Philosophy of
Mathematics
First published Tue 25 Sep, 2007
If
mathematics is regarded as a science, then the philosophy of mathematics can be
regarded as a branch of the philosophy of science, next to disciplines such as
the philosophy of physics and the philosophy of biology. However, because of
its subject matter, the philosophy of mathematics occupies a special place in
the philosophy of science. Whereas the natural sciences investigate entities
that are located in space and time, it is not at all obvious that this is also
the case with respect to the objects that are studied in mathematics. In
addition to that, the methods of investigation of mathematics differ markedly
from the methods of investigation in the natural sciences. Whereas the latter
acquire general knowledge using inductive methods, mathematical knowledge
appears to be acquired in a different way, namely, by deduction from basic
principles. The status of mathematical knowledge also appears to differ from
the status of knowledge in the natural sciences. The theories of the natural
sciences appear to be less certain and more open to revision than mathematical
theories. For these reasons mathematics poses problems of a quite distinctive
kind for philosophy. Therefore philosophers have accorded special attention to
ontological and epistemological questions concerning 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 mathematical logicians 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 highlypraised 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
nonplatonistic 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 amply
deserves a place beside the three traditional schools.
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 antiplatonistic 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 (secondorder) Peano
arithmetic from the basic laws of a system of secondorder 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:
{xFx}={xGx} ≡ ∀x(Fx ≡ Gx),
In words:
the set of the Fs
is identical with the set of the Gs
iff the Fs
are precisely the Gs.
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 both 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 both
ground objects and classes of ground objects, and so on.
Unfortunately,
Russell found that the principles of his typed logic did not suffice to deduce
even the basic laws of arithmetic. Russell 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 secondorder Peano Arithmetic can be broken down into 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 Fs
= the number of the Gs
≡ F≈G,
where F≈G means that the Fs and the Gs stand in
onetoone correspondence with each other. (This relation of onetoone
correspondence can be expressed in secondorder logic.) Then, in a second
stage, the principles of secondorder Peano Arithmetic are derived from Hume's
Principle and the accepted principles of secondorder 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 secondorder Peano arithmetic is reducible
to logic alone. Thus a new form of logicism was born; today this view is known
as neologicism
(Hale & Wright 2001).
Most
philosophers of mathematics today doubt that Hume's Principle is a principle of
logic. Indeed, even Wright has in recent years sought to qualify this claim.
Nevertheless, 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).
Intuitionism
originates in the work of the mathematician L.E.J. Brouwer (van Atten 2004).
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., abstracted from
contingent, physical limitations of the reallife 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. (An exception is made by Brouwer for our intuition of the real
line.) This entails that intuitionism to a large extent rejects the existence
of the actual (or completed) infinite; mostly 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, intuitionists infer to
a revisionist stance in logic and mathematics. They find nonconstructive
existence proofs unacceptable. Nonconstructive 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 nonconstructive existence
proofs as ‘theological’ and ‘metaphysical’. The characteristic feature of
nonconstructive existence proofs is that they make essential use of the
principle of excluded third,
φ ∨ ¬φ,
or one of
its equivalents, such as the principle of double negation,
¬¬φ → φ.
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).
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 abstract entities,
but 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 higherorder 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 in fact 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 a statement would have to be proved in a
‘safe’ part of mathematics, such as 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
Gödel proved that there exist arithmetical statements that are undecidable in
Peano Arithmetic (Gödel 1931). This has become known as Gödel'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, Gödel then quickly realized that, unless
(God forbid!) Peano Arithmetic is inconsistent, the consistency of Peano
Arithmetic is independent of Peano Arithmetic. This is Gödel's second
incompleteness theorem. Gödel'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 (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 Gödel'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 (1987). He
defends the view that in some sense, Peano Arithmetic may be complete after
all. He argues that true sentences undecidable in Peano Arithmetic can only be
proved by means of higherorder 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 settheoretic, and
hence nonarithmetical, concept. If the only ways of proving the consistency of
arithmetic make essential use of notions which arguably belong to higherorder
mathematics, then the consistency of arithmetic, even though it can be
expressed in the language of Peano Arithmetic, is a nonarithmetical 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 Poincaré, he arrived at
the following diagnosis of the Russell paradox. To state the Russell paradox,
the collection C of all mathematical entities that satisfy ¬x∈ x
is defined. The paradox then proceeds by asking whether C itself meets this
condition, and derives a contradiction. The PoincaréRussell diagnosis of the
paradox states that the definition of C 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 which exist independently
from the defined collection. Such definitions are called predicative. As Gödel
later pointed out, a convinced 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. All this led Russell to develop the
simple and the ramified theory of types, in which syntactical restrictions were
built in which make impredicative definitions illformed. In simple type
theory, the free variables in defining formulas range over entities to which
the collection to be defined does not belong. In ramified type theory, it is
required, in addition, that the range of the bound variables in defining
formulas 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
ramified type theory is unsuitable to formalize ordinary mathematical
arguments.
When Russell
turned to other areas of analytical philosophy, Hermann Weyl took up the
predicativist cause (Weyl 1918). Like Poincaré, 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 as an actual infinity. But the concept of arbitrary
subset of the natural numbers was not taken to be immediately given in
mathematical intuition. Only those subsets which are determined by arithmetical
firstorder predicates are taken to be 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, the attention was gradually shifted away from
concern about the settheoretical paradoxes to the role of impredicativity in
mainstream mathematics. On the other hand, Weyl showed that it is often
possible to work around impredicative notions. It even emerged that most of
mainstream nineteenthcentury mathematical analysis could 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 wellordering of the natural numbers.
This calibration of the strength of predicative mathematics, which is due to
Feferman and (independently) Schütte, is nowadays fairly generally accepted.
Feferman then investigated how much of standard mathematical analysis could be
carried out within a predicativist framework. The research of Feferman and
others (most notably Harvey Friedman) shows that most of twentiethcentury
analysis is acceptable from a predicativist point of view.
3. Platonism
In the years
before the second world war it became clear that weighty objections had been
raised against each of the three antiplatonist programs in the philosophy of
mathematics. Predicativism was 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
Gödel's Platonism
Gödel was a
platonist with respect to mathematical objects and with respect to mathematical
concepts (Gödel 1944, 1964). But his platonistic view was more sophisticated
than that of the mathematician in the street.
Gödel 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 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 quasiperceptual 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
foolproof — 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 ZermeloFraenkel set theory
with the Axiom of Choice (ZFC). In Gödel'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 Gödel'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. Gödel says that “success here means fruitfulness in
consequences, particularly in ‘verifiable’ consequences, i.e., consequences
verifiable without the new axiom, whose proofs 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 wellestablished physical theory”
(Gödel 1947, 477). This inspired Gödel 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).
Gödel
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 settheoretical propositions have
determinate truth values. There are versions of platonism that maintain, for
instance, that all theorems of ZFC are made true by determinate settheoretical
facts, but that there are no settheoretical facts that make certain statements
that are highly independent of ZFC truthdeterminate. It seems that the famous
set theorist Paul Cohen held some such view (Cohen 1971).
3.2 Naturalism and Indispensability
Quine
articulated 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).
Since Galileo, 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 Gödel'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
roundingoff 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).
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
largecardinal axioms can be taken to be true. Thus her view is closer to that
of Gödel than to that of Quine. In her more recent work, she isolates two
maxims that seem to be guiding set theorists when contemplating the
acceptability of new settheoretic 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 all structures of mathematics can be instantiated
or modelled. The maxim “maximize” means that set theory should adopt
settheoretic 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 nonplatonistic 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). Tait has worked out
in detail how something like this distinction can be applied to mathematics
(Tait 2005). This has resulted in what might be regarded as a deflationary
version 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 supramathematical 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 Gödelian 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 Gödel.
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
fleshandblood 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 theory of reference, causalhistorical
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, a
question such as the continuum problem does not receive a unique answer: in
some settheoretical 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 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.
As if saddling
platonism with one difficult problem were not enough (Section
3.4), Benacerraf formulated a challenge for settheoretic 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:
I:
0

=

Ø

1

=

{Ø}

2

=

{{Ø}}

3

=

{{{Ø}}}

…



II:
0

=

Ø

1

=

{Ø}

2

=

{Ø, { Ø}}

3

=

{Ø, {Ø}, {Ø, {Ø}}}

…



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
numbercandidates of I and on the numbercandidates 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
settheoretic sense of the word), and isomorphic structures make the same
sentences true (they are elementarily
equivalent). It is only when we ask extraarithmetical questions,
such as 1 ∈ 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 = {{{Ø}}}, whereas according to story II, 3 = {Ø, {Ø},
{Ø, {Ø}}}. If both accounts were correct, then the transitivity of identity
would yield a purely settheoretic 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 reasonablelooking 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 Gödel, for instance, is committed to reducing the natural
numbers to pure sets. It seems that a platonist should be able to uphold the
claim that the natural numbers can be embedded into the settheoretic universe
while maintaining that the embedding should not be seen as an ontological
reduction. Indeed, we have seen that 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 whether there is another way to
elucidate their nature.
Shapiro
draws a useful distinction between algebraic
and nonalgebraic
mathematical theories (Shapiro 1997). Roughly, nonalgebraic
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 nonalgebraic 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 nonalgebraic 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 nonalgebraic ones,
describe structures.
This position is known as structuralism
(Shapiro 1997; Resnik 1997). Structures consist 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 3, 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 nonalgebraic 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 naturalnumber structure. The sets
{{{Ø}}} and {Ø, {Ø}, {Ø, {Ø}}} are equally suitable for playing the role of the
number three. But neither one is
the number 3. For the number 3 is an open place in the naturalnumber
structure, and this open place does not have any internal structure. Systems
typically contain 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 ∈ 4, but not cogently: this question involves a category mistake. The
question mixes two different structures: ∈ is a settheoretical 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
settheoretic definition says that a structure is an ordered ntuple 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
settheoretic 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 settheoretical structure is merely a system that
instantiates a structure that is ontologically prior to it.
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.
Goodman
and Quine tried early on to bite the bullet and 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. But here a
problem arises. 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?
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 the functions on them. He adopted a realist
stance toward the spatial continuum. He 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. In this way 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 to nominalistically
interpret theories of function spaces on the reals, say. But it seems
farfetched 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 he
has thereby taken an important step towards undermining the indispensability
argument for Quinean modest platonism in mathematics — 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 illfounded. 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 ultrafinitism. On most accounts, this
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. It is needless to say that many find such consequences hard to
swallow. But Lavine has developed a sophisticated form of settheoretical
ultrafinitism which is mathematically nonrevisionist (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.
Field's
physicalist interpretation of arithmetic and analysis not only undermines the
QuinePutnam 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 spacetime regions are physical. So we are no longer required to explicate
how fleshandblood mathematicians stand in contact with nonphysical 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
π, 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 φ of the language of analysis is (roughly)
given by:
Every concrete system S which makes RA
true, also makes φ 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 ante rem 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 settheoretic sense of the word can be viewed as a form of in
rebus structuralism.
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
nominaliststructuralist rendering of the content of a sentence φ of the
language of analysis gets the truth conditions of such sentences wrong. For
then for every universally quantified sentence φ, its paraphrase will come out
vacuously true. So an existential assumption to the effect that there exist
concrete physical systems that can serve as models 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 φ of the
language of analysis:
Necessarily,
every concrete system S which makes RA true, also makes φ 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 modalexistential 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.
Again, it
is admittedly not easy to give a satisfying account of how we know that this
modalexistential 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 modalexistential assumption becomes:
It is possible that there exist concrete
physical systems that can serve as models 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 1990). We seem to have no reason to
believe that there could be physical worlds that contain
highlytransfinitelymany 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.
Even this
crudest of descriptions of the fictionalist position immediately opens up the
question what sort of entities fictional entities are. This is a deep
metaphysicalontological problem. Mathematical fictionalists have hitherto not
contributed much to the resolution of this question.
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 entities, as the fictionalist contends, then
Benacerraf's epistemological problem does not arise. Fictionalism shares this
advantage over most forms of platonism with nominalistic reconstructions of
mathematics. But at the same time it shares with platonism the advantage of
respecting the surface logical form of mathematical statements.
Whether
Benacerraf's identification problem is solved is not completely clear. In
general, fictionalism is a nonreductionist 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 π) 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 for 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 are starting to arise.
They evolve in a way that is not completely determined by the “big debates”
about the nature of mathematics. In this concluding section, we look at a few
of these disciplines.
5.1 Philosophy of Set Theory
Many
regard set theory as 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.
One question
that has been important from the beginning of set theory concerns the
difference between sets and proper classes. Cantor's diagonal argument forces
us to recognize that the settheoretical 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 settheoretical 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 settheoretical 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 properclass view seems compelled to recognize in addition a
realm of superproper 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
settheoretical 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. This way of speaking reveals that in our attempts to
understand this notion of absolute potential infinity, we are drawn to temporal
metaphors. It is not surprising that these temporal metaphors cause some
philosophers of mathematics acute discomfort.
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 settheoretical
universe can be thought of as generated from the empty set by means of the
power set operation (Boolos 1971). On the other hand, there is the limitationofsize
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). The
limitationofsize conception motivates other axioms better (such as the
restricted comprehension axiom). Many philosophers of mathematics believe that
we today have 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 largecardinal
axioms. Nowadays, largecardinal hypotheses are really taken to mean
some kind of embedding properties between the set theoretic universe and inner
models of set theory. Most of the time, largecardinal principles entail the
existence of sets that are larger than any sets which can be guaranteed by ZFC
to exist.
Gödel
hoped that on the basis of such largecardinal axioms, important open questions
in set theory could eventually be settled. The most important settheoretic
problem 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 onetoone correspondence between S and the natural
numbers, but too small for there to exist a onetoone correspondence between S and the real
numbers. Despite strenuous efforts, all attempts to settle the continuum
problem failed. Gödel came to suspect that the continuum hypothesis is
independent of the accepted principles of set theory. 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 Gödel's conjecture of the independence of the
continuum hypothesis was eventually confirmed.
But
Gödel's hope that largecardinal axioms could solve the continuum problem
turned out to be unfounded. The continuum hypothesis is independent of ZFC even
in the context of most largecardinal axioms. Nevertheless, largecardinal
principles have managed to settle restricted versions of the continuum
hypothesis (in the affirmative). The existence of socalled 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, 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?
Many of
the researchers who seek to decide the continuum hypothesis on the basis of new
axioms think that there already is significant evidence for the thesis that the
continuum hypothesis is false. But there are also many set theorists and
philosophers of mathematics who believe that the continuum hypothesis is 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). This is
related to the more general question whether any
reasonable bounds can be placed on the extension of the concept of informal
provability. At present, this area of research is wide open.
5.2 Categoricity
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 quasicategorical:
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}. Recently, McGee has
shown that if we consider set theory with Urelements, then the theory is fully categorical
with respect to pure sets if we assume that there are only setmany Urelements
(McGee 1997).
At the
same time, the LöwenheimSkolem theorem says that every firstorder 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
LöwenheimSkolem 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 firstorder formalizations of the basic principles of
arithmetic and analysis. Instead, he informally worked with secondorder
formalizations. The same holds for Zermelo and McGee.
Let us
focus on arithmetic to see what this amounts to. The basic postulates of
arithmetic contain the induction axiom. In firstorder formalizations of
arithmetic, this is formulated as a scheme: for each firstorder arithmetical
formula with one free variable, an 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 firstorder formula. But intuitively, it seems that the
induction principle holds for all
properties of natural numbers. So in a firstorder 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 secondorder
language (Shapiro 1991). Secondorder languages contain not just
firstorder quantifiers that range over elements of the domain, but also
secondorder quantifiers that range over properties (or subsets) of the domain.
In full
secondorder logic, it is insisted that these secondorder quantifiers range
over all
subsets of the domain. If the principles of arithmetic are formulated in a
secondorder 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 secondorder language,
and the secondorder formulation of set theory turns out to be
quasicategorical.
Ante rem structuralism, as well as the modal nominalist
structuralist interpretation of mathematics, could benefit from a secondorder
formulation. If the ante
rem structuralist wants to insists that the naturalnumber
structure is fixed up to isomorphism by the Peano axioms, then she will want to
formulate the Peano axioms in secondorder logic. And the modal nominalist
structuralist will want to insist that the relevant concrete systems for
arithmetic are those that make the secondorder
Peano axioms true (Hellman 1989). Similarly for mathematical analysis and set
theory. Thus the appeal to secondorder logic appears as the final step in the
structuralist project of isolating the intended models of mathematics.
Yet appeal
to secondorder logic in the philosophy of mathematics is by no means
uncontroversial. A first objection is that the ontological commitment of
secondorder logic is higher than the ontological commitment of firstorder
logic. After all, use of secondorder logic seems to commit us to the existence
of abstract objects: classes. In response to this problem, Boolos has
articulated an interpretation of secondorder logic which avoids this
commitment to abstract entities (Boolos 1985). His interpretation spells out
the truth clauses for the secondorder quantifiers in terms of plural
expressions, without invoking classes. For instance, a secondorder expression
of the form ∃XF(X)
is interpreted as: “there are some (firstorder
objects) x
such that they have the property F”.
This interpretation is called the plural
interpretation of secondorder logic. It is clear that an appeal to
such an interpretation of secondorder logic will be tempting for nominalist
versions of structuralism.
A second
objection against secondorder logic can be traced back to Quine (Quine 1970).
This objection states that the interpretation of full secondorder logic is
connected with settheoretical questions. This is already indicated by the fact
that most regimentations of secondorder logic adopt a version of the axiom of
choice as one of its axioms. But more worrisome is the fact that secondorder
logic is inextricably intertwined with deep
problems in set theory, such as the continuum hypothesis. For theories such as
arithmetic, which intend to describe an infinite collection of objects, even a
matter as elementary as the question of the cardinality of the range of the
secondorder quantifiers is equivalent to the continuum problem. Also, it turns
out that there exists a sentence which is a secondorder 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 truthvalueless. If
this is so, then there is an inherent indeterminacy in the very notion of
secondorder 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 secondorder logic is inherently
indeterminate.
If one
does not want to appeal to full secondorder 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 firstorder and
secondorder quantifiers. For instance, one might treat “there are finitely
many x”
as a primitive quantifier. This will allow one, for instance, to 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). A
theorem of Tennenbaum states that all firstorder 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.
5.3 Computation and Proof
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
Hilbertstyle 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 quasiempirical. Attempts to find convincing counterexamples to
Church's Thesis led 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, HerbrandGödel computability, lambdadefinability, …
have been proposed. But these mathematical notions all turn out to be
equivalent. Thus, to use Gödelian 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 Turing's 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 (PourEl
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).
The past
decades have witnessed the first occurrences of mathematical proofs in which
computers appear to play an essential role. The fourcolour 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 doublechecked by humans. The proof
of the fourcolour theorem gave rise to a debate about the question to what
extent computerassisted 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 quasiempirical
knowledge (Tymoczko 1979). In other words, through the advent of computer
proofs the notion of proof has lost its purely a priori character. Others hold
that the empirical factors on which we rely when we accept computer proofs do
not appear as premises in the argument. Hence, computer proofs can yield a
priori knowledge after all (Burge 1998).
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 lowlevel mathematical
concepts. They are notoriously weak at developing deep concepts on their own,
and have difficulties with linking concepts 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?
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Acknowledgments
The editors would
like to thank Christopher von Bülow for carefully reading this entry and
bringing numerous typographical errors to our attention.