, doesn’t apply to chess since it lacks “thoughts”, whose alleged
absence blocks any analogy with mathematics), or they don’t (which for Frege
applies to chess but not to mathematics), in which case the game is
meaningless. Wittgenstein’s point is that Frege overlooked the possibility
that even _a priori_ meaningless symbols might “come alive” by their use, as
governed by the rules they are subject to:404040As noted by Kienzler (1997),
§4a, Wittgenstein overlooks or ignores the fact that Frege (1903) does note
this ‘other possibility’, for in footnote 1 on page 83 (which is part of §71)
he says: ‘Es besteht freilich auch eine Meinung, nach der die Zahlen weder
Zeichen sind, die etwas bedeuten, noch auch unsinnliche Bedeutungen solcher
Zeichen, sondern Figuren, die nach gewissen Regeln gehandhabt werden, etwa wie
Schachfiguren. Danach sind die Zahlen weder Hülfsmittel der Forschung noch
Gegenstände der Betrachtung, sondern Gegenstände der Handhabung. Das werd
später zu prüfen sein.’ (‘Of course, there is also an opinion according to
which numbers are neither symbols that mean something nor nonsensical meanings
of such symbols, but rather figures that are handled according to certain
rules, for example like chess pieces. According to this, the numbers are
neither aids for research nor objects of observation, but rather objects of
handling. This will have to be checked later.’) Indeed, in §95 Frege (1903)
complains that the rules of chess do not endow the chess pieces with any
_content_ that would be the consequence of these rules, ‘like the name
“Sirius” designates a certain fixed star.’ This suggests a stubborn inability
to see that the rules themselves comprise the meaning of chess (even though
the pieces are meaningless); which of course was Wittgenstein’s view.

  * •

For Frege, the use of symbols, i.e., the rules they are subject to, follows
from their meaning. This seems almost reactionary, given the “post-modernist
revolution” in mathematics.

  * •

For Wittgenstein, on the other hand, the rules are primary and whatever
meaning the symbols have follows from these rules. This resonates well with
Hilbert, to whom we now turn.

##  4 Hilbert

Essentially the same idea lies behind Hilbert’s concept of an _implicit
definition_ ,414141Giovannini & Schiemer (2021) call these _structural_ ,
since the words ‘implicit’ and ‘explicit’ are adjectives for certain technical
definitions in logic (which Beth’s definability theorem actually identifies).
See Hodges (1993), §6.6. which comes from Hilbert (1899) but is most clearly
explained in his fascinating correspondence with Frege:424242 The
Frege–Hilbert correspondence lasted from 1895 to 1903, centered around Hilbert
(1899). See for example Gabriel _et al._ (1980), Shapiro (2005), Hallett
(2010), Burke (2015), Blanchette (2018), and Rohr (2023).

> Meine Meinung ist eben die, dass ein Begriff nur durch seine Beziehungen zu
> anderen Begriffen logisch festgelegt werden kann. Diese Beziehungen, in
> bestimmten Aussagen formulirt, nenne ich Axiome und komme so dazu, dass die
> Axiome (ev[tl]. mit Hinzunahme der Namengebungen für die Begriffe) die
> Definitionen der Begriffe sind. Diese Auffassung habe ich mir nicht etwa zur
> Kurzweil ausgedacht, sondern ich sah mich zu derselben gedrängt durch die
> Forderung der Strenge beim logischen Schliessen und beim logischen Aufbau
> einer Theorie. Ich bin zu der Überzeugung gekommen, dass man in der
> Mathematik und den Naturwissenschaften subtilere Dinge nur so mit Sicherheit
> behandeln kann, anderenfalls sich bloss im Kreise dreht. (Hilbert to Frege,
> 22 September 1900; Gabriel _et al,_ , 1980, p. 23).434343 ‘In my opinion, a
> concept can be fixed logically only by its relations to other concepts.
> These relations, formulated in certain statements, I call axioms, thus
> arriving at the view that axioms (perhaps together with propositions
> assigning names to concepts) are the definitions of the concepts. I did not
> think of this view because I had nothing better to do, but I found myself
> forced into it by the requirements of strictness in logical inference and in
> the logical construction of a theory. I have become convinced that the more
> subtle parts of mathematics and the natural sciences can be treated with
> certainty only in this way; otherwise one is only going around in a circle.’

Thus Hilbert’s idea is that some axiom system—like the one given by Hilbert
(1899) himself for Euclidean geometry— _defines_ all non-logical things that
occur in this system (via certain arbitrary symbols).444444Hilbert held this
view at least since 1891, see Blumenthal (1970), pp. 402–403: ‘In einem
Berliner Wartesaal [in 1891] diskutierte er mit zwei Geometrn (wenn ich nicht
irre, A. Schoenflies und E. Kötter) über die Axiomatik der Geometrie und gab
seiner Auffassung das ihm eigentümlich scharfe Gepräge durch den Ausspruch:
‘Man muß jederzeit an Stelle von “Punkte, Geraden, Ebenen” “Tische, Stühle,
Bierseidel” sagen können.’ Seine Einstellung, daß das anschauliche Substrat
der geometrische Begriffe mathematisch belanglos sei und nur ihre Verknüpfung
durch die Axiome in Betracht komme, war also damals bereits fertig.’ ‘In a
Berlin waiting room [in 1891] he discussed the axioms of geometry with two
geometers (if I am not mistaken, A. Schoenflies and E. Kötter) and gave his
opinion a sharp thrust typical for him through the statement: ‘One must be
able to say “tables, chairs, beer mugs” instead of “points, lines, planes” at
any time.’ His attitude that the intuitive substrate of the geometric concepts
was mathematically irrelevant and that only their connection through the
axioms should be taken into account was already present at that time.’  As
illustrated by a famous earlier excerpt from the Frege–Hilbert correspondence,
this view was diametrically opposite to Frege’s, who maintained that the
things in axioms should _already_ be defined in advance, so that the
(compelling) axioms simply record their properties.

> Sie sagen weiter: “Ganz anders sind wohl die Erklärungen in §1, wo die
> Bedeutungen Punkt, Gerade, …nich angegeben, sonders als bekannt
> vorausgesetzt werden.” Hier liegt wohl der Cardinalpunkt des
> Missverständnisses. Ich will nichts als bekannt voraussetzen; ich sehe in
> meiner Erklärung in §I die Definition der Begriffe Punkte, Gerade, Ebenen,
> wenn man wieder die sämtlichen Axiome der Axiomgruppen I–V als die Merkmale
> hinzunimmt. Wenn man nach andern Definitionen für “Punkt”, etwa durch
> Umschreibungen wie ausdehnungslos etc. sucht, so muss ich solchem Beginnen
> allerdings aufs entschiedenste widersprechen; man sucht da etwas, was man
> nie finden kann, weil nichts da ist, und alles verliert sich und wird wirr
> und vage und artet in Versteckspiel aus. (…) Sie sagen meine Begriffe z.B.
> “Punkt”, “zwischen” seien nicht eindeutig festgelegt; z.B. [auf] S. 20 sei
> “zwischen” anders gefasst und dort sei der Punkt ein Zahlenpaar. Ja, es ist
> doch selbstverständlich eine jede Theorie nur ein Fachwerk oder Schema von
> Begriffen nebst ihren nothwendigen Beziehungen zu einander, und die
> Grundelemente können in beliebiger Weise gedacht werden. Wenn ich unter
> meinen Punkten irgendwelche Systeme von Dingen, z.B. das System: Liebe,
> Gesetz, Schornsteinfeger …, denke und dann nur meine sämmtlichen Axiome als
> Beziehungen zwischen diesen Dingen annehme, so gelten meine Sätze, z.B. der
> Pythagoras auch von diesen Dingen.454545 ‘You say further: ‘The explanations
> in sect. 1 are apparently of a very different kind, for here the meanings of
> the words “point”, “line”, …are not given, but are assumed to be known in
> advance.’ This is apparently where the cardinal point of the
> misunderstanding lies. I do not want to assume anything as known in advance;
> I regard my explanation in sect. 1 as the definition of the concepts point,
> line, plane - if one adds again all the axioms of groups I to V as
> characteristic marks. If one is looking for other definitions of a “point”,
> e.g., through paraphrase in terms of extensionless, etc., then I must indeed
> oppose such attempts in the most decisive way; one is looking for something
> one can never find because there is nothing there; and everything gets lost
> and becomes vague and tangled and degenerates into a game of hide-and-seek.
> (…) You say my concepts, e.g., “point”, “between”, are not unambiguously
> determined; e.g., on p. 20, “between” is taken differently and there a point
> is a number-pair. – Yes, it is obvious that any theory is actually only a
> framework or a schema of concepts, together with the necessary relations of
> these concepts to each other, and the base elements can be thought of in an
> arbitrary way. If, as my points, I think of some system of things, e.g., the
> system: love, law, chimney sweep … and then assume my axioms as relations
> between these things, then my theorems, too, hold of these things, e.g., the
> Pythagorean Theorem.’ See Gabriel et al., 1980, pp. 12, 13, and Hallett
> (2010), pp. 453–454.
>  (Hilbert to Frege, 29 December, 1899)

The idea is also clearly stated by von Neumann (1925), p. 220:464646It is no
accident that von Neumann went to Göttingen to work with Hilbert in 1926, soon
after this paper.

> Man wendet, um diesen [naiven] Begriff [der Menge] zu ersetzen, die
> axiomatische Methode an; d. h. man konstruiert eine Reihe von Postulaten, in
> denen das Wort “Menge” zwar vorkommt, aber ohne jede Bedeutung. Unter
> “Menge” wird hier (im Sinne der axiomatischen Methode) nur ein Ding
> verstanden, von dem man nicht mehr weiß und nicht mehr wissen will, als aus
> den Postulaten über es folgt. Die Postulate sind so zu formulieren, daß aus
> ihnen alle erwünschten Sätze der Cantorschen Mengenlehre folgen, die
> Antinomien aber nicht.474747‘To replace this [naive] notion [of a set] the
> axiomatic method is employed; that is, one formulates a number of postulates
> in which, to be sure, the word “set” occurs but without any meaning. Here
> (in the spirit of the axiomatic method) one understands by “set” nothing but
> an object of which one knows no more and wants to know no more than what
> follows about it from the postulates. The postulates are to be formulated in
> such a way that all the desired theorems of Cantor’s set theory follow from
> them, but not the antinomies.’ See also Muller (2004) and Schlimm (2013).

Unnecessarily conservatively, Hilbert did initially interpret the logical
connectives in the traditional way; but as suggested by his student Gentzen,
even these are implicitly defined by the rules for logical inference.484848See
Giovannini & Schiemer (2021), §4.2, and references therein, as well as von
Plato (2017). This idea is especially clear in Gentzen’s own proof system
called _Natural Deduction_ , where each logical symbol has an introduction
rule and an elimination rule. For example, the elimination rule for the
implication sign $\rightarrow$ is _modus ponens_ (i.e., $A$ and $A\rightarrow
B$ imply $B$), whilst its introduction rule states that $A\rightarrow B$
follows if from $A$ _as a hypothesis_ one can deduce $B$. In compensation, it
was one of Hilbert’s deepest conceptual insights that:

_Both the rigour and the applicability of mathematics originate in
axiomatization._

> Der Mathematik kommt hierbei eine zweifache Aufgabe zu: Einerseits gilt es,
> die Systeme von Relationen zu entwickeln und auf ihre logischen Konsequenzen
> zu untersuchen, wie dies ja in den rein mathematischen Disziplinen
> geschieht. Dies ist die _progressive Aufgabe_ der Mathematik. Andererseits
> kommt es darauf an, den an Hand der Erfahrung gebildeten Theorien ein
> festeres Gefüge und eine möglichst einfache Grundlage zu geben. Hierzu ist
> es nötig, die Voraussetzungen deutlich herauszuarbeiten, und überall genau
> zu unterscheiden, was Annahme und was logische Folgerung ist. Dadurch
> gewinnt man insbesondere auch Klarheit über alle unbewußt gemachten
> Voraussetzungen, und man erkennt die Tragweite der verschiedenen Annahmen,
> so daß man übersehen kann, was für Modifikationen sich ergeben, falls eine
> oder die andere von diesen Annahmen aufgehoben werden muß. Dies ist die
> _regressieve Aufgabe_ der Mathematik.494949 ‘Mathematics has a two-fold task
> here: On the one hand, it is necessary to develop the systems of relations
> and examine their logical consequences, as happens in purely mathematical
> disciplines. This is the _progressive task_ of mathematics. On the other
> hand, it is important to give the theories formed on the basis of experience
> a firmer structure and a basis that is as simple as possible. For this it is
> necessary to clearly work out the prerequisites and to differentiate exactly
> what is an assumption and what is a logical conclusion. In this way, one
> gains clarity about all unconsciously made assumptions, and one recognizes
> the significance of the various assumptions, so that one can overlook what
> modifications will arise if one or the other of these assumptions has to be
> eliminated. This is the _regressive task_ of mathematics.’ (Hilbert, 1992,
> pp. 17–18).

By axiomatization,505050 Primary sources for Hilbert’s program of
axiomatization include Sauer & Majer (2009, 2024), and Hilbert (1900, 1918).
For secondary literature see especially Corry (2004, 2018), Majer (2001, 2006,
2014) and Majer & Sauer (2014). Unfortunately, the power of this program for
the philosophy of mathematics got somewhat lost because of what is generally
perceived to be the failure of “Hilbert’s program” (i.e. of proving the
consistency of classical mathematics using finitistic methods). Here,
important primary sources are Ewald & Sieg (2013), Hilbert (1926), and Hilbert
& Bernays (1934, 1939). See also Volkert (2015), Hallett & Majer (2004), and
Ewald et al. (2012) for Hilbert’s early formalism, which predated his
“program” by about two decades. The large secondary literature on “Hilbert’s
program” includes Detlefsen (1986), Franks (2009), Sieg (2013), Tapp (2013),
and Zach (2023), and references therein. Proof theory (von Plato, 2018;
Rathjen & Sieg, 2022) is still seen as a positive outcome of this program, but
its interest seems limited to logic. We emphasize the relevance of Hilbert’s
program (without scare quotes!) of axiomatization for _applied mathematics_ ,
which—following Hilbert—we take to include the “quantitative” and “spatial”
sciences altogether.  Hilbert meant the identification of certain sentences
(becoming axioms) that form the foundation of a specific field in the sense
that its theoretical structure (_Fachwerk_) can be (re)constructed from the
axioms via logical principles. The epistemological status of the axioms
differs between fields. For example, Hilbert considered geometry initially a
natural science that emerged from the observation of nature (i.e. experience),
which then turned into a mathematical science through axiomatization (Corry,
2004, p. 90). This does not mean that he treated the axioms of geometry as
“true” (as Euclid had done): Hilbert often stressed the tentative and
malleable nature of axiom systems—just look at the seven editions of
_Grundlagen der Geometrie_!

> Wie man aus dem bisher Gesagten ersieht, wird in den physikalischen Theorien
> die Beseitigung sich einstellender Widersprüehe stets durch veränderte Wahl
> der Aziome erfolgen müssen und die Schwierigkeit besteht darin, die Auswahl
> so zu treffen, daß alle beobachteten physikalischen Gesetze logische Folgen
> der ausgewällten Axiome sind.515151 ‘As can be seen from what has been said
> so far, in physical theories the elimination of contradictions that arise
> will always have to be done by changing the choice of axioms and the
> difficulty lies in making the selection in such a way that all observed
> physical laws are logical consequences of the selected axioms.’
>  (Hilbert, 1918, p. 411)

Or, as historian Majer powerfully summarized Hilbert’s view on the
axiomatization of physics:

> physical theories live, as it were, on the border of inconsistency (Majer,
> 2014, p. 72)

For Hilbert, the axiomatization of physical theories is therefore never a
static process: it moves on as physics itself moves on (Corry, 2004; Majer,
2014). Axiomatization may lead to the exposure of contradictions via a purely
logical analysis, whose removal is then an important step forward.

Axiomatization, then, contributes in two very different ways to the _rigour_
of mathematics:

  1. 1.

via syntactic proofs from the axioms (whose symbols remains uninterpreted);

  2. 2.

via the axiomatization of sufficiently mature informal theories of
mathematics.525252Even in mathematics itself Hilbert grudgingly had to
acknowledge the appearance of contradictions as a historical phenomenon. But
unlike physics, he apparently found contradictions unacceptable in
mathematics, give his obsession of proving the consistency of classical
mathematics. This marks, incidentally, a major difference with Wittgenstein,
whose cheerful acceptance of inconsistent theories, his repeated comments on
the indeterminateness of decimal expansions (e.g. of $\pi$), his relaxed
attitude towards the possibility of rejecting a correct proof, and his
insisting that proofs change the nature of what was proved, sound out of touch
with modern mathematics and hence inappropriate for our program.

Similarly, axiomatization is also the key to the _applicability_ of
mathematics, namely:

  1. 3.

via the axiomatization of sufficiently mature theories of physics, space,
quantity, etc.

In fact, it seems neither possible nor necessary to sharply distinguish
between the second and third activities: for example, are Euclid’s axioms
(more precisely: his so-called postulates and common notions–whatever their
clarity and worth from a modern point of view) attempts to axiomatize earlier
informal geometry, or some physical theory of space? Or, for a more recent
example, did the axiomatization of number theory by Dedekind, Peano, and
others in the late 19th century serve to make earlier informal mathematics
rigorous (at least by the standards of the time) or did it formalize non-
mathematical theories of quantity? Even the axiomatization of set theory in
the early twentieth century tried to bring rigour into both the informal set
theories of Riemann, Dedekind, and Cantor, and the genuine efforts by Frege,
Russell, and others to understand sets as ingredients of the physical universe
or at least the human mind (Ferreirós, 2008); see also §A.3.

The following excellent paragraphs by Wittgenstein,535353For comparisons of
Hilbert and Wittgenstein see Muller (2004), Mühlhölzer (2006, 2008, 2010,
2012) and Friederich (2011, 2014). There is no evidence that Hilbert read the
_Tractatus_. Wittgenstein’s middle and late work came too late for Hilbert,
but it is unlikely that he would have been moved by it. Conversely, even in
the 1930s Wittgenstein’s mathematical education and reflections still relied
on Frege and Russell. Mühlhölzer (2010), p. 10 writes that ‘Auch in BGM III
wird hin und wieder ein gewisser Mangel an relevantem mathematischen Wissen
deutlich, und W.s weitgehende Konzentration auf die Schriften Freges und
Russells verleiht seinen Überlegungen manchmal etwas Antiquirtes.’
Nonetheless, Wittgenstein was well aware of Hilbert and his program
(Mühlhölzer, 2006, 2008, 2010; Floyd, 2023). For example, BGM III, §81
reflects on Hilbert (1922a) (without citation, as always). But Wittgenstein
regarded Hilbert’s aim of proving the consistency of classical mathematics as
‘completely vapid’ (Shanker, 1987, Chapter 6).  originally meant to counter
one of Frege’s objections to the analogy between mathematics and chess, as
well as a similar point raised by Hardy (who as we saw in part endorsed the
said analogy) confirm the artificiality of any sharp distinction between nos.
2 and 3, and hence between pure and applied mathematics:

> Wenn es Menschen auf dem Mars gäbe und sie so Krieg miteinander führten wie
> die Figuren auf dem Schachfeld, dann würde der Generalstab die Regeln des
> Schachspiels zum Prophezeien benutzen. Es wäre dann eine wissenschaftliche
> Frage, ob sich der König bei einer bestimmten Spielkonstellation matt setzen
> läßt, ob er sich in drie Zügen matt setzen läßt und so
> weiter.545454Translation: If there were people on Mars and they waged war
> with each other like the pieces on a chess board, then the General Staff
> would use the rules of chess to prophesy. It would then be a scientific
> question whether the king can be checkmated in a certain game constellation,
> whether he can be checkmated in three moves, and so on. (Wittgenstein,
> 1984b, p. 104)
>
> Wenn die menschliche Kriegsführung dem Schachspiel ähnlicher wäre als sie
> tatsächlich ist so könnte man versuchen eine Schlacht auf dem Schachbrett
> darzustellen und mathematische Probleme die die Möglichkeiten der Schlacht
> betreffen auf dem Schachbrett zu lösen. Freilich nur mathematische Probleme,
> denn Experimente über den Vorgang der Schlacht könnte man mit den
> Schachfiguren nicht vornehmen da sie sich anders verhalten als die Menschen.
> Wenn also das Problem gelöst würde etwa von einer bestimmten Position
> ausgehend den Anderen in $N$ Zügen matt zu setzen, so wäre das die Lösung
> eines mathematischen Problems des Krieges.555555 See
> https://wab.uib.no/transform/wab.php?modus=opsjoner.). Quoted by Max (2020),
> p. 198. Translation: If human warfare were more similar to chess than it
> actually is, one could try to represent a battle on the chessboard and solve
> mathematical problems relating to the possibilities of the battle on the
> chessboard. Of course, only mathematical problems, because you couldn’t
> carry out experiments on the battle process with the chess pieces because
> they behave differently from people. So if the problem were solved, for
> example starting from a certain position, to checkmate the other person in
> $N$ moves, then that would be the solution to a mathematical problem of war.
> (Wittgenstein, MS 108: 162 f.)

_Thus the key difference is between numbers 1 on the one hand and 2–3 on the
other_ : the first is formal and focuses on proofs, whereas numbers 2 and 3
both take us outside (formal) mathematics.565656 We follow Tait (1986) in
seeing models in set theory (or even topos theory) as internal to mathematics,
and hence the distinction between syntax (as e.g. in formulating mathematical
sentences in first-order logic) and their interpretation in set theory is
irrelevant for our theme. See also Baldwin (2018) and Button & Walsh (2018).
Combining (allegedly) “pure” and “applied” mathematics was natural for
Hilbert, who emphasized the _unity_ of mathematics and the mathematical
sciences throughout his career:575757He did so against many others, including
Frege, who claimed well into the 19th century that e.g. arithmetic and
geometry had a fundamentally different epistemological status.

> Die genannten Probleme sind nur Proben von Problemen; sie genügen jedoch, um
> uns vor Augen zu führen, wie reich, wie mannigfach und wie ausgedehnt die
> mathematische Wissenschaft schon heute ist und es drängt sich uns die Frage
> auf, ob der Mathematik einst bevorsteht, was anderen Wissenschaften längst
> widerfahren ist, nämlich daß sie in einzelne Teilwissenschaften zerfällt,
> deren Vertreter kaum noch einander verstehen und deren Zusammenhang daher
> immer loser wird. Ich glaube und wünsche dies nicht; die mathematische
> Wissenschaft ist meiner Ansicht nach ein unteilbares Ganze, ein Organismus,
> dessen Lebensfähigkeit durch den Zusammenhang seiner Teile bedingt wird.
> Denn bei aller Verschiedenheit des mathematischen Wissenstoffes im
> Einzelnen, gewahren wir doch sehr deutlich die Gleichheit der logischen
> Hülfsmittel, die Verwandtschaft der Ideenbildungen in der ganzen Mathematik
> und die zahlreichen Analogieen in ihren verschiedenen Wissensgebieten. Auch
> bemerken wir: je weiter eine mathematische Theorie ausgebildet wird, desto
> harmonischer und einheitlicher gestaltet sich ihr Aufbau und ungeahnte
> Beziehungen zwischen bisher getrennten Wissenszweigen, werden entdeckt. So
> kommt es, daß mit der Ausdehnung der Mathematik ihr einheitlicher Charakter
> nicht verloren geht, sondern desto deutlicher offenbar wird.585858‘The
> problems mentioned are merely samples of problems, yet they will suffice to
> show how rich, how manifold and how extensive the mathematical science of
> today is, and the question is urged upon us whether mathematics is doomed to
> the fate of those other sciences that have split up into separate branches,
> whose representatives scarcely understand one another and whose connection
> becomes ever more loose. I do not believe this nor wish it. Mathematical
> science is in my opinion an indivisible whole, an organism whose vitality is
> conditioned upon the connection of its parts. For with all the variety of
> mathematical knowledge, we are still clearly conscious of the similarity of
> the logical devices, the relationship of the ideas in mathematics as a whole
> and the numerous analogies in its different departments. We also notice
> that, the farther a mathematical theory is developed, the more harmoniously
> and uniformly does its construction proceed, and unsuspected relations are
> disclosed between hitherto separate branches of the science. So it happens
> that, with the extension of mathematics, its organic character is not lost
> but only manifests itself the more clearly.’  (Hilbert, 1900, pp. 296–297)
>
> So ordnen sich die geometrische Tatsachen zu einer Geometrie, die
> arithmetischen Tatsachen zu einer Zahlentheorie, die statischen,
> mechanischen, elektrodynamischen Tatsachen zu einer Theorie der Statik,
> Mechanik, Elektrodynamik oder die Tatsachen aus der Physik der Gase zu einer
> Gastheorie. Ebenso ist es mit den Wissensgebieten der Thermodynamik, der
> geometrischen Optik, der elementaren Strahlungstheorie, der Wärmeleitung
> oder auch mit der Wahrscheinlichkeitsrechnung und der Mengenlehre. Ja es
> gilt von speziellen rein mathematischen Wissensgebieten wie Fläichentheorie,
> Galoisscher Gleichungstheorie, Theorie der Primzahlen nicht weniger als für
> manche der Mathematik fern liegende Wissensgebiete wie gewisse Abschnitte
> der Psychophysik oder die Theorie des Geldes.595959‘Thus the geometric facts
> are arranged into some geometry, the arithmetic facts into some number
> theory, the static, mechanical, electrodynamic facts into some theory of
> statics, mechanics, electrodynamics or the facts from the physics of gases
> into some gas theory. And similarly with the disciplines of thermodynamics,
> geometric optics, elementary radiation theory, heat conduction, or even
> probability calculation and set theory. Yes, it applies as much to special
> purely mathematical areas of knowledge such as the theory of surfaces,
> Galois’ theory of equations, and the theory of prime numbers, as to some
> areas of knowledge that are remote from mathematics, such as certain
> sections of psychophysics or the theory of money.’
>  (Hilbert, 1918, pp. 405–406.

Hilbert (implicitly) plays two different language games with _the same_
axioms. The cleanest one is the first in the list 1, 2, 3 above: this is his
famous emphasis on the meaninglessness of mathematical symbols _in so far as
proofs and other formal aspects of axiom systems are concerned_ (such as
consistency and completeness), cf. §A.2. The analogy between mathematics and
chess is particularly clear in this language game, given the formal, deductive
notion of proof shared by Frege, Russell, and Hilbert, in which (unlike in
Euclid—let alone 17th and 18th century mathematics) not only the axioms but
also the rules of deduction are formalized and clearly stated. This is, of
course, a pristine example of rule following. Thus formalizing proofs is one
of the language games played in mathematics—in which one ignores whatever
meaning the symbols may have.

##  5 Applied mathematics

A very different language game underlies nos. 2–3 in §4. This one is sometimes
played in mathematics itself (no. 2), but it is especially important in
applied mathematics and mathematical physics (no. 3), or, in other words, in
understanding the relationship between axiom systems and the physical world.
Conceptually, this game is much more complicated than the previous one
(technically, it is the other way round!). Indeed, it is not at all clear what
it means to extract some mathematical structure from natural phenomena through
axiomatization. According to (late) Wittgenstein,606060See also Steiner
(2009). Although Wittgenstein said this about mathematics in general, we only
use this idea for applied mathematics and mathematical physics, where we
include elementary arithmetic and geometry in the former.

> Die Rechtfertigung des Satzes $25\times 25=625$ ist natürlich daß, wer so
> und so abgerichtet wurde, unter normalen Umständen bei der Multiplikation
> $25\times 25=625$ erhält. Der arithmetische Satz abes sagt nicht _dies_ aus.
> Er ist so zu sagen ein zur Regel verhärteter Erfahrungssatz. Er bestimmt,
> daß der Regel nur dann gefolgt wurde, wenn dies das Resultat des
> Mulitplizierens ist. Er ist also der Kontrolle durch die Erfahrung entzogen,
> dient aber nur dazu, die Erfahrung zu beurteilen.616161 ‘The justification
> of the proposition $25\times 25=625$ is, naturally, that if anyone has been
> trained in such-and-such a way, then under normal circumstances he gets 625
> as the result of multiplying 25 by 25. But the arithmetical proposition does
> not assert that. It is so to speak an empirical proposition hardened into a
> rule. It stipulates that the rule has been followed only when that is the
> result of the multiplication. It is thus withdrawn from being checked by
> experience, but now serves as a paradigm for judging experience.’  (BGM,
> §VI.23c, p. 325)
>
> Wie ist es mit dem Satz “die Winkelsumme im Dreieck ist $180^{\circ}$”? (…)
> ich werde wenn sie sich bei einer Messung nicht als $180^{\circ}$ erweist
> einen Messungsfehler annehmen. Der Satz ist also ein Postulat über die Art
> und Weise der Beschreibung.626262‘What about the sentence “The sum of the
> angles in the triangle is $180^{\circ}$’? If it does not turn out to be
> $180^{\circ}$’ in a measurement, I will assume a measurement error. The
> sentence is therefore a postulate about the manner of description.’
> Typescripts Ts- (or manuscripts Ms-) can be found online via
> http://www.wittgensteinsource.org.  (Ts-212,XV-114-5[1])
>
> Mathematical truth isn’t established by their all agreeing that it’s true—as
> if they were witnesses to it. _Because_ they all agree in what they do, we
> lay it down as a rule, and put it in the archives. (LFM, Lecture IX, p. 107)

Thus the claim that _if we give 25 apples to 25 people, then in total they
hold 625 apples,_ just means that the operation we carried out was really
multiplication and that we didn’t make a mistake. This is very different from
the platonic view that $25\times 25=625$ is a description of some external
reality about ethereal numbers. Similarly, the theorem from Euclidean geometry
stating that the sum $\alpha+\beta+\gamma$ of three angles in a triangle is
$180^{\circ}$ means the following for an actually drawn triangle: if the sum
of its angles equals $180^{\circ}$ (presumably within some error bound
determined by the accuracy of the drawing and our ability to measure angles),
then Euclidean geometry and in particular its concept of an angle has been
used correctly to represent the situation. Thus Gauss’s measurement of the sum
of the angles in the triangle with vertices Brocken, Inselberg, and Hohenhagen
(near Göttingen) between 1821–1825, resulting in $180^{\circ}$ within his
measurement accuracy (Bühler, 1981) showed that Euclidean geometry could be
used to represent the local geography as appropriate.

This is in the spirit of §§130–131 of the _Philosophical Investigations_
quoted earlier, with the difference that those paragraphs described the role
of language games as objects of comparison for some natural language, whereas
here we are talking about (say) the applied side of Euclidean geometry as a
mathematical language game being applied to the physical world as a
yardstick.636363 Baker & Hacker (1989a), §VII.4) write that mathematical
theorems are ‘ _norms of representation_ ’, in the sense that theorems are not
(primarily) _descriptive_ , as in a platonic view; instead they are _normative
for possible descriptions_. Similarly, Mühlhölzer (2012), p. 109, says that
mathematical propositions are _preparatory_ for descriptions of empirical
states of affair, instead of being _about_ such states.

However, Hilbert talked about axioms, whereas Wittgenstein talks about
theorems (or ‘propositions’). We follow Friederich (2011, 2014) in
transferring Wittgenstein’s interpretation of _theorems_ to _axioms_ , and
hence in using axioms as “irreducible” yardsticks. After all, axioms are
special instances of theorems, and conversely, they imply the theorems. It
should therefore be enough to ‘harden’ certain _key_ empirical regularities
into the _axioms_ of a physical theory.646464In classical physics one may
think of (partial) differential equations like those of Newton, Maxwell, or
Einstein. As noted by Friederich, this resonates particularly well with
Hilbert’s implicit definitions (see §4).

Even so, neither Hilbert nor Wittgenstein explained how empirical phenomena
acquire a mathematical structure, or what this structure (once in place)
refers to.656565Hilbert never got beyond the vague notion of “pre-established
harmony”, a philosophical concept originally going back to Leibniz. See e.g.
Pyenson, (1982), Kragh (2015), and Corry (2004), pp. 393–394, who even claims
that it was ‘one of the most basic concepts that underlay the whole scientific
enterprise in Göttingen’, adding that ‘Hilbert, like all his colleagues in
Göttingen, was never really able to explain, in coherent philosophical terms,
its meaning and the possible basis of its putative pervasiveness, except by
alluding to “a miracle”.’  This requires more analysis. As the most
appropriate corresponding approach to the philosophy of science we suggest
_constructive empiricism_ (van Fraassen, 1980, 1992, 2008; Morton & Mohler,
2021). In particular:

> The two poles of scientific understanding, for the empiricist, are the
> observable phenomena on the one hand and the theoretical models on the
> other. The former are the target of scientific representation and the latter
> its vehicle. But those theoretical models are abstract structures, even in
> the case of the practical sciences such as materials science, geology, and
> biology—let alone in the advanced forms of physics. All abstract structures
> are mathematical structures, in the contemporary sense of “mathematical”,
> which is not restricted to the traditional number-oriented forms. And
> mathematical structures, as Weyl so emphatically pointed out, are not
> distinguished beyond isomorphism—to know the structure of a mathematical
> object is to know all there is to know. (…) Essential to an empiricist
> structuralism is the following core construal of the slogan that all we know
> is structure:
>
>   1. 1.
>
> Science represents the empirical phenomena as embeddable in certain
> _abstract structures_ (theoretical models).
>
>   2. 2.
>
> Those abstract structures are describable only up to structural isomorphism.
>
>

>
> (…) How can we answer the question of how a theory or model relates to the
> phenomena by pointing to a relation between theoretical and data models,
> both of them abstract entities? The answer has to be that the data model
> represent the phenomena; but why does that not just push the problem
> [namely: _what is the relation between the theoretical model and the
> phenomena it models_] one step back? The short answer is this: construction
> of a data model is precisely the selective relevant depiction of the
> phenomena _by the user of the theory_ required for the possibility of
> representation of the phenomenon. (van Fraassen, 2008, pp. 238, 253)

Thus in this view the link between mathematical formalism and “reality”
consists of three steps:

  1. 1.

A mathematical representation of natural phenomena by some (“surface”) _data
model_ ;666666Think of a numerical table in which the position of a certain
planet in the sky is recorded on a daily basis. One may equally well think of
the result of some calculation of $25\times 25$ or a measurement of
$\alpha+\beta+\gamma$ as data models.

  2. 2.

A mathematical _theory_ of this data model, consisting of some _abstract
structure_.676767The mathematical theory in which the data model of the
previous footnote is embedded could be Kepler’s laws, or Newton’s. As van
Fraassen states, data models do not stand on their own but are typically
informed by such theories.

  3. 3.

A “user” (which may be an entire team of scientists!) acting like a “middle
man” between the mathematical theory and the (_a priori_ non-mathematical)
natural phenomena.

The realization that some mathematical theory is related to the phenomena it
tries to describe via an intermediate data model constructed by some user
obviates the need for chimerical philosophical constructions like platonism or
some other introduction of universals. Like Newton’s absolute space and time,
some platonic realm housing numbers and perfect circles may be postulated, but
it is unnecessary and leads to insurmountable ontological and epistemological
problems (cf. §1).

In sum: Wittgenstein asks about the outcome of the multiplication $25\times
25$ or the addition $\alpha+\beta+\gamma$ _according to the standard rules of
arithmetic or Euclidean geometry, respectively_. Van Fraassen (1992), pp. 3–4,
asks _what the world would be like according to some (mathematical) theory_.
If it meets the “empirical regularities” the theory is accepted—which word is
literally used by van Fraassen, as opposed to “believed”, which a realist
would use—as a valid representation of the phenomenon in question (i.e., the
performance of someone doing the multiplication or the outcome of appropriate
measurements on a particular landscape). If it doesn’t, it is
rejected.686868Something similar happens in the brain according to what is
called _predictive processing theory_ (also called _predictive coding_). This
theory replaces traditional views (according to which the brain just _records_
sense data) by the idea that all the time the brain actively produces
_predictions_ about the world, which are compared with incoming sense data,
and then updated if necessary. What one consciously experiences then results
from a combination of both the “outgoing” predictions and the “incoming” sense
data. See e.g. Hohwy (2013) and Millidge, Seth, & Buckley (2022).

##  6 Truth

What does the “formalist” language game of §3 imply for the concept of _truth_
in mathematics, and how would that resonate with the “applied” language game
of §5? This is a difficult question, since although we acknowledge both the
existence of an external world and the fact that mathematics has endless
representational capacities with regard to it, we have just seen that such
representations are much more subtle than naive realism would suggest. As
noted before, Wittgenstein clearly realized the tension between the abstract
nature of mathematics (and many other language games) and the worldly nature
of the world (or of some natural language) described by it (Kuusela, 2019a).
At the same time, like chess, mathematics has historically always been based
on certain _man-made_ rules, for which there are even many different
possibilities; and this is the case whether or not these rules are inspired by
empirical phenomena. So where could there be any room for truth in
mathematics, if not in the real world (which is _a priori_ unmathematical, let
alone platonic) or in man?

Her is our view. Much as it is meaningless to say that a position in chess is
“true”, it makes no sense to say that mathematical theorems are true either,
since there is no objective state of affairs they could describe
correctly.696969Tarski-style truth in the sense of model theory is irrelevant
here; it lies within mathematics. See footnote 56. Truth in chess can only
reside in the claim that some position is _legal_ , in that it arose from a
game played according to the rules. Similarly, truth in mathematics can only
reside in the claim “ $T\vdash\varphi$ ” that some sentence $\varphi$ is a
theorem within the theory $T$ in which $\varphi$ is expressed as a well-formed
formula.707070This might be varied by defining $T\vdash\varphi$ to be true if
a proof of $\varphi$ _is known_ , as in intuitionistic mathematics. And this
is the case (by definition) iff there exists a proof of $\varphi$ according to
the rules of $T$ (which is supposed to include rules of inference).

Thus the only thing we can, in our opinion, say about mathematical truth is
this:717171 This is idea is part of what is called _deductivism_ in the
philosophy of mathematics (Paseau & Pregel, 2023). In general philosophy
(Künne, 2003; David, 2022), it is an example of a _disquotational definition
of truth_.

_Mathematical truth resides not _in theorems_ but in claims _that some
sentence is a theorem_. _

This makes a proof of $\varphi$ in $T$ the truth-maker of the truth-bearer
$T\vdash\varphi$. This idea should be distinguished from what e.g. Dieudonné
(1971), Tait (1986), Weir (2010), and others propose, namely that a sentence
$\varphi$ _itself_ (as opposed to $T\vdash\varphi$) is true iff $\varphi$ has
a proof. Against this:

  * •

Unless one believes (with e.g. Gödel) that there is a single “true”
foundational system for all of mathematics (such as ZF set theory with
additional axioms) such proposals endorse a _coherence theory of truth_
(Young, 2018), in which each such system would come with its own set of
truths. We leave this to politicians. On our proposal, although people may
differ about the worths of different foundational systems, given an
unambiguous concept of proof they cannot reasonably differ about the theorems
in each of these systems.

  * •

One encounters difficulties with (ironically) Gödel’s first incompleteness
theorem,727272This theorem is also used _against_ deductivism, cf. Paseau &
Pregel (2023), §9, the idea being that the Gödel sentence in some formal
system is “seen” to be true although it cannot be proved in the given system.
But it can duly be proved in the standard interpretation of this system in the
natural numbers Franzén (2005).  according to which (under the usual
assumptions) there are sentences $\varphi$ such that neither $\varphi$ nor
$\neg\varphi$ is provable. Yet at least in classical logic
$\varphi\vee\neg\varphi$ is provable for every formula $\varphi$. If this is
taken to mean that $\varphi\vee\neg\varphi$ is true, then this can be the case
(namely for undecidable $\varphi$) without either $\varphi$ or $\neg\varphi$
being true. But there is no problem if $\vdash(\varphi\vee\neg\varphi)$ is
true without either $\vdash\varphi$ or $\vdash\neg\varphi$ being true, since
$\vdash(\varphi\vee\neg\varphi)$ is quite different from
$(\vdash\varphi)\vee\vdash(\neg\varphi)$.

Nonetheless, one needs to get used to the idea that say Kant’s famous example
$7+5=12$ is not true (but neither is it false; it is just not the kind of
mathematical statement that has a truth value), whereas
$\mathrm{PA}\vdash(7+5=12)$ _is_ true, i.e., the claim that $7+5=12$ is a
theorem of Peano arithmetic.

Adding to the arguments above, which were related to the “formal” language
game, let us therefore try to explain why $7+5=12$ has no truth value from the
point of view of our “applied” language game either, cf. §5. Truth should come
from the alleged fact that $7+5=12$, but what could that mean? We reject the
existence of platonic numbers 7 and 5 that can be platonically added to yield
a similarly ethereal number 12. But even if this were to make any sense, proof
would be the only access to the alleged truth of $7+5=12$; yet proof by
construction establishes $\mathrm{PA}\vdash(7+5=12)$ rather than $7+5=12$
itself.737373This is the “Truth/Proof problem”, which forms another challenge
to platonism (Tait, 1986).

Let us try again: don’t seven apples add up with five apples to yield twelve
apples? They do; take this as our data model. But this by itself expresses
neither $7+5=12$ nor $\mathrm{PA}\vdash(7+5=12)$: it says that seven _apples_
add up with five _apples_ to yield twelve _apples_. On the constructive
empiricist account above, the user of the mathematical theory PA compares the
latter theorem with the stated data model for counting and adding apples, and
finds a match. This match is between a data model and a formal theory; again,
there is no need for abstract/platonic numbers.

As far as we can judge, this analysis seems reinforced by what Wittgenstein
taught in 1939:

> one asks such a thing as what mathematics is about—and someone replies that
> it is about numbers. Then someone comes along and says that it is not about
> numbers but about numerals; for numbers seem very mysterious things. And
> then it seems the mathematical propositions are about scratches on the black
> board. That must seem ridiculous even to those who hold it, bu they hold it
> because there seems to be no way out—I am trying to show in a very general
> way how the misunderstanding of supposing a mathematical proposition to be
> like an experiential proposition leads to the misunderstanding of supposing
> that a mathematical proposition is about scratches on the blackboard.
>
> Take “$20+15=35$”. We say this is about numbers. Now is it about the
> symbols, the scratches? That is absurd. It couldn’t be called a statement or
> proposition about them; if we have to say that it is a so-and-so about them,
> we could say that it is a rule or convention about them.—One might say,
> “Could it not be a statement about how people use symbols?” I should reply
> that that is not in fact how it is used—any more than as a declaration of
> love.
>
> One might say that it is a statement about numbers. Is it wrong to say that?
> Not at all; that is what we call a statement about numbers. But this gives
> the impression that it’s not about some coarse thing like scratches, but
> about something very thin and gaseous.—Well, what is a number, then? I can
> show you what a numeral is. But when I say it is a statement about numbers
> it seems as though we were introducing some new entity somewhere.
>  (LFM, Lecture XII, p. 112)

The unity of mathematics emphasized by Hilbert provides an additional argument
for the lack of truth of $7+5=12$. If it were true, then, being the content of
a theorem, every theorem in mathematics should be true. This leads to a
problem discussed earlier: since different foundational systems may yield
contradictory results, just one of these systems could be “true”. The history
of mathematics suggests this is dubious. Even if only one of them ultimately
comes out be correct (e.g. since the others unexpectedly are inconsistent),
putting esoteric result in ZFC set theory about crazy cardinals on a par with
$7+5=12$ as both being “true” sounds equally wrong. The only way to get around
these problems is to—indeed—treat all theorems from all foundational systems
on a par; but instead of declaring them all true, the ensuing notion of truth
is expressed much better by saying that the claim $T\vdash\varphi$ that
$\varphi$ can be deduced from $T$ is true, rather than $\varphi$ itself.

##  7 Certainty

The notion of _necessity_ (or Wittgenstein’s “logical must”) adds nothing to
our deductivist concept of mathematical truth. The _certainty_ of mathematical
truth, on the other hand, is far from trivial.

For both Hilbert and Wittgenstein the certainty of mathematics originates in
proofs, but proof is not enough: both add the requirement that these proofs be
_surveyable_ (Shanker, 1987, Mühlhölzer, 2006; Floyd, 2023). But they mean
very different things by this. Starting with Hilbert:

> Wie wir sahen, hat sich das abstrakte Operieren mit allgemeinen
> Begriffsumfängen und Inhalten als unzulänglich und unsicher herausgestellt.
> Als Vorbedingung für die Anwendung logischer Schlüsse und die Betätigung
> logischer Operationen muß vielmehr schon etwas in der Vorstellung gegeben
> sein: gewisse außerlogische diskrete Objekte, die anschaulich als
> unmittelbares Erlebnis vor allem Denken da sind. Soll das logische Schließen
> sicher sein, so müssen sich diese Objekte vollkommen in allen Teilen
> überblieken lassen und ihre Aufweisung, ihre Unterscheidung, ihr
> Aufeinanderfolgen ist mit den Objekten zugleich unmittelbar anschaulich für
> uns da als etwas, das sich nicht noch auf etwas anderes reduzieren läßt. (…)
> Hierin liegt die feste philosophische Einstellung, die ich zur Begründung
> der reinen Mathematik — wie überhaupt zu allem wissenschaftlichen Denken,
> Verstehen und Mitteilen — für erforderlich halte: _am Anfang — so heißt es
> hier — ist das Zeichen_.747474 ‘As we have seen, abstract operation with
> general concept-scopes and contents has proved to be inadequate and
> uncertain. Instead, as a precondition for the application of logical
> inferences and for the activation of logical operations, something must
> already be given in representation: certain extra-logical discrete objects,
> which exist intuitively as immediate experience before all thought. If
> logical inference is to be certain, then these objects must be capable of
> being completely surveyed in all their parts, and their presentation, their
> diff erence, their succession (like the objects themselves) must exist for
> us immediately, intuitively, as something that cannot be reduced to
> something else. The solid philosophical attitude that I think is required
> for the grounding of pure mathematics — as well as for all scientific
> thought, understanding, and communicatio — is this: _In the beginning was
> the sign_.’ (Hilbert, 1922b, pp. 162–163).
>
> Vielmehr ist als Vorbedingung für die Anwendung logischer Schlüsse und für
> die Betätigung logischer Operationen schon etwas in der Vorstellung gegeben:
> gewisse, außer-logische Konkrete Objekte, die anschaulich als unmittelbares
> Erlebnis vor allem Denken da sind. Soll das logische Schließen sicher sein,
> so müssen sich diese Objekte vollkommen in allen Teilen überblicken lassen
> und ihre Aufweisung, ihre Unterscheidung, ihr Aufeinanderfolgen oder
> Nebeneinandergereihtsein ist mit den Objekten zugleich unmittelbar
> anschaulich gegeben als etwas, das sich noch auf etwas anderes reduzieren
> läßoder einer Reduktion bedarf.Dies ist die philosophische Grundeinstellung,
> die ich für die Mathematik wie überhaupt zu allem wisseschaftlichen Denken,
> Verstehen und Mitteilen für erforderlich halte. Und insbesondere in der
> Mathematik sind Gegenstand unserer Einstellung zufolge unmittelbar deutlich
> und wiedererkennbar ist.757575 ‘As a further precondition for using logical
> deduction and carrying out logical operations, something must be given in
> conception, viz., certain extra-logical concrete objects which are intuited
> as directly experienced prior to all thinking. For logical deduction to be
> certain, we must be able to see every aspect of these objects, and their
> properties, differences, sequences, and contiguities must be given, together
> with the objects themselves, as something which cannot be reduced to
> something else and which requires no reduction. This is the basic philosophy
> which I find necessary, not just for mathematics, but for all scientific
> thinking, understanding, and communicating. The subject matter of
> mathematics is, in accordance with this theory, the concrete symbols
> themselves whose structure is immediately clear and recognizable.’ (Hilbert,
> 1926, p. 171)

Thus Hilbert’s understanding of the certainty of mathematics relies on the
certainty of logical inference, which in turn relies on the use of very simple
signs. This was also the basis of his eventual “finitism”, which called for a
final mark of certainty in the form of a finitist proof of the consistency of
classical mathematics (which goal was never achieved or even well
defined).767676See the references in footnote 50 on Hilbert’s program.

For the present discussion it is a moot point if Wittgenstein was a finitist,
too (cf. Marion, 1998). What matters is that his notion of ‘surveyability’ of
a proof was almost the opposite of Hilbert’s. He explains this notion in
§III.1 of his _Remarks on the Foundations of Mathematics_ :

> ‘Ein Mathematischer Beweis muß übersichtlich sein.’ “Beweis” nennen wir nur
> eine Struktur, deren Reproduktion eine leicht lösbare Aufgabe ist. Es muß
> sich mit Sicherheit entscheiden lassen, ob wir hier wirklich zweimal den
> gleichen Beweis vor uns haben, oder nicht. Der Beweis muß ein Bild sein,
> welches sich mit Sicherheit genau reproduzieren läßt. Oder auch: was dem
> Beweise wesentlich ist muß sich mit Sicherheit genau reproduzieren lassen.
> Er kann z.B. in zwei verschiedenen Handschriften oder Farben
> niedergeschrieben sein. Zur Reproduktion eines Beweises soll nichts gehören,
> was von der Art einer genauen Reproduktion eines Farbtones oder einer
> Handschrift ist.
>
> Es muß leicht sein, _genau_ diesen Beweis wieder anfzuschreiben. 777777 ‘A
> mathematical proof must be perspicuous.’ Only a structure whose reproduction
> is an easy task is called a “proof”. It must be possible to decide with
> certainty whether we really have the same proof twice over, or not. The
> proof must be a configuration whose exact reproduction can be certain. Or
> again: we must be sure we can exactly reproduce what is essential to the
> proof. It may for example be written down in two different handwritings or
> colours. What goes to make the reproduction of a proof is not anything like
> an exact reproduction of a shade of colour or a hand-writing. It must be
> easy to write down exactly this proof again.’
>  (Wittgenstein, 1984d, p. 143)

These conditions are studied in detail by Mühlhölzer (2006), who summarizes
them as follows:

>   1. S1
>
> The surveyability of a proof consists in its possibility of reproduction.
>
>   2. S2
>
> This reproduction must be an easy task.
>
>   3. S3
>
> We must be able to decide with certainty whether the reproduction produces
> the same proof.
>
>   4. S4
>
> The reproduction of a proof is of the sort of a reproduction of a picture.
> (p.59)
>
>

The feature that for Hilbert ensured surveyability of a proof, namely the
consistent use of elementary signs or strokes, was precisely the reason why
Wittgenstein considered such proofs _non_ -surveyable and hence not even
proofs. And likewise for the infamous proof of $1+1=2$ given by Russel and
Whitehead (1910), *54.43, which including all preparation takes hundreds of
pages.787878The blog by Dominus (2006) gives a very nice discussion of this
proof.

This theme influenced and subsequently became strongly influenced by Turing’s
concept of computability (Floyd, 2023). Four decades onwards, computer-
assisted proofs, like the famous one of the four-colour theorem (Appel &
Haken, 1977; Robertson _et al._ , 1997; Haken, 2006; Gonthier, 2008) also
changed the debate (Tymoczko, 1979; Shanker, 1987, pp. 143–160). Indeed, such
proofs can hardly be called surveyable in either Hilbert’s or Wittgenstein’s
sense; and this is also the case for proofs entirely done by hand but
involving large teams of mathematicians publishing their work in dozens of
papers whose length adds up to thousands of pages, like the classification of
finite simple groups,797979Wikipedia gives an excellent summary of this
classification. The special issue on formal proof of the _Notices of the AMS_
, December 2008, available at
https://www.ams.org/notices/200811/200811FullIssue.pdf, covers both computer-
verified and computer-assisted proofs. See also
https://www.cs.ru.nl/~freek/100/index.html. or the stability of space-time in
general relativity.808080See Dafermos _et al._ (2019), and Dafermos _et al._
(2021), and references therein.

Without mentioning Hilbert, Wittgenstein, or the concept of surveyability,
Avigad (2021) nonetheless describes the essence of their opposition. An
informal proof is more likely to be understandable (and would arguably be
surveyable in the sense of Wittgenstein), but alas, it is not rigorous. A
formal Hilbert-style proof, on the other hand, will hardly be understandable
if only because of its length, which also enormously increases the probability
of error. Thus the universal habit among “working mathematicians” of not
writing out proofs according to the rules of logic is essential to their
readability (and often even reliability), but this habit obviously sacrifices
rigour. Conversely, formal rigour sacrifices readability and in the worst case
introduces the errors Hilbert so desperately hoped to avoid. See also Thurston
(1994), Weir (2014), Hamami (2018), Ording (2019), Burgess & De Toffoli
(2022), Stillwell (2022), and Hamami & Morris (2023) for perspectives on the
wide variety of styles of proof in mathematics. Even within the formal
setting, Dutilh Novaes (2011) identifies eight different ways in which
formality and rules can be interpreted.

How do we navigate between Scylla and Charybdis? Returning to Wittgenstein,
the key property of a proof that is relevant for the certainty of mathematics
is, quite simply, the following:

> “Der Beweis muß übersehbar sein” — heißt das nicht: daß es ein Beweis ist,
> muß zu sehen sein.818181 “‘The proof must be surveyable” — doesn’t this
> mean: it should be visible that a proof is a proof.’ (Wittgenstein, MS 122:
> p. 105; Mühlhölzer, 2010, p. 574)

On a relaxed interpretation of ‘sehen’ (seeing), Wittgenstein and Hilbert may
both get their way if computer-verified proofs (which they did not live to
see) are invoked.828282 To avoid confusion, we note that computer-_verified_
proofs are produced by so-called proof _assistants_ , which have nothing to do
with computer-_assisted_ proofs of the kind mentioned above. See e.g. Geuvers
(2009). These greatly enhance the certainty of mathematics—perhaps without
driving it up to full certainty. The underlying proof assistants rely on a so-
called logical kernel, which ultimately has to be trusted (verifying it would
lead to infinite regress).838383 See
https://en.wikipedia.org/wiki/Proof_assistant. Their list of proof assistants
also tells us if the kernel is ‘small’: the smaller, the better. The authors
are grateful to Freek Wiedijk for information about this topic. If it contains
bugs, this “probably” would have been noted in verifying the dozens of
theorems whose proofs have now been checked.

Of course, this argument is as weak as the analogous argument for the
consistency of set theory (namely that _so far_ it has not produced a
contradiction). Bugs in the compiler for the programming language in which the
proof assistant is written are also possible, but the likelihood of such
errors can be reduced by using a number of different compilers, as has indeed
been done to good effect.

In sum: short of absolute guarantees, the jury is still out on the certainty
of mathematics. But if it applies, it must come from proof, as both Hilbert
and Wittgenstein maintained. Interestingly, the historical circle closes at
this point, since proof assistants (obviously) rely on a complete
formalization of mathematics as envisaged by Hilbert (Nederpelt & Geuvers,
2014).

All of this confirms the language-game view of mathematics; like a game it is
based on rules, but one has to take the _actual practice of rule following_
into account to get a complete picture (whereas in a game like chess,
deviations from the rules would end the game at once):

> Warum nenne ich die Regeln des Kochens nicht willkürlich; und warum bin ich
> versucht, die Regeln der Grammatik willkürlich zu nennen? Weil ‘Kochen’
> durch seinen Zweck definiert ist, dagegen ‘Sprechen’ nicht. Darum ist der
> Gebrauch der Sprache in einem gewissen Sinne autonom, in dem das Kochen und
> Waschen es nicht ist. Wer sich beim Kochen nach andern als den richtigen
> Regeln richtet, kocht schlecht; aber wer sich nach andern Regeln als denen
> des Schach richtet, spielt _ein anderes Spiel_ ; und wer sich nach andern
> grammatischen Regeln richtet, als den unsern, spricht darum nichts Falsches,
> sondern _von etwas Anderm_.848484 ‘Why don’t I call the rules of cooking
> arbitrary; and why am I tempted to call the rules of grammar arbitrary?
> Because ‘cooking’ is defined by its purpose, whereas ‘speaking’ is not. That
> is why the use of language is autonomous in a sense in which cooking and
> washing are not. Anyone who follows rules other than the correct ones when
> cooking cooks poorly; but anyone who follows rules other than those of chess
> is playing a different game; and anyone who follows grammatical rules other
> than ours is not speaking falsely, but rather about something different.’
> (Wittgenstein, Ts-228,108[3]et109[1])

##  Appendix A Formalism, axioms, definitions, and proofs

###  A.1 Formalism

Though often associated with Hilbert,858585The classical exposition of
formalism by von Neumann (1931) is entirely devoted to Hilbert’s program.
there isn’t a canonical notion of “formalism” in the philosophy of
mathematics. Here is the rather sterile version defined e.g. by Linnebø
(2017):

> _Formalism_ is the view that mathematics has no need for semantic notions,
> or at least none that cannot be reduced to syntactic ones. (Linnebø, 2017,
> p. 39)

This is the version attacked by Frege (cf. the extracts from his
correspondence with Hilbert above). But it is a straw man that certainly
shouldn’t be associated with Hilbert, for whom it is only _in the context of
proofs and in the analysis of axiom systems regarding consistency etc._ (and
hence, in metamathematics) that mathematics is seen as a deductive enterprise
in which symbols have no meaning (outside the rules they are subject to). A
much broader view of formalism is promoted by Detlefsen (2005), who
characterizes it by five key components (some of historical interest
only):868686We put Detlefsen’s fourth criterion (which he traces this back to
Berkeley) first, since it is the key point.

  1. 1.

Formalism rejected a representational role for mathematical language, making
mathematical reasoning independent of the actual content of the words and
symbols in this language:

> The pivotal commitment of formalism (…) is a view concerning the nature of
> language—namely, that it can serve as a guide to thought even when it does
> not function descriptively. (Detlefsen, 2005, p. 251)

  2. 2.

Formalism rejected the traditional (Aristotelian) _division_ of the
mathematical sciences into arithmetic (as the science of multitude), geometry
(as the science of magnitude), and the mixed sciences (notably optics,
harmonics, mechanics, and astronomy).878787See also Katz (2018) and Mendell
(2019).

  3. 3.

It also rejected the traditional (Euclidean) concept of _proof_ (which is
sometimes called _genetic_), which was based on the _construction_ of some
object claimed to “exist”, followed by an explicit verification of the
properties claimed by the theorem to be proved.

  4. 4.

As a corollary of the previous point, against the previous standards of proof
going back to Euclid (residing in visualization and intuition, based on the
actual meaning of the symbolic situation), formalism advocated rigour through
abstraction and algebraic deduction.

  5. 5.

Finally, formalists emphasized the almost unlimited freedom and creative force
of mathematicians to propose (consistent) mathematical theories.

None of these ideas truly originated with Hilbert,888888 For example, a
telling quotation is: ‘They who are acquainted with the present state of the
theory of Symbolic Algebra, are aware of the validity of the processes of
analysis does not depend upon the interpretation of the symbols which are
employed, but solely upon the laws of their combination.’ (Boole, 1847,
Preface). Indeed, the full title of the book from which this is taken is _The
Mathematical Analysis of Logic: Being an Essay Towards a Calculus of Deductive
Reasoning_. See also Mehrtens (1990), Heintz (2000), Gray (2008), and Maddy
(2008). but he was certainly the most important mathematician who broadly
championed formalism (in the sense described by Detlefsen above), and thus
took mathematics from the 19th into the 20th century. We already discussed the
first and second points. At least since his instantly famous proofs of the
Basis theorem (1890) and the _Nullstellensatz_ (1893), Hilbert was a non-
constructivist.898989His later adoption of intuitionistic logic and even
finitism was restricted to _metamathematics_ and was intended to justify non-
constructive proofs in classical mathematics. Hilbert (1899) made the third
and fourth ingredients widely known and acceptable, and the fifth was stressed
by him throughout his career.

Wittgenstein’s take on formalism has been analyzed in detail by Mühlhölzer
(2008, 2010). It should be obvious that Wittgenstein supported ingredient 1 of
formalism (in the above list from 1 to 5), i.e. the representational role of
mathematical language (as opposed to its _constitutive_ role); this may even
be said to be the whole point of his late philosophy. He did not (as far as we
know) comment on the historically oriented second ingredient on the division
of mathematics. He might be ambiguous on the nature of proof as meant in the
third point, since on the one hand he is often seen as a constructivist,
whereas on the other his outspoken aim was to leave mathematical _practice_ as
it is (and just provide therapy for those who are confused about its
_foundations_).909090Here is an example of the first point: ‘Ich glaube, die
Mathematik wird, wenn die Grundlagenstreit beendent sein wird, das Gesicht
annehmen, das sie auf der Volksschule hat (…) Er braucht in keiner Weise
verbesster zu werden’ (Wittgenstein, 1984b, pp. 105–106). ’I believe that when
the foundational dispute is over, mathematics will take on the appearance that
it has in elementary school (…) It does not need to be improved in any way.’
Writings from his middle phase, where his calculus-conception of mathematics
reigned (Gerrard, 1991), cf. §2, seem to support the fourth goal of increasing
the rigour of proofs.919191All this is blurred by various ideas Wittgenstein
forwarded on mathematical proof that are out of touch with modern mathematics,
like the idea that the meaning of a theorem lies in its proof, or that a proof
changes the meaning of a theorem. As noted before Wittgenstein did not connect
to Hilbert-style proofs and proof theory. According to Mühlhölzer (2005), p.
128, Wittgenstein was not so much interested in formal proofs but in proofs
that are carried out in (non-logical) mathematical practice.  Finally, the
unlimited freedom and creative force of mathematics also seems compatible with
his views. It therefore seems fair to say that although Wittgenstein opposed
all “isms”, such as logicism, platonism, intuitionism, and formalism,929292For
example: ‘In dem Kampf zwischen dem ‘Formalismus’ & der ‘inhaltlichen
Mathematik’, was behauptet denn jeder Teil? Dieser Streit ist so ähnlich dem
zwischen Realismus & Idealismus! Darin z.B., daß er sehr bald obsolet geworden
sein wird & daß beide Parteien entgegen ihrer täglichen Praxis
Ungerechtigkeiten behaupten.’ (Ms-112,15v[2], quoted in Mühlhölzer (2010), p.
72, from BT, p. 535). ‘In the battle between ‘formalism’ & ‘contentful
mathematics’, what does each side claim? This dispute is similar to that
between realism and idealism! For example, in that it will very soon become
obsolete and that both parties, contrary to their daily practice, claim
injustice.’  the latter was nonetheless closest to his heart, also because of
his sympathy for the closely related analogies with chess (Max, 2020a;
Mühlhölzer, 2010, §I.7). See also §§2–4.

###  A.2 Axioms

Hilbert’s main tool, which Detlefsen (2005) might (and perhaps should) have
included in the above list, was _axiomatization_ (Corry, 2004; Majer, 2001,
2006, 2014; Schlimm, 2013; Majer & Sauer, 2014; Steingart, 2023). As already
mentioned (and illustrated by quotations), Hilbert’s program of axiomatization
was meant to secure both the _rigor_ and the _applicability_ of mathematics:

  1. 1.

the former partly by formalizing the notion of a mathematical proof (which led
to his glib identification by a number of philosophers as a sterile
formalist), and partly by the axiomatization of what might be called immature
mathematical theories;

  2. 2.

the latter by the axiomatization of physics.

All of this was in place around 1900 (and even earlier in Hilbert’s lectures,
cf. footnote 50). Both aspects of the first point were manifest in Hilbert
(1899), as well as in his correspondence with Frege from 1899 and 1900. The
second came out into the public especially through his _sixth problem_ (from
the famous list of 23 problems in 1900):939393See e.g. Corry (2004, 2018) and
references therein. It is puzzling that Hilbert did not mention Newton’s
_Principia_ in this light, which was surely the first explicit and successful
axiomatization of physics.

> Mathematische Behandlung der Axiome der Physik.
>
> _Durch die Untersuchungen über die Grundlagen der Geometrie wird uns die
> Aufgabe nahe gelegt, nach diesem Vorbilde diejenigen physikalischen
> Disciplinen axiomatisch zu behandeln, in denen schon heute die Mathematik
> eine hervorragende Rolle spielt; dies sind in erster Linie die
> Wahrscheinlichkeitsrechnung und die Mechanik._ 949494 ‘Mathematical
> Treatment of the Axioms of Physics. _The investigations on the foundations
> of geometry suggest the problem: To treat in the same manner, by means of
> axioms, those physical sciences in which already today mathematics plays an
> important part; in the first rank are probability theory and mechanics._ ’
> (Hilbert, 1900, p. 272)

In the next two decades Hilbert extended this ambition quite significantly,
e.g.,

> Ich glaube: Alles, was Gegenstand des wissenschaftlichen Denkens überhaupt
> sein kann, verfällt, sobald es zur Bildung einer Theorie reif ist, der
> axiomatischen Methode und damit mittelbar der Mathematik. Durch Vordringen
> zu immer tieferliegender Schichten von Axiomen im vorhin dargelegten Sinne
> gewinnen wir auch in das Wesen des wissenschaftlichen Denkens selbst immer
> tiefere Einblicke und werden uns der Einheit unseres Wissens immer mehr
> bewußt. In dem Zeichen der axiomatischen Methode erscheint die Mathematik
> berufen zu einer führenden Rolle in der Wissenschaft überhaupt.959595‘I
> believe this: as soon as it is ripe for theory building, anything that can
> be the subject of scientific thought at all falls under the scope of the
> axiomatic method and hence indirectly of mathematics. By penetrating into
> ever deeper layers of axioms in the sense outlined earlier we also gain
> insight into the nature of scientific thought by itself and become steadily
> more aware of the unity of our knowledge. Under the header of the axiomatic
> method mathematics appears to be called into a leading role in science in
> general.’ (Hilbert, 1918, p. 414).

Hilbert (1918) _begins_ his essay on axiomatic thought (of which the above
words are the _end_) by stressing the importance of the connection between
mathematics and neigbouring fields like physics and epistemology, introducing
the axiomatic method as the key to this connection:

> Wie im Leben der Völker das einzelne Volk nur dann gedeihen kann wenn es
> such allen Nachbarvökern gut geht, und wie das Interesse der Staaten es
> erheischt, daß nicht nur innerhalb jedes einzelnen Staates Ordnung herrsche,
> sondern auch die Beziehungen der Staaten unter sich gut geordnet worden
> müssen, so ist es auch im Leben der Wissenschaften. In richtiger Erkenntnis
> dessen haben die bedeutendsten Träger der mathematischen Gedankens stets
> groß es Interesse an den Gesetzen und der Ordnung in den
> Nachbarwissenschaften bewiesen und vor allem zu Gunsten der Mathematik
> selbst von jeher die Beziehungen zu den Nachbarwissenschaften, insbesondere
> zu den großen Reichen der Physik und der Erkenntnistheorie, gepflegt. Das
> Wesen dieser Beziehungen und der Grund ihrer Fruchtbarkeit glaube ich, wird
> am besten deutlich, wenn ich lhnen diejenige allgemeine Forschungsmethode
> schildere, die in der neueren Mathematik mehr und mehr zur Geltung zu kommen
> scheint: ich meine die _axiomatische Methode_.969696 ‘Just as in the life of
> nations the individual nation can only thrive when all neighbouring nations
> are in good health; and just as the interest of states demands, not only
> that order prevail within every individual state, but also that the
> relationships of the states among themselves be in good order; so it is in
> the life of the sciences. In due recognition of this fact the most important
> bearers of mathematical thought have always evinced great interest in the
> laws and the structure of the neighbouring sciences; above all for the
> benefit of mathematics itself they have always cultivated the relations to
> the neighbouring sciences, especially to the great empires of physics and
> epistemology. I believe that the essence of these relations, and the reason
> for their fruitfulness, will appear most clearly if I describe for you the
> general method of research which seems to be coming more and more into its
> own in modern mathematics: I mean the _axiomatic method_.’ (Hilbert, 1918,
> p. 405).

###  A.3 Definitions

The concept of a definition goes back to Socrates, Plato, and
Aristotle,979797Further to the recent _Stanford Encyclopedia of Philosophy_
article by Gupta & Mackereth (2023), the _Encyclopedia of Philosophy_ entry by
Abelson (1967) also remains worth reading, as is the even older monograph by
Robinson (1954). who tried to define things (words or concepts) by finding
their _essence_.989898Cellucci (2018) regards ‘Galileo’s decision to abandon
Aristotle’s aim to penetrate the essence of natural substances’ and replace
this aim by stipulative definitions via mathematics as the beginning of modern
science. Turning to our three protagonists, Frege promoted various concepts of
definition. In his _Begrifsschrift_ from 1879, a definition was an
abbreviation:999999In this quote, ‘(69)’ refers to a certain symbolic
definition: if some object $\delta$ has some property $F$ and for some
function $f$ the object $f(\delta)$ also has the property $F$ (in case this is
defined), then we say that the ‘$f$-sequence’ $(\delta,f(\delta),\ldots)$
inherits $F$. The details are irrelevant for what follows, including the
Kantian terminology. All that matters is that the notion of ‘inheritage’
(expressed symbolically) is defined in terms of known things (expressed
symbolically).

> Wäre nun (69) ein synthetisches Urteil, so wären es auch die daraus
> abgeleiteten Sätzen. Man kann aber die durch diesen Satz eingeführten
> Bezeichnungen und daher ihn selbst als ihre Erklärung entbehren: nichts
> folgt aus ihm, was nicht auch ohne ihn erschlossen werden könnte. Solche
> Erklärungen haben nur den Zweck, durch Festsetzung einer Abkürzung eine
> äusserliche Erleichterung herbeizuführen. Ausserdem dienen sie dazu eine
> besondere Verbindung von Zeichen aus der Fülle der möglichen hervorzuheben,
> um daran einen festern Anhalt für die Vorstellung zu gewinnen.100100100 ‘If
> (69) had been a synthetic judgement, then so would have been the theorems
> derived from it. But we can do without the notation introduced by this
> sentence, and hence without the sentence itself as its definition; nothing
> follows from the sentence that could not also be inferred without it. Our
> sole purpose in introducing such definitions is to bring about an extrinsic
> simplification by stipulating an abbreviation. Apart from this, such
> definitions serve the purpose of highlighting special combinations of
> symbols from the wealth of all possibilities, so as to anchor them in our
> imagination.’  (Frege, 1879, p. 56).

In other words: A definition is an arbitrary stipulation by which a new sign
is introduced to take the place of a complex expression whose meaning we
already know. But also note the last sentence! All of this is echoed in
_Principa Mathematica_ (whose debt to Frege is generally enormous):

> Theoretically, it is unnecessary ever to give a definition: we might always
> use the definiens instead, and thus wholly dispense with the definiendum.
> Thus although we employ definitions and do not define “definition,” yet
> “definition” does not appear among our primitive ideas, because the
> definitions are no part of our subject, but are, strictly speaking, mere
> typographical conveniences. Practically, of course, if we introduced no
> definitions, our formulae would very soon become so lengthy as to be
> unmanageable; but theoretically, all definitions are superfluous. In spite
> of the fact that definitions are theoretically superfluous, it is
> nevertheless true that they often convey more important information than is
> contained in the propositions in which they are used. This arises from two
> causes. First, a definition usually implies that the definiens is worthy of
> careful consideration. Hence the collection of definitions embodies our
> choice of subjects and our judgment as to what is most important. Secondly,
> when what is defined is (as often occurs) something already familiar, such
> as cardinal or ordinal numbers, the definition contains an analysis of a
> common idea, and may therefore express a notable advance. Cantor’s
> definition of the continuum illustrates this: his definition amounts to the
> statement that what he is defining is the object which has the properties
> commonly associated with the word “continuum,” though what precisely
> constitutes these properties had not before been known. In such cases, a
> definition is a “making definite”: it gives definiteness to an idea which
> had previously been more or less vague. For these reasons, it will be found,
> in what follows, that the definitions are what is most important, and what
> most deserves the reader’s prolonged attention. (Russell & Whitehead, 1910,
> pp. 11–12).

Returning to Frege, despite the quotation preceding the one just given, his
_Grundlagen der Arithmetik_ from 1884 is a sustained quest for the essence of
number, although the author explicitly declares himself to be a logicist
rather than an essentialist. Starting with the very first sentence (which
states that ‘the number one is a thing’ is unsatisfactory because the first
article is definite whereas the second is not),101101101‘Auf die Frage, was
die Zahl Eins sei, oder was das Zeichen 1 bedeute, wird man meistens die
Antwort erhalten: nun, ein Ding. Und wenn man dann darauf aufmerksam macht,
dass der Satz “die Zahl Eins ist ein Ding” keine Definition ist, weil auf der
einen Seite der bestimmte Artikel, auf der anderen der unbestimmte steht, dass
er nur besagt, die Zahl Eins gehöre zu den Dingen, aber nicht, welches Ding
sie sei, so wird man vielleicht aufgefordert, sich irgendein Ding zu wählen,
das man Eins nennen wolle.’ (Frege, 1884, p. I). Ironically, even Frege’s
definition of the number one, which eventually relied on his entire subsequent
program, turned out to be wrong in view of Russell’s paradox.  the
_Grundlagen_ contains a large number of examples of what definitions should
_not_ be. The only positive characterization is that a ‘a definition must be
logical’ (p. IX). And finally, in a more general context Frege defined (_sic_)
definitions by their extension:

> Eine Definition eines Begriffes (möglichen Prädikates) muss vollständig
> sein, sie muss für jeden Gegenstand unzweideutig bestimmen, ob er unter den
> Begriff falle (ob das Prädikat mit Wahrheit von ihm ausgesagt werden könne)
> oder nicht. Es darf also keinen Gegenstand geben, für den es nach der
> Definition zweifelhaft bliebe, ob er unter den Begriff fiele, wenn es auch
> für uns Menschen bei unsern mangelhaften Wissen nicht immer möglich sein
> mag, die Frage zu entscheiden. Man kann dies Bildlich so ausdrücken: der
> Begriff muss scharf begrenzt sein.102102102 ‘A definition of a concept (of a
> possible predicate) must be complete; it must unambiguously determine, as
> regards any object, whether or not it falls under the concept (whether or
> not the predicate is truly assertible of it). Thus there must not be any
> object as regards which the definition leaves in doubt whether it falls