## 1. Philosophy of Mathematics, Logic, and the Foundations of Mathematics



On the one hand, philosophy of mathematics is concerned with problems
that are closely related to central problems of metaphysics and
epistemology. At first blush, mathematics appears to study abstract
entities. This makes one wonder what the nature of mathematical
entities consists in and how we can have knowledge of mathematical
entities. If these problems are regarded as intractable, then one
might try to see if mathematical objects can somehow belong to the
concrete world after all.



On the other hand, it has turned out that to some extent it is
possible to bring mathematical methods to bear on philosophical
questions concerning mathematics. The setting in which this has been
done is that of *mathematical logic* when it is broadly
conceived as comprising proof theory, model theory, set theory, and
computability theory as subfields. Thus the twentieth century has
witnessed the mathematical investigation of the consequences of what
are at bottom philosophical theories concerning the nature of
mathematics.



When professional mathematicians are concerned with the foundations of
their subject, they are said to be engaged in foundational research.
When professional philosophers investigate philosophical questions
concerning mathematics, they are said to contribute to the philosophy
of mathematics. Of course the distinction between the philosophy of
mathematics and the foundations of mathematics is vague, and the more
interaction there is between philosophers and mathematicians working
on questions pertaining to the nature of mathematics, the better.


## 2. Four schools



The general philosophical and scientific outlook in the nineteenth
century tended toward the empirical: platonistic aspects of
rationalistic theories of mathematics were rapidly losing support.
Especially the once highly praised faculty of rational intuition of
ideas was regarded with suspicion. Thus it became a challenge to
formulate a philosophical theory of mathematics that was free of
platonistic elements. In the first decades of the twentieth century,
three non-platonistic accounts of mathematics were developed:
logicism, formalism, and intuitionism. There emerged in the beginning
of the twentieth century also a fourth program: predicativism. Due to
contingent historical circumstances, its true potential was not
brought out until the 1960s. However it deserves a place beside the
three traditional schools that are discussed in most standard
contemporary introductions to philosophy of mathematics, such as
(Shapiro 2000) and (Linnebo 2017).


### 2.1 Logicism



The logicist project consists in attempting to reduce mathematics to
logic. Since logic is supposed to be neutral about matters
ontological, this project seemed to harmonize with the
anti-platonistic atmosphere of the time.



The idea that mathematics is logic in disguise goes back to Leibniz.
But an earnest attempt to carry out the logicist program in detail
could be made only when in the nineteenth century the basic principles
of central mathematical theories were articulated (by Dedekind and
Peano) and the principles of logic were uncovered (by Frege).



Frege devoted much of his career to trying to show how mathematics can
be reduced to logic (Frege 1884). He managed to derive the principles
of (second-order) Peano arithmetic from the basic laws of a system of
second-order logic. His derivation was flawless. However, he relied on
one principle which turned out not to be a logical principle after
all. Even worse, it is untenable. The principle in question is
Frege's *Basic Law V*:

 

\[ \{x|Fx\}=\{x|Gx\} \text{ if and only if } \forall x(Fx \equiv Gx), \]


In words: the set of the *F*s is identical with the
set of the *G*s iff the *F*s are
precisely the *G*s.



In a famous letter to Frege, Russell showed that Frege's Basic
Law V entails a contradiction (Russell 1902). This argument has come
to be known as *Russell's paradox* (see
 section 2.4).



Russell himself then tried to reduce mathematics to logic in another
way. Frege's Basic Law V entails that corresponding to every
property of mathematical entities, there exists a class of
mathematical entities having that property. This was evidently too
strong, for it was exactly this consequence which led to
Russell's paradox. So Russell postulated that only properties of
mathematical objects that have already been shown to exist, determine
classes. Predicates that implicitly refer to the class that they were
to determine if such a class existed, do not determine a class. Thus a
typed structure of properties is obtained: properties of ground
objects, properties of ground objects and classes of ground objects,
and so on. This typed structure of properties determines a layered
universe of mathematical objects, starting from ground objects,
proceeding to classes of ground objects, then to classes of ground
objects and classes of ground objects, and so on.



Unfortunately, Russell found that the principles of his typed logic
did not suffice for deducing even the basic laws of arithmetic. He
needed, among other things, to lay down as a basic principle that
there exists an infinite collection of ground objects. This could
hardly be regarded as a logical principle. Thus the second attempt to
reduce mathematics to logic also faltered.



And there matters stood for more than fifty years. In 1983, Crispin
Wright's book on Frege's theory of the natural numbers
appeared (Wright 1983). In it, Wright breathes new life into the
logicist project. He observes that Frege's derivation of
second-order Peano Arithmetic can be broken down in two stages. In a
first stage, Frege uses the inconsistent Basic Law V to derive what
has come to be known as *Hume's Principle*:



The number of the *F*s = the number of the *G*s
if and only if \(F\approx G\),



where \(F \approx G\) means that the *F*s and the *G*s
stand in one-to-one correspondence with each other.
(This relation of one-to-one correspondence can be expressed in
second-order logic.) Then, in a second stage, the principles of
second-order Peano Arithmetic are derived from Hume's Principle
and the accepted principles of second-order logic. In particular,
Basic Law V is *not* needed in the second part of the
derivation. Moreover, Wright conjectured that in contrast to
Frege's Basic Law V, Hume's Principle is consistent.
George Boolos and others observed that Hume's Principle is
indeed consistent (Boolos 1987).



Wright went on to claim that Hume's Principle can be regarded as
a truth of logic. If that is so, then at least second-order Peano
arithmetic is reducible to logic alone. Thus a new form of logicism
was born; today this view is known as *neo-logicism* (Hale
& Wright 2001). Most philosophers of mathematics today doubt that
Hume's Principle is a principle of *logic*. Indeed, even
Wright later sought to qualify this claim. Nonetheless, many
philosophers of mathematics feel that the introduction of natural
numbers through Hume's Principle is attractive from an
ontological and from an epistemological point of view. Linnebo argues
that because the left-hand-side of Hume's Principle merely
*re-carves* the content of its right-hand-side, not much is
needed from the world to make Hume's Principle true. For this
reason, he calls natural numbers and mathematical objects that can be
introduced in a similar way *light* mathematical objects
(Linnebo 2018).



Wright's work has drawn the attention of philosophers of
mathematics to the *kind* of principles of which Basic Law V
and Hume's Principle are examples. These principles are called
*abstraction principles*. At present, philosophers of
mathematics attempt to construct general theories of abstraction
principles that explain which abstraction principles are acceptable
and which are not, and why (Weir 2003; Fine 2002). Also, it has
emerged that in the context of weakened versions of second-order
logic, Frege's Basic Law V is consistent. But these weak
background theories only allow very weak arithmetical theories to be
derived from Basic Law V (Burgess 2005).


### 2.2 Intuitionism



Intuitionism originates in the work of the mathematician L.E.J.
Brouwer (van Atten 2004), and it is inspired by Kantian views of what
objects are (Parsons 2008, chapter 1). According to intuitionism,
mathematics is essentially an activity of construction. The natural
numbers are mental constructions, the real numbers are mental
constructions, proofs and theorems are mental constructions,
mathematical meaning is a mental construction... Mathematical
constructions are produced by the *ideal* mathematician, i.e.,
abstraction is made from contingent, physical limitations of the real
life mathematician. But even the ideal mathematician remains a finite
being. She can never complete an infinite construction, even though
she can complete arbitrarily large finite initial parts of it. This
entails that intuitionism resolutely rejects the existence of the
actual (or completed) infinite; only potentially infinite collections
are given in the activity of construction. A basic example is the
successive construction in time of the individual natural numbers.



From these general considerations about the nature of mathematics,
based on the condition of the human mind (Moore 2001), intuitionists
infer to a revisionist stance in logic and mathematics. They find
non-constructive existence proofs unacceptable. Non-constructive
existence proofs are proofs that purport to demonstrate the existence
of a mathematical entity having a certain property without even
implicitly containing a method for generating an example of such an
entity. Intuitionism rejects non-constructive existence proofs as
'theological' and 'metaphysical'. The
characteristic feature of non-constructive existence proofs is that
they make essential use of the *principle of excluded
third*

 

\[ \phi \vee \neg \phi, \]


or one of its equivalents, such as the principle of double
negation

 

\[ \neg \neg \phi \rightarrow \phi \]


In classical logic, these principles are valid. The logic of
intuitionistic mathematics is obtained by removing the principle of
excluded third (and its equivalents) from classical logic. This of
course leads to a revision of mathematical knowledge. For instance,
the classical theory of elementary arithmetic, *Peano
Arithmetic*, can no longer be accepted. Instead, an intuitionistic
theory of arithmetic (called *Heyting Arithmetic*) is proposed
which does not contain the principle of excluded third. Although
intuitionistic elementary arithmetic is weaker than classical
elementary arithmetic, the difference is not all that great. There
exists a simple syntactical translation which translates all classical
theorems of arithmetic into theorems which are intuitionistically
provable.



In the first decades of the twentieth century, parts of the
mathematical community were sympathetic to the intuitionistic critique
of classical mathematics and to the alternative that it proposed. This
situation changed when it became clear that in higher mathematics, the
intuitionistic alternative differs rather drastically from the
classical theory. For instance, intuitionistic mathematical analysis
is a fairly complicated theory, and it is very different from
classical mathematical analysis. This dampened the enthusiasm of the
mathematical community for the intuitionistic project. Nevertheless,
followers of Brouwer have continued to develop intuitionistic
mathematics onto the present day (Troelstra & van Dalen 1988).


### 2.3 Formalism



David Hilbert agreed with the intuitionists that there is a sense in
which the natural numbers are basic in mathematics. But unlike the
intuitionists, Hilbert did not take the natural numbers to be mental
constructions. Instead, he argued that the natural numbers can be
taken to be *symbols*. Symbols are strictly speaking abstract
objects. Nonetheless, it is essential to symbols that they can be
embodied by concrete objects, so we may call them
*quasi-concrete* objects (Parsons 2008, chapter 1). Perhaps
physical entities could play the role of the natural numbers. For
instance, we may take a concrete ink trace of the form | to be the
number 0, a concretely realized ink trace || to be the number 1, and
so on. Hilbert thought it doubtful at best that higher mathematics
could be directly interpreted in a similarly straightforward and
perhaps even concrete manner.



Unlike the intuitionists, Hilbert was not prepared to take a
revisionist stance toward the existing body of mathematical knowledge.
Instead, he adopted an instrumentalist stance with respect to higher
mathematics. He thought that higher mathematics is no more than a
formal game. The statements of higher-order mathematics are
uninterpreted strings of symbols. Proving such statements is no more
than a game in which symbols are manipulated according to fixed rules.
The point of the 'game of higher mathematics' consists, in
Hilbert's view, in proving statements of elementary arithmetic,
which do have a direct interpretation (Hilbert 1925).



Hilbert thought that there can be no reasonable doubt about the
soundness of classical Peano Arithmetic -- or at least about the
soundness of a subsystem of it that is called *Primitive Recursive
Arithmetic* (Tait 1981). And he thought that every arithmetical
statement that can be proved by making a detour through higher
mathematics, can also be proved directly in Peano Arithmetic. In fact,
he strongly suspected that *every* problem of elementary
arithmetic can be decided from the axioms of Peano Arithmetic. Of
course solving arithmetical problems in arithmetic is in some cases
practically impossible. The history of mathematics has shown that
making a "detour" through higher mathematics can sometimes
lead to a proof of an arithmetical statement that is much shorter and
that provides more insight than any purely arithmetical proof of the
same statement.



Hilbert realized, albeit somewhat dimly, that some of his convictions
can actually be considered to be mathematical conjectures. For a proof
in a formal system of higher mathematics or of elementary arithmetic
is a finite combinatorial object which can, modulo coding, be
considered to be a natural number. But in the 1920s the details of
coding proofs as natural numbers were not yet completely
understood.



On the formalist view, a minimal requirement of formal systems of
higher mathematics is that they are at least consistent. Otherwise
*every* statement of elementary arithmetic can be proved in
them. Hilbert also saw (again, dimly) that the consistency of a system
of higher mathematics entails that this system is at least partially
arithmetically sound. So Hilbert and his students set out to prove
statements such as the consistency of the standard postulates of
mathematical analysis. Of course such statements would have to be
proved in a 'safe' part of mathematics, such as elementary
arithmetic. Otherwise the proof does not increase our conviction in
the consistency of mathematical analysis. And, fortunately, it seemed
possible in principle to do this, for in the final analysis
consistency statements are, again modulo coding, arithmetical
statements. So, to be precise, Hilbert and his students set out to
prove the consistency of, e.g., the axioms of mathematical analysis in
classical Peano arithmetic. This project was known as
*Hilbert's program* (Zach 2006). It turned out to be more
difficult than they had expected. In fact, they did not even succeed
in proving the consistency of the axioms of Peano Arithmetic in Peano
Arithmetic.



Then Kurt Godel proved that there exist arithmetical statements
that are undecidable in Peano Arithmetic (Godel 1931). This has
become known as his Godel's *first incompleteness
theorem*. This did not bode well for Hilbert's program, but
it left open the possibility that the consistency of higher
mathematics is not one of these undecidable statements. Unfortunately,
Godel then quickly realized that, unless (God forbid!) Peano
Arithmetic is inconsistent, the consistency of Peano Arithmetic is
independent of Peano Arithmetic. This is Godel's *second
incompleteness theorem*. Godel's incompleteness
theorems turn out to be generally applicable to all sufficiently
strong but consistent recursively axiomatizable theories. Together,
they entail that Hilbert's program fails. It turns out that
higher mathematics cannot be interpreted in a purely instrumental way.
Higher mathematics can prove arithmetical sentences, such as
consistency statements, that are beyond the reach of Peano
Arithmetic.



All this does not spell the end of formalism. Even in the face of the
incompleteness theorems, it is coherent to maintain that mathematics
is the science of formal systems.



One version of this view was proposed by Curry (Curry 1958). On this
view, mathematics consists of a collection of formal systems which
have no interpretation or subject matter. (Curry here makes an
exception for metamathematics.) Relative to a formal system, one can
say that a statement is true if and only if it is derivable in the
system. But on a fundamental level, *all* mathematical systems
are on a par. There can be at most pragmatical reasons for preferring
one system over another. Inconsistent systems can prove all statements
and therefore are pretty useless. So when a system is found to be
inconsistent, it must be modified. It is simply a lesson from
Godel's incompleteness theorems that a sufficiently strong
consistent system cannot prove its own consistency.



There is a canonical objection against Curry's formalist
position. Mathematicians do not in fact treat all apparently
consistent formal systems as being on a par. Most of them are
unwilling to admit that the preference of arithmetical systems in
which the arithmetical sentence expressing the consistency of Peano
Arithmetic are derivable over those in which its negation is
derivable, for instance, can ultimately be explained in purely
pragmatical terms. Many mathematicians want to maintain that the
perceived correctness (incorrectness) of certain formal systems must
ultimately be explained by the fact that they correctly (incorrectly)
describe certain subject matters.



Detlefsen has emphasized that the incompleteness theorems do not
preclude that the consistency of *parts* of higher mathematics
that are in practice used for solving arithmetical problems that
mathematicians are interested in can be arithmetically established
(Detlefsen 1986). In this sense, something can perhaps be rescued from
the flames even if Hilbert's instrumentalist stance towards all
of higher mathematics is ultimately untenable.



Another attempt to salvage a part of Hilbert's program was made
by Isaacson (Isaacson 1987). He defends the view that *in some
sense*, Peano Arithmetic may be complete after all (Isaacson
1987). He argues that true sentences undecidable in Peano Arithmetic
can only be proved by means of *higher-order concepts*. For
instance, the consistency of Peano Arithmetic can be proved by
induction up to a transfinite ordinal number (Gentzen 1938). But the
notion of an ordinal number is a set-theoretic, and hence
non-arithmetical, concept. If the only ways of proving the consistency
of arithmetic make essential use of notions which arguably belong to
higher-order mathematics, then the consistency of arithmetic, even
though it can be expressed in the language of Peano Arithmetic, is a
non-arithmetical problem. And generalizing from this, one can wonder
whether Hilbert's conjecture that *every* problem of
arithmetic can be decided from the axioms of Peano Arithmetic might
not still be true.


### 2.4 Predicativism



As was mentioned earlier, predicativism is not ordinarily described as
one of the schools. But it is only for contingent reasons that before
the advent of the second world war predicativism did not rise to the
level of prominence of the other schools.



The origin of predicativism lies in the work of Russell. On a cue of
Poincare, he arrived at the following diagnosis of the Russell
paradox. The argument of the Russell paradox defines the collection C
of all mathematical entities that satisfy \(\neg x\in x\). The
argument then proceeds by asking whether C itself meets this
condition, and derives a contradiction.



The Poincare-Russell diagnosis of this argument states that
this definition does not pick out a collection at all: it is
impossible to define a collection S by a condition that implicitly
refers to S itself. This is called the *vicious circle
principle*. Definitions that violate the vicious circle principle
are called *impredicative*. A sound definition of a collection
only refers to entities that exist independently from the defined
collection. Such definitions are called *predicative*. As
Godel later pointed out, a platonist would find this line of
reasoning unconvincing. If mathematical collections exist
independently of the act of defining, then it is not immediately clear
why there could not be collections that can *only* be defined
impredicatively (Godel 1944).



All this led Russell to develop the simple and the ramified theory of
types, in which syntactical restrictions were built in that make
impredicative definitions ill-formed. In simple type theory, the free
variables in defining formulas range over entities to which the
collection to be defined do not belong. In ramified type theory, it is
required in addition that the range of the bound variables in defining
formulas do not include the collection to be defined. It was pointed
out in
 section 2.1
 that Russell's type theory cannot be seen as a reduction of
mathematics to logic. But even aside from that, it was observed early
on that especially in ramified type theory it is too cumbersome to
formalize ordinary mathematical arguments.



When Russell turned to other areas of analytical philosophy, Hermann
Weyl took up the predicativist cause (Weyl 1918). Like
Poincare, Weyl did not share Russell's desire to reduce
mathematics to logic. And right from the start he saw that it would be
in practice impossible to work in a ramified type theory. Weyl
developed a philosophical stance that is in a sense intermediate
between intuitionism and platonism. He took the collection of natural
numbers as unproblematically given. But the concept of an arbitrary
subset of the natural numbers was not taken to be immediately given in
mathematical intuition. Only those subsets which are determined by
arithmetical (i.e., first-order) predicates are taken to be
predicatively acceptable.



On the one hand, it emerged that many of the standard definitions in
mathematical analysis are impredicative. For instance, the minimal
closure of an operation on a set is ordinarily defined as the
intersection of all sets that are closed under applications of the
operation. But the minimal closure itself is one of the sets that are
closed under applications of the operation. Thus, the definition is
impredicative. In this way, attention gradually shifted away from
concern about the set-theoretical paradoxes to the role of
impredicativity in mainstream mathematics. On the other hand, Weyl
showed that it is often possible to bypass impredicative notions. It
even emerged that most of mainstream nineteenth century mathematical
analysis can be vindicated on a predicative basis (Feferman 1988).



In the 1920s, History intervened. Weyl was won over to Brouwer's
more radical intuitionistic project. In the meantime, mathematicians
became convinced that the highly impredicative transfinite set theory
developed by Cantor and Zermelo was less acutely threatened by
Russell's paradox than previously suspected. These factors
caused predicativism to lapse into a dormant state for several
decades.



Building on work in generalized recursion theory, Solomon Feferman
extended the predicativist project in the 1960s (Feferman 2005). He
realized that Weyl's strategy could be iterated into the
transfinite. Also those sets of numbers that can be defined by using
quantification over the sets that Weyl regarded as predicatively
justified, should be counted as predicatively acceptable, and so on.
This process can be propagated along an ordinal path. This ordinal
path stretches as far into the transfinite as the *predicative
ordinals* reach, where an ordinal is predicative if it measures
the length of a provable well-ordering of the natural numbers. This
calibration of the strength of predicative mathematics, which is due
to Feferman and (independently) Schutte, is nowadays fairly
generally accepted. Feferman then investigated how much of standard
mathematical analysis can be carried out within a predicativist
framework. The research of Feferman and others (most notably Harvey
Friedman) shows that most of twentieth century analysis is acceptable
from a predicativist point of view. But it is also clear that not all
of contemporary mathematics that is generally accepted by the
mathematical community is acceptable from a predicativist standpoint:
transfinite set theory is a case in point.


## 3. Platonism



In the years before the second world war it became clear that weighty
objections had been raised against each of the three anti-platonist
programs in the philosophy of mathematics. Predicativism was perhaps
an exception, but it was at the time a program without defenders. Thus
room was created for a renewed interest in the prospects of
platonistic views about the nature of mathematics. On the platonistic
conception, the subject matter of mathematics consists of *abstract
entities*.


### 3.1 Godel's Platonism



Godel was a platonist with respect to mathematical objects and
with respect to mathematical concepts (Godel 1944; Godel
1964). But his platonistic view was more sophisticated than that of
the mathematician in the street.



Godel held that there is a strong parallelism between plausible
theories of mathematical objects and concepts on the one hand, and
plausible theories of physical objects and properties on the other
hand. Like physical objects and properties, mathematical objects and
concepts are not constructed by humans. Like physical objects and
properties, mathematical objects and concepts are not reducible to
mental entities. Mathematical objects and concepts are as objective as
physical objects and properties. Mathematical objects and concepts
are, like physical objects and properties, postulated in order to
obtain a good satisfactory theory of our experience. Indeed, in a way
that is analogous to our perceptual relation to physical objects and
properties, through *mathematical intuition* we stand in a
quasi-perceptual relation with mathematical objects and concepts. Our
perception of physical objects and concepts is fallible and can be
corrected. In the same way, mathematical intuition is not fool-proof
-- as the history of Frege's Basic Law V shows-- but
it can be trained and improved. Unlike physical objects and
properties, mathematical objects do not exist in space and time, and
mathematical concepts are not instantiated in space or time.



Our mathematical intuition provides *intrinsic evidence* for
mathematical principles. Virtually all of our mathematical knowledge
can be deduced from the axioms of *Zermelo-Fraenkel set theory with
the Axiom of Choice* (ZFC). In Godel's view, we have
compelling intrinsic evidence for the truth of these axioms. But he
also worried that mathematical intuition might not be strong enough to
provide compelling evidence for axioms that significantly exceed the
strength of ZFC.



Aside from intrinsic evidence, it is in Godel's view also
possible to obtain *extrinsic evidence* for mathematical
principles. If mathematical principles are successful, then, even if
we are unable to obtain intuitive evidence for them, they may be
regarded as probably true. Godel says that:



> 
> ... success here means fruitfulness in consequences, particularly
> in 'verifiable' consequences, i.e. consequences verifiable
> without the new axiom, whose proof with the help of the new axiom,
> however, are considerably simpler and easier to discover, and which
> make it possible to contract into one proof many different proofs
> [...] There might exist axioms so abundant in their verifiable
> consequences, shedding so much light on a whole field, yielding such
> powerful methods for solving problems [...] that, no matter
> whether or not they are intrinsically necessary, they would have to be
> accepted at least in the same sense as any well-established physical
> theory. (Godel 1947, p. 477)
> 



This inspired Godel to search for new axioms which can be
extrinsically motivated and which can decide questions such as the
*continuum hypothesis* which are highly independent of ZFC (cf.
 section 5.1).



Godel shared Hilbert's conviction that all mathematical
questions have definite answers. But platonism in the philosophy of
mathematics should not be taken to be ipso facto committed to holding
that all set-theoretical propositions have determinate truth values.
There are versions of platonism that maintain, for instance, that all
theorems of ZFC are made true by determinate set-theoretical facts,
but that there are no set-theoretical facts that make certain
statements that are highly independent of ZFC truth-determinate. It
seems that the famous set theorist Paul Cohen held some such view
(Cohen 1971).


### 3.2 Naturalism and Indispensability



Quine formulated a methodological critique of traditional philosophy.
He suggested a different philosophical methodology instead, which has
become known as *naturalism* (Quine 1969). According to
naturalism, our best theories are our best *scientific*
theories. If we want to obtain the best available answer to
philosophical questions such as *What* *do we know?* and
*Which kinds of entities exist?*, we should not appeal to
traditional epistemological and metaphysical theories. We should also
refrain from embarking on a fundamental epistemological or
metaphysical inquiry starting from first principles. Rather, we should
consult and analyze our best scientific theories. They contain, albeit
often implicitly, our currently best account of what exists, what we
know, and how we know it.



Putnam applied Quine's naturalistic stance to mathematical
ontology (Putnam 1972). At least since Galilei, our best theories from
the natural sciences are mathematically expressed. Newton's
theory of gravitation, for instance, relies heavily on the classical
theory of the real numbers. Thus an ontological commitment to
mathematical entities seems inherent to our best scientific theories.
This line of reasoning can be strengthened by appealing to the Quinean
thesis of confirmational holism. Empirical evidence does not bestow
its confirmatory power on any one individual hypothesis. Rather,
experience globally confirms the theory in which the individual
hypothesis is embedded. Since mathematical theories are part and
parcel of scientific theories, they too are confirmed by experience.
Thus, we have empirical confirmation for mathematical theories. Even
more appears true. It seems that mathematics is indispensable to our
best scientific theories: it is not at all obvious how we
*could* express them without using mathematical vocabulary.
Hence the naturalist stance commands us to accept mathematical
entities as part of our philosophical ontology. This line of
argumentation is called an *indispensability argument* (Colyvan
2001).



If we take the mathematics that is involved in our best scientific
theories at face value, then we appear to be committed to a form of
platonism. But it is a more modest form of platonism than
Godel's platonism. For it appears that the natural sciences
can get by with (roughly) function spaces on the real numbers. The
higher regions of transfinite set theory appear to be largely
irrelevant to even our most advanced theories in the natural sciences.
Nevertheless, Quine thought (at some point) that the sets that are
postulated by ZFC are acceptable from a naturalistic point of view;
they can be regarded as a generous rounding off of the mathematics
that is involved in our scientific theories. Quine's judgement
on this matter is not universally accepted. Feferman, for instance,
argues that all the mathematical theories that are essentially used in
our currently best scientific theories are predicatively reducible
(Feferman 2005). Maddy even argues that naturalism in the philosophy
of mathematics is perfectly compatible with a non-realist view about
sets (Maddy 2007, part IV).



In Quine's philosophy, the natural sciences are the ultimate
arbiters concerning mathematical existence and mathematical truth.
This has led Charles Parsons to object that this picture makes the
obviousness of elementary mathematics somewhat mysterious (Parsons
1980). For instance, the question whether every natural number has a
successor ultimately depends, in Quine's view, on our best
empirical theories; however, somehow this fact appears more immediate
than that. In a kindred spirit, Maddy notes that mathematicians do not
take themselves to be in any way restricted in their activity by the
natural sciences. Indeed, one might wonder whether mathematics should
not be regarded as a science in its own right, and whether the
ontological commitments of mathematics should not be judged rather on
the basis of the rational methods that are implicit in mathematical
practice.



Motivated by these considerations, Maddy set out to inquire into the
standards of existence implicit in mathematical practice, and into the
implicit ontological commitments of mathematics that follow from these
standards (Maddy 1990). She focussed on set theory, and on the
methodological considerations that are brought to bear by the
mathematical community on the question which large cardinal axioms can
be taken to be true. Thus her view is closer to that of Godel
than to that of Quine. In more recent work, she isolates two maxims
that seem to be guiding set theorists when contemplating the
acceptability of new set theoretic principles: *unify* and
*maximize* (Maddy 1997). The maxim "unify" is an
instigation for set theory to provide a single system in which all
mathematical objects and structures of mathematics can be instantiated
or modelled. The maxim "maximize" means that set theory
should adopt set theoretic principles that are as powerful and
mathematically fruitful as possible.


### 3.3 Deflating Platonism



Bernays observed that when a mathematician is at work she
"naively" treats the objects she is dealing with in a
platonistic way. Every working mathematician, he says, is a platonist
(Bernays 1935). But when the mathematician is caught off duty by a
philosopher who quizzes her about her ontological commitments, she is
apt to shuffle her feet and withdraw to a vaguely non-platonistic
position. This has been taken by some to indicate that there is
something wrong with philosophical questions about the nature of
mathematical objects and of mathematical knowledge.



Carnap introduced a distinction between questions that are internal to
a framework and questions that are external to a framework (Carnap
1950). It has been argued that Carnap's distinction in some
guise survives the demise of the logical empiricist framework in which
it was first articulated (Burgess 2004b). Tait has attempted to work
out in detail how the resulting distinction can be applied to
mathematics (Tait 2005). This has resulted in what might be regarded
as a deflationary versions of platonism.



According to Tait, questions of existence of mathematical entities can
only be sensibly asked and reasonably answered from within (axiomatic)
mathematical frameworks. If one is working in number theory, for
instance, then one can ask whether there are prime numbers that have a
given property. Such questions are then to be decided on purely
mathematical grounds. Philosophers have a tendency to step outside the
framework of mathematics and ask "from the outside"
whether mathematical objects *really* exist and whether
mathematical propositions are *really* true. In this question
they are asking for supra-mathematical or metaphysical grounds for
mathematical truth and existence claims. Tait argues that it is hard
to see how any sense can be made of such external questions. He
attempts to deflate them, and bring them back to where they belong: to
mathematical practice itself. Of course not everyone agrees with Tait
on this point. Linsky and Zalta have developed a systematic way of
answering precisely the sort of external questions that Tait
approaches with disdain (Linsky & Zalta 1995).



It comes as no surprise that Tait has little use for Godelian
appeals to mathematical intuition in the philosophy of mathematics, or
for the philosophical thesis that mathematical objects exist
"outside space and time". More generally, Tait believes
that mathematics is not in need of a philosophical foundation; he
wants to let mathematics speak for itself. In this sense, his position
is reminiscent of the (in some sense Wittgensteinian) *natural
ontological attitude* that is advocated by Arthur Fine in the
realism debate in the philosophy of science.


### 3.4 Benacerraf's Epistemological Problem



Benacerraf formulated an epistemological problem for a variety of
platonistic positions in the philosophy of science (Benacerraf 1973).
The argument is specifically directed against accounts of mathematical
intuition such as that of Godel. Benacerraf's argument
starts from the premise that our best theory of knowledge is the
causal theory of knowledge. It is then noted that according to
platonism, abstract objects are not spatially or temporally localized,
whereas flesh and blood mathematicians are spatially and temporally
localized. Our best epistemological theory then tells us that
knowledge of mathematical entities should result from causal
interaction with these entities. But it is difficult to imagine how
this could be the case.



Today few epistemologists hold that the causal theory of knowledge is
our best theory of knowledge. But it turns out that Benacerraf's
problem is remarkably robust under variation of epistemological
theory. For instance, let us assume for the sake of argument that
reliabilism is our best theory of knowledge. Then the problem becomes
to explain how we succeed in obtaining reliable beliefs about
mathematical entities.



Hodes has formulated a semantical variant of Benacerraf's
epistemological problem (Hodes 1984). According to our currently best
semantic theory, causal-historical connections between humans and the
world of concreta enable our words to refer to physical entities and
properties. According to platonism, mathematics refers to abstract
entities. The platonist therefore owes us a plausible account of how
we (physically embodied humans) are able to refer to them. On the face
of it, it appears that the causal theory of reference will be unable
to supply us with the required account of the 'microstructure of
reference' of mathematical discourse.


### 3.5 Plenitudinous Platonism



A version of platonism has been developed which is intended to provide
a solution to Benacerraf's epistemological problem (Linsky &
Zalta 1995; Balaguer 1998). This position is known as
*plenitudinous platonism*. The central thesis of this theory is
that every logically consistent mathematical theory
*necessarily* refers to an abstract entity. Whether the
mathematician who formulated the theory knows that it refers or does
not know this, is largely immaterial. By entertaining a consistent
mathematical theory, a mathematician automatically acquires knowledge
about the subject matter of the theory. So, on this view, there is no
epistemological problem to solve anymore.



In Balaguer's version, plenitudinous platonism postulates a
multiplicity of mathematical universes, each corresponding to a
consistent mathematical theory. Thus, in particular a question such as
the continuum problem (cf.
 section 5.1)
 does not receive a unique answer: in some set-theoretical universes
the continuum hypothesis holds, in others it fails to hold. However,
not everyone agrees that this picture can be maintained. Martin has
developed an argument to show that multiple universes can always to a
large extent be "accumulated" into a single universe
(Martin 2001).



In Linsky and Zalta's version of plenitudinous platonism, the
mathematical entity that is postulated by a consistent mathematical
theory has exactly the mathematical properties which are attributed to
it by the theory. The abstract entity corresponding to ZFC, for
instance, is *partial* in the sense that it neither makes the
continuum hypothesis true nor false. The reason is that ZFC neither
entails the continuum hypothesis nor its negation. This does not
entail that all ways of consistently extending ZFC are on a par. Some
ways may be fruitful and powerful, others less so. But the view does
deny that certain consistent ways of extending ZFC are preferable
because they consist of true principles, whereas others contain false
principles.


## 4. Structuralism and Nominalism



Benacerraf's work motivated philosophers to develop both
structuralist and nominalist theories in the philosophy of mathematics
(Reck & Price 2000). And since the late 1980s, combinations of
structuralism and nominalism have also been developed.


### 4.1 What Numbers Could Not Be



As if saddling platonism with one difficult problem were not enough
 (section 3.4),
 Benacerraf formulated a challenge for set-theoretic platonism
(Benacerraf 1965). The challenge takes the following form.



There exist infinitely many ways of identifying the natural numbers
with pure sets. Let us restrict, without essential loss of generality,
our discussion to two such ways:

 

\[\begin{align\*} \mathrm{I}{:} & \\ 0 &= \varnothing \\ 1 &= \{\varnothing\} \\ 2 &= \{\{\varnothing\}\} \\ 3 &= \{\{\{\varnothing\}\}\} \\ \vdots&\\ &\\ \mathrm{II}{:} & \\ 0 &= \varnothing \\ 1 &= \{\varnothing \} \\ 2 &= \{\varnothing , \{ \varnothing \}\}\\ 3 &= \{\varnothing , \{\varnothing \}, \{\varnothing , \{\varnothing \}\}\} \\ \vdots& \end{align\*}\]


The simple question that Benacerraf asks is:



Which of these consists solely of true identity statements: I or
II?



It seems very difficult to answer this question. It is not hard to see
how a successor function and addition and multiplication operations
can be defined on the number-candidates of I and on the
number-candidates of II so that all the arithmetical statements that
we take to be true come out true. Indeed, if this is done in the
natural way, then we arrive at *isomorphic* structures (in the
set-theoretic sense of the word), and isomorphic structures make the
same sentences true (they are *elementarily equivalent*). It is
only when we ask extra-arithmetical questions, such as '\(1 \in
3\)?' that the two accounts of the natural numbers yield
diverging answers. So it is impossible that both accounts are correct.
According to story I, \(3 = \{\{\{\varnothing \}\}\}\), whereas
according to story II, \(3 = \{\varnothing , \{\varnothing \},
\{\varnothing , \{\varnothing \}\}\}\). If both accounts were correct,
then the transitivity of identity would yield a purely set theoretic
falsehood.



Summing up, we arrive at the following situation. On the one hand,
there appear to be no reasons why one account is superior to the
other. On the other hand, the accounts cannot both be correct. This
predicament is sometimes called labelled Benacerraf's
*identification problem*.



The proper conclusion to draw from this conundrum appears to be that
neither account I nor account II is correct. Since similar
considerations would emerge from comparing other reasonable-looking
attempts to reduce natural numbers to sets, it appears that natural
numbers are not sets after all. It is clear, moreover, that a similar
argument can be formulated for the rational numbers, the real
numbers... Benacerraf concludes that they, too, are not sets at
all.



It is not at all clear whether Godel, for instance, is committed
to reducing the natural numbers to pure sets. A platonist can uphold
the claim that the natural numbers can be embedded into the
set-theoretic universe while maintaining that the embedding should not
be seen as an ontological reduction. Indeed, on Linsky and
Zalta's plenitudinous platonist account, the natural numbers
have no properties beyond those that are attributed to them by our
theory of the natural numbers (Peano Arithmetic). But then it seems
that platonists would have to take a similar line with respect to the
rational numbers, the complex numbers, .... Whereas maintaining
that the natural numbers are sui generis admittedly has some appeal,
it is perhaps less natural to maintain that the complex numbers, for
instance, are also sui generis. And, anyway, even if the natural
numbers, the complex numbers, ... are in some sense not reducible
to anything else, one may wonder if there may not be another way to
elucidate their nature.


### 4.2 *Ante Rem* Structuralism



Shapiro draws a useful distinction between *algebraic* and
*non-algebraic mathematical theories* (Shapiro 1997). Roughly,
non-algebraic theories are theories which appear at first sight to be
about a unique model: the *intended* model of the theory. We
have seen examples of such theories: arithmetic, mathematical
analysis... Algebraic theories, in contrast, do not carry a prima
facie claim to be about a unique model. Examples are group theory,
topology, graph theory...



Benacerraf's challenge can be mounted for the objects that
non-algebraic theories appear to describe. But his challenge does not
apply to algebraic theories. Algebraic theories are not interested in
mathematical objects per se; they are interested in structural aspects
of mathematical objects. This led Benacerraf to speculate whether the
same could not be true also of non-algebraic theories. Perhaps the
lesson to be drawn from Benacerraf's identification problem is
that even arithmetic does not describe specific mathematical objects,
but instead only describes structural relations?



Shapiro and Resnik hold that all mathematical theories, even
non-algebraic ones, describe *structures*. This position is
known as structuralism (Shapiro 1997; Resnik 1997). Structures
consists of places that stand in structural relations to each other.
Thus, derivatively, mathematical theories describe places or positions
in structures. But they do not describe objects. The number three, for
instance, will on this view not be an object but a place in the
structure of the natural numbers.



*Systems* are instantiations of structures. The systems that
instantiate the structure that is described by a non-algebraic theory
are isomorphic with each other, and thus, for the purposes of the
theory, equally good. The systems I and II that were described in
 section 4.1
 can be seen as instantiations of the natural number structure.
\(\{\{\{\varnothing \}\}\}\) and \(\{\varnothing , \{\varnothing \},
\{\varnothing , \{\varnothing \}\}\}\) are equally suitable for
playing the role of the number three. But neither *are* the
number three. For the number three is an open place in the natural
number structure, and this open place does not have any internal
structure. Systems typically contain structural properties over and
above those that are relevant for the structures that they are taken
to instantiate.



Sensible identity questions are those that can be asked from within a
structure. They are those questions that can be answered on the basis
of structural aspects of the structure. Identity questions that go
beyond a structure do not make sense. One can pose the question
whether \(3 \in 4\), but not cogently: this question involves a
category mistake. The question mixes two different structures: \(\in\)
is a set-theoretical notion, whereas 3 and 4 are places in the
structure of the natural numbers. This seems to constitute a
satisfactory answer to Benacerraf's challenge.



In Shapiro's view, structures are not ontologically dependent on
the existence of systems that instantiate them. Even if there were no
infinite systems to be found in Nature, the structure of the natural
numbers would exist. Thus structures as Shapiro understands them are
abstract, platonic entities. Shapiro's brand of structuralism is
often labeled *ante rem* structuralism.



In textbooks on set theory we also find a notion of structure.
Roughly, the set theoretic definition says that a structure is an
ordered \(n+1\)-tuple consisting of a set, a number of relations on
this set, and a number of distinguished elements of this set. But this
cannot be the notion of structure that structuralism in the philosophy
of mathematics has in mind. For the set theoretic notion of structure
presupposes the concept of set, which, according to structuralism,
should itself be explained in structural terms. Or, to put the point
differently, a set-theoretical structure is merely a *system*
that instantiates a structure that is ontologically prior to it.



Nonetheless, the motivation for extending ante rem structuralism even
to the most encompassing mathematical discipline (set theory) is not
entirely evident (Burgess 2015). Recall that the main motivation for
arriving at a structuralist understanding of a mathematical discipline
lies in Benacerraf's identification problem. For set theory, it
seems hard to mount an identification challenge: sets are not usually
defined in terms of more primitive concepts.



It appears that *ante rem* structuralism describes the notion
of a structure in a somewhat circular manner. A structure is described
as places that stand in relation to each other, but a place cannot be
described independently of the structure to which it belongs. Yet this
is not necessarily a problem. For the *ante rem* structuralist,
the notion of structure is a primitive concept, which cannot be
defined in other more basic terms. At best, we can construct an
axiomatic theory of mathematical structures.



But Benacerraf's epistemological problem still appears to be
urgent. Structures and places in structures may not be objects, but
they are abstract. So it is natural to wonder how we succeed in
obtaining knowledge of them. This problem has been taken by certain
philosophers as a reason for developing a nominalist theory of
mathematics and then to reconcile this theory with basic tenets of
structuralism.


### 4.3 Mathematics Without Abstract Entities



Goodman and Quine tried early on to bite the bullet: they embarked on
a project to reformulate theories from natural science without making
use of abstract entities (Goodman & Quine 1947). The nominalistic
reconstruction of scientific theories proved to be a difficult task.
Quine, for one, abandoned it after this initial attempt. In the past
decades many theories have been proposed that purport to give a
nominalistic reconstruction of mathematics. (Burgess & Rosen 1997)
contains a good critical discussion of such views.



In a nominalist reconstruction of mathematics, concrete entities will
have to play the role that abstract entities play in platonistic
accounts of mathematics, and concrete relations (such as the
part-whole relation) have to be used to simulate mathematical
relations between mathematical objects. But here problems arise.
First, already Hilbert observed that, given the discretization of
nature in quantum mechanics, the natural sciences may in the end claim
that there are only finitely many concrete entities (Hilbert 1925).
Yet it seems that we would need infinitely many of them to play the
role of the natural numbers -- never mind the real numbers. Where
does the nominalist find the required collection of concrete entities?
Secondly, even if the existence of infinitely many concrete objects is
assumed, it is not clear that even elementary mathematical theories
such as Primitive Recursive Arithmetic can be "simulated"
by means of nominalistic relations (Niebergall 2000).



Field made an earnest attempt to carry out a nominalistic
reconstruction of Newtonian mechanics (Field 1980). The basic idea is
this. Field wanted to use concrete surrogates of the real numbers and
functions on them. He adopted a realist stance toward the spatial
continuum, and took regions of space to be as physically real as
chairs and tables. And he took regions of space to be concrete (after
all, they are spatially located). If we also count the very
disconnected ones, then there are as many regions of Newtonian space
as there are subsets of the real numbers. And then there are enough
concrete entities to play the role of the natural numbers, the real
numbers, and functions on the real numbers. And the theory of the real
numbers and functions on them is all that is needed to formulate
Newtonian mechanics. Of course it would be even more interesting to
have a nominalistic reconstruction of a truly contemporary scientific
theory such as Quantum Mechanics. But given that the project can be
carried out for Newtonian mechanics, some degree of initial optimism
seems justified.



This project clearly has its limitations. It may be possible
nominalistically to interpret theories of function spaces on the real
numbers, say. But it seems far-fetched to think that along Fieldian
lines a nominalistic interpretation of set theory can be found.
Nevertheless, if it is successful within its confines, then
Field's program has really achieved something. For it would mean
that, to some extent at least, mathematical entities appear to be
dispensable after all. He would thereby have taken an important step
towards undermining the indispensability argument for Quinean modest
platonism in mathematics, for, to some extent, mathematical entities
appear to be dispensable after all.



Field's strategy only has a chance of working if Hilbert's
fear that in a very fundamental sense our best scientific theories may
entail that there are only finitely many concrete entities, is
ill-founded. If one sympathizes with Hilbert's concern but does
not believe in the existence of abstract entities, then one might bite
the bullet and claim that there are only finitely many
*mathematical* entities, thus contradicting the basic
principles of elementary arithmetic. This leads to a position that has
been called *ultra-finitism* (Essenin-Volpin 1961).



On most accounts, ultra-finitism leads, like intuitionism, to
revisionism in mathematics. For it would seem that one would then have
to say that there is a largest natural number, for instance. From the
outside, a theory postulating only a finite mathematical universe
appears proof-theoretically weak, and therefore very likely to be
consistent. But Woodin has developed an argument that purports to show
that from the ultra-finitist perspective, there are no grounds for
asserting that the ultra-finitist theory is likely to be consistent
(Woodin 2011).



Regardless of this argument (the details of which are not discussed
here), many already find the assertion that there is a largest number
hard to swallow. But Lavine has articulated a sophisticated form of
set-theoretical ultra-finitism which is mathematically non-revisionist
(Lavine 1994). He has developed a detailed account of how the
principles of ZFC can be taken to be principles that describe
determinately finite sets, if these are taken to include indefinitely
large ones.


### 4.4 *In Rebus* structuralism



Field's physicalist interpretation of arithmetic and analysis
not only undermines the Quine-Putnam indispensability argument. It
also partially provides an answer to Benacerraf's
epistemological challenge. Admittedly it is not a simple task to give
an account of how humans obtain knowledge of spacetime regions. But at
least according to many (but not all) philosophers spacetime regions
are physically real. So we are no longer required to explicate how
flesh and blood mathematicians stand in contact with non-physical
entities. But Benacerraf's identification problem remains. One
may wonder why one spacetime point or region rather than another plays
the role of the number \(\pi\), for instance.



In response to the identification problem, it seems attractive to
combine a structuralist approach with Field's nominalism. This
leads to versions of *nominalist structuralism*, which can be
outlined as follows. Let us focus on mathematical analysis. The
nominalist structuralist denies that any concrete physical system is
the unique intended interpretation of analysis. All concrete physical
systems that satisfy the basic principles of Real Analysis (RA) would
do equally well. So the content of a sentence \(\phi\) of the language
of analysis is (roughly) given by:



Every concrete system S that makes RA true, also makes \(\phi\)
true.



This entails that, as with *ante rem* structuralism, only
structural aspects are relevant to the truth or falsehood of
mathematical statements. But unlike *ante rem* structuralism,
no *abstract* structure is postulated above and beyond concrete
systems.



According to *in rebus* structuralism, no abstract structures
exist over and above the systems that instantiate them; structures
exist only *in* the systems that instantiate them. For this
reason nominalist *in rebus* structuralism is sometimes
described as "structuralism without structures".
Nominalist structuralism is a form of *in rebus* structuralism.
But *in rebus* structuralism is not exhausted by nominalist
structuralism. Even the version of platonism that takes mathematics to
be about structures in the set-theoretic sense of the word can be
viewed as a form of *in rebus* structuralism.



In mathematical discourse, non-algebraic structures (such as
'the' natural numbers) and mathematical objects (such as
'the' number 1) are referred to by definite descriptions.
This strongly suggests that mathematical symbols (N, 1) have a unique
reference rather than a 'distributed' one as *in
rebus* structuralism would have it. But *in rebus*
structuralists argue that such mathematical symbols function as
*dedicated variables* in much the same way as in 'Tommy
needs his letters from home', a world war II slogan, the name
'Tommy' is chosen to stand for some arbitrary concrete
soldier, and re-used on many occasions without changing its reference
(Pettigrew 2008).



If Hilbert's worry is wellfounded in the sense that there are no
concrete physical systems that make the postulates of mathematical
analysis true, then the above nominalist structuralist rendering of
the content of a sentence \(\phi\) of the language of analysis gets
the truth conditions of such sentences wrong. For then for
*every* universally quantified sentence \(\phi\), its
paraphrase will come out vacuously true. So an existential assumption
to the effect that there exist concrete physical systems that can
serve as a model for RA is needed to back up the above analysis of the
content of mathematical statements. Perhaps something like
Field's construction fits the bill.



Putnam noticed early on that if the above explication of the content
of mathematical sentences is modified somewhat, a substantially weaker
background assumption is sufficient to obtain the correct truth
conditions (Putnam 1967). Putnam proposed the following *modal*
rendering of the content of a sentence \(\phi\) of the language of
analysis:



*Necessarily*, every concrete system S that makes RA true, also
makes \(\phi\) true.



This is a stronger statement than the nonmodal rendering that was
presented earlier. But it seems equally plausible. And an advantage of
this rendering is that the following modal existential background
assumption is sufficient to make the truth conditions of mathematical
statements come out right:



*It is possible* that there exists a concrete physical system
that can serve as a model for RA.



('It is possible that' here means 'It is or might
have been the case that'.) Now Hilbert's concern seems
adequately addressed. For on Putnam's account, the truth of
mathematical sentences no longer depends on physical assumptions about
the actual world.



It is admittedly not easy to give a satisfying account of how we
*know* that this modal existential assumption is fulfilled. But
it may be hoped that the task is less daunting than the task of
explaining how we succeed in knowing facts about abstract entities.
And it should not be forgotten that the structuralist aspect of this
(modal) nominalist position keeps Benacerraf's identification
challenge at bay.



Putnam's strategy also has its limitations. Chihara sought to
apply Putnam's strategy not only to arithmetic and analysis but
also to set theory (Chihara 1973). Then a crude version of the
relevant modal existential assumption becomes:



*It is possible* that there exist concrete physical systems
that can serve as a model for ZFC.



Parsons has noted that when possible worlds are needed which contain
collections of physical entities that have large transfinite
cardinalities or perhaps are even too large to have a cardinal number,
it becomes hard to see these as possible concrete or physical systems
(Parsons 1990a). We seem to have no reason to believe that there could
be *physical* worlds that contain highly transfinitely many
entities.


### 4.5 Fictionalism



According to the previous proposals, the statements of ordinary
mathematics are true when suitably, i.e., nominalistically,
interpreted. The nominalistic account of mathematics that will now be
discussed holds that all existential mathematical statements are false
simply because there are no mathematical entities. (For the same
reason all universal mathematical statements will be trivially
true.)



Fictionalism holds that mathematical theories are like fiction stories
such as fairy tales and novels. Mathematical theories describe
fictional entities, in the same way that literary fiction describes
fictional characters. This position was first articulated in the
introductory chapter of (Field 1989), and has in recent years been
gaining in popularity.



This crude description of the fictionalist position immediately opens
up the question what sort of entities fictional entities are. This
appears to be a deep metaphysical ontological problem. One way to
avoid this question altogether is to deny that there exist fictional
entities. Mathematical theories should be viewed as invitations to
participate in games of pretence, in which we act as if certain
mathematical entities exist. Pretence or make-believe operators shield
their propositional objects from existential exportation (Leng
2010).



Anyway, as said above, on the fictionalist view, a mathematical theory
isn't literally true. Nonetheless, mathematics is used to get
truths across. So we must *subtract* something from what is
literally said when we assert a physical theory that involves
mathematics, if we want to get at the truth. But this requires a
*theory* of how this subtraction of content works. Such a
theory has been developed in (Yablo, 2014).



If the fictionalist thesis is correct, then one demand that must be
imposed on mathematical theories is surely consistency. Yet Field adds
to this a second requirement: mathematics must be
*conservative* over natural science. This means, roughly, that
whenever a statement of an empirical theory can be derived using
mathematics, it can in principle also be derived without using any
mathematical theories. If this were not the case, then an
indispensability argument could be played out against fictionalism.
Whether mathematics is in fact conservative over physics, for
instance, is currently a matter of controversy. Shapiro has formulated
an incompleteness argument that intends to refute Field's claim
(Shapiro 1983).



If there are indeed no mathematical (fictional) entities, as one form
of fictionalism has it, then Benacerraf's epistemological
problem does not arise. Fictionalism then shares this advantage over
most forms of platonism with nominalistic reconstructions of
mathematics. But the appeal to pretence operators entails that the
logical form of mathematical sentences then differs somewhat from
their surface form. If there are fictional objects, then the surface
form of mathematical sentences can be taken to coincide with their
logical form. But if they exist as abstract entities, then
Benacerraf's epistemological problem reappears.



Whether Benacerraf's identification problem is solved is not
completely clear. In general, fictionalism is a non-reductionist
account. Whether an entity in one mathematical theory is identical
with an entity that occurs in another theory is usually left
indeterminate by mathematical "stories". Yet Burgess has
rightly emphasized that mathematics differs from literary fiction in
the fact that fictional characters are usually confined to one work of
fiction, whereas the same mathematical entities turn up in diverse
mathematical theories (Burgess 2004). After all, entities with the
same *name* (such as \(\pi)\) turn up in different theories.
Perhaps the fictionalist can maintain that when mathematicians develop
a new theory in which an "old" mathematical entity occurs,
the entity in question is made more precise. More determinate
properties are ascribed to it than before, and this is all right as
long as overall consistency is maintained.



The canonical objection to formalism seems also applicable to
fictionalism. The fictionalists should find some explanation of the
fact that extending a mathematical theory in one way, is often
considered preferable over continuing it in a another way that is
incompatible with the first. There is often at least an appearance
that there is a right way to extend a mathematical theory.


## 5. Special Topics



In recent years, subdisciplines of the philosophy of mathematics have
started to arise. They evolve in a way that is not completely
determined by the "big debates" about the nature of
mathematics. In this section, we look at a few of these
disciplines.


### 5.1 Foundations and Set Theory



Many regard set theory as in some sense the foundation of mathematics.
It seems that just about any piece of mathematics can be carried out
in set theory, even though it is sometimes an awkward setting for
doing so. In recent years, the philosophy of set theory is emerging as
a philosophical discipline of its own. This is not to say that in
specific debates in the philosophy of set theory it cannot make an
enormous difference whether one approaches it from a formalistic point
of view or from a platonistic point of view, for instance.



The thesis that set theory is most suitable for serving as the
foundations of mathematics is by no means uncontroversial. Over the
past decades, *category theory* has presented itself as a rival
for this role. Category theory is a mathematical theory that was
developed in the middle of the twentieth century. Unlike in set
theory, in category theory mathematical objects are *only*
defined up to isomorphism. This means that Benacerraf's
identification problem cannot be raised for category theoretical
concepts and 'objects'. At the same time, (roughly)
everything that can be done in set theory can be done in category
theory (but not always in a natural manner), and vice versa (again not
always in a natural manner). This means that for a structuralist
perspective, category theory is an attractive candidate for providing
the foundations of mathematics (McLarty 2004).



One question that has been important from the beginning of set theory
concerns the difference between sets and proper classes. (This
question has a natural counterpart for category theory: the difference
between small and large categories.) Cantor's diagonal argument
forces us to recognize that the set-theoretical universe as a whole
cannot be regarded as a set. Cantor's Theorem shows that the
power set (i.e., the set of all subsets) of any given set has a larger
cardinality than the given set itself. Now suppose that the
set-theoretical universe forms a set: the set of all sets. Then the
power set of the set of all sets would have to be a subset of the set
of all sets. This would contradict the fact that the power set of the
set of all sets would have a larger cardinality than the set of all
sets. So we must conclude that the set-theoretical universe cannot
form a set.



Cantor called pluralities that are too large to be considered as a set
*inconsistent multiplicities* (Cantor 1932). Today,
Cantor's inconsistent multiplicities are called *proper
classes*. Some philosophers of mathematics hold that proper
classes still constitute unities, and hence can be seen as a sort of
collection. They are, in a Cantorian spirit, just collections that are
too large to be sets. Nevertheless, there are problems with this view.
Just as there can be no set of all sets, there can for diagonalization
reasons also not be a proper class of all proper classes. So the
proper class view seems compelled to recognize in addition a realm of
super-proper classes, and so on. For this reason, Zermelo claimed that
proper classes simply do not exist. This position is less strange than
it looks at first sight. On close inspection, one sees that in ZFC one
never needs to quantify over entities that are too large to be sets
(although there exist systems of set theory that do quantify over
proper classes). On this view, the set-theoretical universe is
potentially infinite in an absolute sense of the word. It never exists
as a completed whole, but is forever growing, and hence forever
unfinished (Zermelo 1930). This way of speaking indicates that in our
attempts to understand this notion of potential infinity, we are drawn
to temporal metaphors. It is not surprising that these temporal
metaphors cause some philosophers of mathematics acute discomfort. For
this reason, contemporary philosophers of mathematics who are
sympathetic to Zermelo's potentialist interpretation of the set
theoretic universe, tend to regard the modality involved in this
interpretation as a non-temporal one: the nature of this modality is
hotly debated (Linnebo 2013, Studd 2019).



A second subject in the philosophy of set theory concerns the
justification of the accepted basic principles of mathematics, i.e.,
the axioms of ZFC. An important historical case study is the process
by which the Axiom of Choice came to be accepted by the mathematical
community in the early decades of the twentieth century (Moore 1982).
The importance of this case study is largely due to the fact that an
open and explicit discussion of its acceptability was held in the
mathematical community. In this discussion, general reasons for
accepting or refusing to accept a principle as a basic axiom came to
the surface. On the systematic side, two conceptions of the notion of
set have been elaborated which aim to justify all axioms of ZFC in one
fell swoop. On the one hand, there is the *iterative
conception* of sets, which describes how the set-theoretical
universe can be thought of as generated from the empty set by means of
the power set operation (Boolos 1971, Linnebo 2013). On the other
hand, there is the *limitation of size* conception of sets,
which states that every collection which is not too big to be a set,
is a set (Hallett 1984). The iterative conception motivates some
axioms of ZFC very well (the power set axiom, for instance), but fares
less well with respect to other axioms, such as the replacement axiom
(Potter 2004, Part IV). The limitation of size conception motivates
other axioms better (such as the restricted comprehension axiom). It
seems fair to say that there is no *uniform* conception that
clearly justifies all axioms of ZFC.



The motivation of putative axioms that go beyond ZFC constitutes a
third concern of the philosophy of set theory (Maddy 1988; Martin
1998). One such class of principles is constituted by the *large
cardinal axioms*. Nowadays, large cardinal hypotheses are really
taken to mean some kind of embedding properties between the set
theoretic universe and inner models of set theory (Kanamori 2009).
Most of the time, large cardinal principles entail the existence of
sets that are larger than any sets which can be guaranteed by ZFC to
exist.



The weaker of the large cardinal principles are supported by intrinsic
evidence (see
 section 3.1).
 They follow from what are called *reflection principles*.
These are principles that state that the set theoretic universe as a
whole is so rich that it is very similar to some set-sized initial
segment of it. The stronger of the large cardinal principles hitherto
only enjoy extrinsic support. Many researchers are skeptical about the
possibility that reflection principles, for instance, can be found
that support them (Koellner 2009); others, however, disagree (Welch
& Horsten 2016).



Godel hoped that on the basis of such large cardinal axioms, the
most important open question of set theory could eventually be
settled. This is the *continuum problem*. The *continuum
hypothesis* was proposed by Cantor in the late nineteenth century.
It states that there are no sets S which are too large for there to be
a one-to-one correspondence between S and the natural numbers, but too
small for there to exist a one-to-one correspondence between S and the
real numbers. Despite strenuous efforts, all attempts to settle the
continuum problem failed. Godel came to suspect that the
continuum hypothesis is independent of the accepted principles of set
theory (ZFC). Around 1940, he managed to show that the continuum
hypothesis is consistent with ZFC. A few decades later, Paul Cohen
proved that the negation of the continuum hypothesis is also
consistent with ZFC. Thus Godel's conjecture of the
independence of the continuum hypothesis was eventually confirmed.



But Godel's hope that large cardinal axioms could solve the
continuum problem turned out to be unfounded. The continuum hypothesis
is independent of ZFC even in the context of large cardinal axioms.
Nevertheless, large cardinal principles have manage to settle
restricted versions of the continuum hypothesis (in the affirmative).
The existence of so-called Woodin cardinals ensures that sets
definable in analysis are either countable or the size of the
continuum. Thus the *definable continuum problem* is
settled.



In recent years, attempts have been focused on finding principles of a
different kind which might be justifiable and which might yet decide
the continuum hypothesis (Woodin 2001a, Woodin 2001b). One of the more
general philosophical questions that have emerged from this research
is the following: which conditions have to be satisfied in order for a
principle to be a putative basic axiom of mathematics?



Some of the researchers who seek to decide the continuum hypothesis
think that it is true; others think that it is false. But there are
also many set theorists and philosophers of mathematics who believe
that the continuum hypothesis not just undecidable in ZFC but
*absolutely undecidable*, i.e. that it is neither provable (in
the informal sense of the word) nor disprovable (in the informal sense
of the word) because it is neither true nor false. If the mathematical
universe is a *set theoretic multiverse*, for instance, then
there are equally models that make the continuum hypothesis true and
equally good models that make it false, and there is no more to be
said (Hamkins, 2015).


### 5.2 Categoricity and Pluralism



In the second half of the nineteenth century Dedekind proved that the
basic axioms of arithmetic have, up to isomorphism, exactly one model,
and that the same holds for the basic axioms of Real Analysis. If a
theory has, up to isomorphism, exactly one model, then it is said to
be *categorical*. So modulo isomorphisms, arithmetic and
analysis each have exactly one intended model. Half a century later
Zermelo proved that the principles of set theory are
"almost" categorical or *quasi-categorical*: for
any two models \(M\_1\) and \(M\_2\) of the principles of set theory,
either \(M\_1\) is isomorphic to \(M\_2\), or \(M\_1\) is isomorphic to a
strongly inaccessible rank of \(M\_2\), or \(M\_2\) is isomorphic to a
strongly inaccessible rank of \(M\_1\) (Zermelo 1930). In recent years,
attempts have been made to develop arguments to the effect that
Zermelo's conclusion can be strengthened to a full categoricity
assertion (McGee 1997; Martin 2001), but we will not discuss these
arguments here.



At the same time, the Lowenheim-Skolem theorem says that every
first-order formal theory that has at least one model with an infinite
domain, must have models with domains of all infinite cardinalities.
Since the principles of arithmetic, analysis and set theory had better
possess at least one infinite model, the Lowenheim-Skolem theorem
appears to apply to them. Is this not in tension with Dedekind's
categoricity theorems?



The solution of this conundrum lies in the fact that Dedekind did not
even implicitly work with first-order formalizations of the basic
principles of arithmetic and analysis. Instead, he informally worked
with *second-order* formalizations.



Let us focus on arithmetic to see what this amounts to. The basic
postulates of arithmetic contain the induction axiom. In first-order
formalizations of arithmetic, this is formulated as a scheme: for each
first-order arithmetical formula of the language of arithmetic with
one free variable, one instance of the induction principle is included
in the formalization of arithmetic. Elementary cardinality
considerations reveal that there are infinitely many properties of
natural numbers that are not expressed by a first-order formula. But
intuitively, it seems that the induction principle holds for
*all* properties of natural numbers. So in a first-order
language, the full force of the principle of mathematical induction
cannot be expressed. For this reason, a number of philosophers of
mathematics insist that the postulates of arithmetic should be
formulated in a *second-order language* (Shapiro 1991).
Second-order languages contain not just first-order quantifiers that
range over elements of the domain, but also second-order quantifiers
that range over properties (or subsets) of the domain. In
*full* second-order logic, it is insisted that these
second-order quantifiers range over *all* subsets of the
domain. If the principles of arithmetic are formulated in a
second-order language, then Dedekind's argument goes through and
we have a categorical theory. For similar reasons, we also obtain a
categorical theory if we formulate the basic principles of real
analysis in a second-order language, and the second-order formulation
of set theory turns out to be quasi-categorical.



*Ante rem* structuralism, as well as the modal nominalist
structuralist interpretation of mathematics, could benefit from a
second-order formulation. If the *ante rem* structuralist wants
to insists that the natural number structure is fixed up to
isomorphism by the Peano axioms, then she will want to formulate the
Peano axioms in second-order logic. And the modal nominalist
structuralist will want to insist that the relevant concrete systems
for arithmetic are those that make the *second-order* Peano
axioms true (Hellman 1989). Similarly for real analysis and set
theory. Thus the appeal to second-order logic appears as the final
step in the structuralist project of isolating the intended models of
mathematics.



Yet appeal to second-order logic in the philosophy of mathematics is
by no means uncontroversial. A first objection is that the ontological
commitment of second-order logic is higher than the ontological
commitment of first-order logic. After all, use of second-order logic
seems to commit us to the existence of abstract objects: classes. In
response to this problem, Boolos has articulated an interpretation of
second-order logic which avoids this commitment to abstract entities
(Boolos 1985). His interpretation spells out the truth clauses for the
second-order quantifiers in terms of plural expressions, without
invoking classes. For instance, an second-order expression of the form
\(\exists x F(x)\) is interpreted as: "there are some
(*first-order* objects) *x* such that they
have the property *F*". This interpretation is
called the *plural interpretation* of second-order logic. It is
controversial whether there is a real difference between the
mathematical use of pluralities and of sets (Linnebo 2003).
Nevertheless it is clear that an appeal to the plural interpretation
of second-order logic will be tempting for nominalist versions of
structuralism.



A second objection against second-order logic can be traced back to
Quine (Quine 1970). This objection states that the interpretation of
full second-order logic is connected with set-theoretical questions.
This is already indicated by the fact that most regimentations of
second-order logic adopt a version of the axiom of choice as one of
its axioms. But more worrisome is the fact that second-order logic is
inextricably intertwined with deep problems in set theory, such as the
continuum hypothesis. For theories such as arithmetic that intend to
describe an infinite collection of objects, even a matter as
elementary as the question of the cardinality of the range of the
second-order quantifiers, is equivalent to the continuum problem.
Also, it turns out that there exists a sentence which is a
second-order logical truth if and only if the continuum hypothesis
holds (Boolos 1975). We have seen that the continuum problem is
independent of the currently accepted principles of set theory. And
many researchers believe it to be absolutely truth-valueless. If this
is so, then there is an inherent indeterminacy in the very notion of
second-order infinite model. And many contemporary philosophers of
mathematics take the latter not to have a determinate truth value.
Thus, it is argued, the very notion of an (infinite) model of full
second-order logic is inherently indeterminate.



If one does not want to appeal to full second-order logic, then there
are other ways to ensure categoricity of mathematical theories. One
idea would be to make use of quantifiers which are somehow
intermediate between first-order and second-order quantifiers. For
instance, one might treat "there are finitely many *x*"
as a primitive quantifier. This will allow one
to, for instance, construct a categorical axiomatization of
arithmetic.



But ensuring categoricity of mathematical theories does not require
introducing stronger quantifiers. Another option would be to take the
informal concept of algorithmic computability as a primitive notion
(Halbach & Horsten 2005; Horsten 2012). A theorem of Tennenbaum
states that all first-order models of Peano Arithmetic in which
addition and multiplication are computable functions, are isomorphic
to each other. Now *our* operations of addition and
multiplication are computable: otherwise we could never have learned
these operations. This, then, is another way in which we may be able
to isolate the intended models of our principles of arithmetic.
Against this account, however, it may be pointed out that it seems
that the categoricity of intended models for real analysis, for
instance, cannot be ensured in this manner. For computation on models
of the principles of real analysis, we do not have a theorem that
plays the role of Tennenbaum's theorem.



If one accepts a certain open-endedness of the collection of
arithmetical predicates, then a categoricity theorem of sorts for
arithmetic can be obtained without overstepping the bounds of
first-order logic and without appealing to an informal concept of
computability. Suppose that there are two mathematicians, A and B, who
both assert the first-order Peano-axioms in their own idiolect.
Suppose furthermore that A and B regard the collection of predicates
for which mathematical induction is permissible as open-ended, and are
both willing to accept the other's induction scheme as true.
Then A and B have the wherewithal to convince themselves that both
idiolects describe isomorphic structures (Parsons 1990b). Such
arguments are called internal categoricity arguments. They are widely
debated in contempory philosophy of mathematics: see for instance
(Button & Walsh 2019).



Many of those who are sceptical of the philosophical use of
categoricity argments in the philosophy of mathematics take all of our
consistent mathematical theories to have many structurally different
models, and take all or many of those models to be on a par with one
another. As we saw in the previous sub-section, the set theoretic
multiverse view is a case in point, and so is set theoretic
potentialism. But one can go further, and defend the thesis that any
consistent mathematical theory describes a free-standing mathematical
universe, and that no such theory is more true than any other (Linsky
& Zalta 1995, Bueno 2011).



These theories belong to a family of views that is called
*mathematical pluralism*, which is an increasingly prominent
theme in the philosophy of mathematics. Historically, this
constellation of views has roots in the work of Hilbert and of Carnap.
In a debate with Frege, Hilbert insisted that consistency suffices for
a mathematical theory to have a subject matter (Resnik 1974); Carnap
argued that choice between alternative large-scale theories
(frameworks) is ultimately never more than a *pragmatic* matter
(Carnap 1950).



As is everywhere the case in philosophy, there is disagreement here:
for a critique of the doctrine that mathematical truth is an
irrevocably use-relative notion, see (Koellner 2009b), and for a
retort, see (Warren 2015). Some react to mathematical pluralism by
taking it one step further still, and argue that also all inconsistent
mathematical theories should be regarded as true (in a relativised
sense). Moreover, some mathematical theories that are trivial in the
sense of being inconsistent, are commonly taken to be just as
*valuable* as many venerable consistent ones:
"Historically, there are three [to the author's knowledge]
mathematical theories which had a profound impact on mathematics and
logic, and were found to be trivial. There are Cantor's naive
set theory, Frege's formal theory of logic and the first version
of Church's formal theory of mathematical logic. All three had
profound reprecussions on subsequent mathematics" (Friend 2013,
p. 294).


### 5.3 Computation



Until fairly recently, the subject of computation did not receive much
attention in the philosophy of mathematics. This may be due in part to
the fact that in Hilbert-style axiomatizations of number theory,
computation is reduced to proof in Peano Arithmetic. But this
situation has changed in recent years. It seems that along with the
increased importance of computation in mathematical practice,
philosophical reflections on the notion of computation will occupy a
more prominent place in the philosophy of mathematics in the years to
come.



Church's Thesis occupies a central place in computability
theory. It says that every algorithmically computable function on the
natural numbers can be computed by a Turing machine.



As a principle, Church's Thesis has a somewhat curious status.
It appears to be a *basic* principle. On the one hand, the
principle is almost universally held to be true. On the other hand, it
is hard to see how it can be mathematically proved. The reason is that
its antecedent contains an informal notion (algorithmic computability)
whereas its consequent contains a purely mathematical notion (Turing
machine computability). Mathematical proofs can only connect purely
mathematical notions--or so it seems. The received view was that
our evidence for Church's Thesis is quasi-empirical. Attempts to
find convincing counterexamples to Church's Thesis have come to
naught. Independently, various proposals have been made to
mathematically capture the algorithmically computable functions on the
natural numbers. Instead of Turing machine computability, the notions
of general recursiveness, Herbrand-Godel computability,
lambda-definability... have been proposed. But these mathematical
notions all turn out to be equivalent. Thus, to use Godelian
terminology, we have accumulated extrinsic evidence for the truth of
Church's Thesis.



Kreisel pointed out long ago that even if a thesis cannot be formally
proved, it may still be possible to obtain intrinsic evidence for it
from a rigorous but informal analysis of intuitive notions (Kreisel
1967). Kreisel calls these exercises in *informal rigour*.
Detailed scholarship by Sieg revealed that the seminal article (Turing
1936) constitutes an exquisite example of just this sort of analysis
of the intuitive concept of algorithmic computability (Sieg 1994).



Currently, the most active subjects of investigation in the domain of
foundations and philosophy of computation appear to be the following.
First, energy has been invested in developing theories of algorithmic
computation on structures other than the natural numbers. In
particular, efforts have been made to obtain analogues of
Church's Thesis for algorithmic computation on various
structures. In this context, substantial progress has been made in
recent decades in developing a theory of effective computation on the
real numbers (Pour-El 1999). Second, attempts have been made to
explicate notions of computability other than algorithmic
computability by humans. One area of particular interest here is the
area of *quantum computation* (Deutsch *et al*.
2000).


### 5.4 Mathematical Proof



We know much about the concepts of *formal proof* and
*formal provability*, their connection with algorithmic
computability, and the principles by which these concepts are
governed. We know, for instance, that the proofs of a formal system
are computably enumerable, and that provability in a sound (strong
enough) formal system is subject to Godel's incompleteness
theorems. But a mathematical proof as you find it in a mathematical
journal is not a formal proof in the sense of the logicians: it is a
(rigorous) *informal proof* (Myhill 1960, Detlefsen 1992,
Antonutti 2010).



First, whereas the collection of sentences provable in a formal system
is always computably enumerable, we know much less about the
*extension* of the concept of informal provability. Lucas
(Lucas 1961), and later Penrose (Penrose 1989, 1994), have argued that
informal mathematical provability outstrips provability in any given
formal system. But their arguments are widely regarded as
unpersuasive. Benacerraf has argued against Lucas and Penrose that it
cannot be excluded that there is a formal system \(T\) such that in fact
mathematical provability extensionally coincides with provability in
\(T\), even though we cannot know that it does (Benacerraf 1967). Others
have argued that the concept of informal mathematical provability is
not even clear enough for the question whether its extension is
computably enumerable to have a definite answer (Horsten & Welch
2016).



Second, there is no agreement about what the *standard* is for
an argument to qualify as a mathematical proof. According to what may
be called the received view, a mathematical argument for a statement \(p\)
constitutes an informal mathematical proof if the argument allows a
competent mathematician to transform it into a *formal
deduction* of \(p\) from generally accepted mathematical axioms
(Avigad 2021). An informal mathematical proof can then be taken to be
a *derivation-indicator* for \(p\) (Azzouni 2004). But the received
view of the standard of mathematical proof has come under attack in
recent years. It has been argued, for instance, that the
interpolations of reasons in an informal mathematical proof until a
logically correct and non-elliptical first-order derivation is
reached, can be an *infinite* process (Rav 1999, p.14-15).
Others are mounting a defence of the received view, so that there is a
lively debate about these issues at the moment (Tatton-Brown forthcoming,
Di Toffoli 2021).



The past decades have witnessed the first occurrences of mathematical
proofs in which computers appear to play an essential role. The
four-colour theorem is one example. It says that for every map, only
four colours are needed to colour countries in such a way that no two
countries that have a common border receive the same color. This
theorem was proved in 1976 (Appel et al. 1977). But the proof
distinguishes many cases which were verified by a computer. These
computer verifications are too long to be double-checked by humans.
The proof of the four colour theorem gave rise to a debate about the
question to what extent computer-assisted proofs count as proofs in
the true sense of the word.



The received view has it that mathematical proofs yield a priori
knowledge. Yet when we rely on a computer to generate part of a proof,
we appear to rely on the proper functioning of computer hardware and
on the correctness of a computer program. These appear to be empirical
factors. Thus one is tempted to conclude that computer proofs yield
*quasi-empirical* knowledge (Tymoczko 1979). In other words,
through the advent of computer proofs the notion of proof has lost its
purely a priori character. Burge, in contrast, held the view that
because the empirical factors on which we rely when we accept computer
proofs do not appear as premises in the argument, computer proofs can
yield a priori knowledge after all (Burge 1998). (Burge later
retracted this claim: see (Burge 2013, p.31).)


## 6. The Future



In the twentieth century, research in the philosophy of mathematics
revolved mostly around the nature of mathematical objects, the
fundamental laws that govern them, and how we acquire mathematical
knowledge about them. These are *foundational concerns* that
are intimately connected with traditional metaphysical and
epistemological questions.



In the second half of the twentieth century, research in the
philosophy of science to a significant extent moved away from
foundational concerns. Instead, philosophical questions relating to
the growth of scientific knowledge and of scientific understanding
became more central. As early as the 1970s, there were voices that
argued that a similar shift of attention should take place in the
philosophy of mathematics. Lakatos initiated the philosophical
*investigation of the evolution of mathematical concepts*
(Lakatos 1976). He argued that the content of a mathematical concept
evolves in roughly the following way. A mathematician formulates a
deep conjecture, but is unable to prove it. Then counterexamples
against the conjecture are found. In response, the definition of one
or more central concepts in the conjecture is changed in such a way
that the counterexamples are at least eliminated. Still the thus
revised conjecture cannot be proved, and gradually new counterexamples
appear. The procedure of revising the definition of one or more
central concepts is applied again and again, until a proof of the
conjecture is found. Lakatos calls this procedure *concept
stretchin*g. In recent decades, Lakatos' model of concept
change in mathematics has been revised and refined (Mormann 2002).



For some decades, the view that the philosophy of mathematics should
take a historical and sociological turn remained restricted to a
somewhat marginal school of thought in the philosophy of mathematics.
However, in recent years the opposition between this new movement of
mathematical practice on the one hand, and 'mainstream'
philosophy of mathematics on the other hand, is softening.
Philosophical questions relating to mathematical practice, the
evolution of mathematical theories, and mathematical explanation and
understanding have become more prominent, and have been related to
more traditional themes from the philosophy of mathematics (Mancosu
2008). This trend will doubtlessly continue in the years to come.



For an example, let us briefy return to the subject of computer proofs
(see
 section 5.3).
 The source of the discomfort that mathematicians experience when
confronted with computer proofs appears to be the following. A
"good" mathematical proof should do more than to convince
us that a certain statement is true. It should also explain
*why* the statement in question holds. And this is done by
referring to deep relations between deep mathematical concepts that
often link different mathematical domains (Manders 1989). Until now,
computer proofs typically only employ fairly low level mathematical
concepts. They are notoriously weak at developing deep concepts on
their own, and have difficulties with linking concepts in from
different mathematical fields. All this leads us to a philosophical
question which is just now beginning to receive the attention that it
deserves: what is mathematical understanding?