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Conceived of philosophically, the foundations of mathematics concern various metaphysical and epistemological problems
raised by mathematical practice, its results and applications. Most of these problems are of ancient vintage; two, in particular,
have been of perennial concern. These are its richness of content and its necessity. Important too, though not so prominent in
the history of the subject, is the problem of application, or how to account for the fact that mathematics has given rise to such
an extensive, important and varied body of applications in other disciplines.
The Greeks struggled with these questions. So, too, did various medieval and modern thinkers. The ideas of many of these
continue to influence foundational thinking to the present day.
During the nineteenth and twentieth centuries, however, the most influential ideas have been those of Kant. In one way or
another and to a greater or lesser extent, the main currents of foundational thinking during this period - the most active and
fertile period in the entire history of the subject - are nearly all attempts to reconcile Kant’s foundational ideas with various
later developments in mathematics and logic.
These developments include, chiefly, the nineteenth-century discovery of non-Euclidean geometries, the vigorous development
of mathematical logic, the development of rigorous axiomatizations of geometry, the arithmetization of analysis and the
discovery (by Dedekind and Peano) of an axiomatization of arithmetic. The first is perhaps the most important. It led to
widespread acceptance of the idea that space was not merely a Kantian ‘form’ of intuition, but had an independence from our
intellect that made it different in kind from arithmetic. This asymmetry between geometry and arithmetic became a major
premise of more than one of the main ‘isms’ of twentieth-century philosophy of mathematics. The intuitionists retained Kant’s
conception of arithmetic and took the same view of that part of geometry which could be reduced to arithmetic. The logicists
maintained arithmetic to be ‘analytic’ but differed over their view of geometry. Hilbert’s formalist view endorsed a greater part
of Kant’s conception.
The second development carried logic to a point well beyond where it had been in Kant’s day and suggested that his views on
the nature of mathematics were in part due to the relatively impoverished state of his logic. The third indicated that geometry
could be completely formalized and that intuition was therefore not needed for the sake of conducting inferences within proofs.
The fourth and fifth, finally, provided for the codification of a large part of classical mathematics - namely analysis and its
neighbours - within a single axiomatic system - namely (second-order) arithmetic. This confirmed the views of those (for
example, the intuitionists and the logicists) who believed that arithmetic had a special centrality within human thinking. It also
provided a clear reductive target for such later anti-Kantian enterprises as Russell’s logicism.
The major movements in the philosophy of mathematics during this period all drew strength from post-Kantian developments in
mathematics and logic. Each, however, also encountered serious difficulties soon after gaining initial momentum. Frege’s
logicism was defeated by Russell’s paradox; Russell’s logicism, in turn, made use of such questionable (from a logicist
standpoint) items as the axioms of infinity and reduction. Both logicism and Hilbert’s formalist programme came under heavy
attack from Gِdel’s incompleteness theorems. And finally, intuitionism suffered from its inability to produce a body of
mathematics comparable in richness to classical mathematics.
Despite the failure of these non-Kantian programmes, however, movement away from Kant continued in the mid- and late
twentieth century. From the 1930s on this has been driven mainly by a revival of empiricist and naturalist ideas in philosophy,
prominent in the writings of both the logical empiricists and the later influential work of Quine, Putnam and Benacerraf. This
continues as perhaps the major force shaping work in the philosophy of mathematics.
1 Kant’s views; reactions
The ‘Problematik’ that Kant established for the epistemology of pure mathematics focused on the reconciliation of two
apparently incompatible features of pure mathematics: (1) the problem of necessity, or how to explain the apparent fact that
mathematical statements (for example, statements such as that or that the sum of the interior angles of a
Euclidean triangle is equal to two right angles) should appear to be not only true but necessarily true and independent of
empirical evidence; and (2) the problem of cognitive richness, or how to account for the fact that pure mathematics should
yield subjects as rich and deep in content and method, as robust in growth and as replete with surprising discoveries as the
history of mathematics demonstrates.
In mathematics, Kant said, we find a ‘great and established branch of knowledge’ - a cognitive domain so ‘wonderfully large’
and with promise of such ‘unlimited future extension’ that it would appear to arise from sources other than those of pure
unaided (human) reason (1783: §§6, 7). At the same time, it carries with it a certainty or necessity that is typical of judgments
of pure reason. The problem, then, is to explain these apparently conflicting characteristics. Kant’s explanation was that
mathematical knowledge arises from certain standing conditions or ‘forms’ which shape our experience of space and time -
forms which, though they are part of the innate cognitive apparatus that we bring to experience, none the less shape our
experience in a way that goes beyond mere logical processing.
To elaborate this hypothesis, Kant sorted judgments/propositions in two different ways: first, according to whether they
required appeal to sensory experience for their justification; and, second, according to whether their predicate concepts were
‘contained in’ their subject concepts. A judgment or proposition was ‘a priori’ if it could be justified without appeal to sensory
content. If not, it was ‘a posteriori’. It was ‘analytic’ if the very act of thinking the subject concept contained, as a constituent
part, the thinking of the predicate concept. If not, it was either false or ‘synthetic’. In synthetic a priori judgment - the type of
judgment Kant regarded as characteristic of mathematics - the predicate concept was thought not through the mere thinking
of the subject concept, but through its ‘construction in intuition’. He took a similar view of mathematical inference, believing it
to involve an intuition that goes beyond the mere logical connection of premises and conclusions (1781/1787: A713-19/B741-7).
Kant erected his mathematical epistemology upon these distinctions and, famously, maintained that mathematical judgment and
inference is synthetic a priori in character. In this way, he intended to account for both the necessity and cognitive richness of
mathematics, its necessity reflecting its a priority, its cognitive richness its syntheticity.
Kant applied this basic outlook to both arithmetic and geometry (and also to pure mechanics). He did not regard them as
entirely identical, however, since he saw them as resting on different a priori intuitions. Neither did he see them as possessing
precisely the same universality (1781/1787: A163-5, 170-1, 717, 734/B204-6, 212, 745, 762). None the less, he regarded their
similarities as more important than their differences and therefore took them to be of essentially the same epistemic type -
namely, synthetic a priori. In the end, it was this inclusion of geometry and arithmetic within the same basic epistemic type
rather than his more central claim concerning the existence of synthetic a priori knowledge that gave rise to the sternest
challenges to his views.
In the decades following the publication of the first Critique (1781/1787), the principal source of concern regarding its views
was the growing evidence for and eventual discovery of non-Euclidean geometries. This led many to question whether
geometry and arithmetic are of the same basic epistemic character.
The serious possibility of non-Euclidean geometries went back to the work of Lambert and others in the eighteenth century.
Building on this work, some - in particular, Gauss (1817, 1829) - stated their opposition to Kant’s views even before the actual
discovery of non-Euclidean geometries by Bolyai and Lobachevskii in the 1820s. Gauss’ reasoning was essentially this: number
seems to be purely a product of the intellect and, so, something of which we can have purely a priori knowledge. Space, on the
other hand, seems to have a reality external to our minds that prohibits a purely a priori knowledge of it. Arithmetic and
geometry are therefore not on an epistemological par with one another.
This reasoning became a potent force shaping nineteenth- and twentieth-century foundational thinking. Another such force was
the dramatic development of logic and the axiomatic method in the mid- to late nineteenth century and early twentieth century.
This included the introduction of algebraic methods by Boole and De Morgan, the improved treatment of relations by Peirce,
Schrِder and Peano, the replacement of the subject-predicate conception of propositional form with Frege’s more fecund
functional conception, and the advances in axiomatization and formalization brought about by the work of Frege, Pasch, Peano,
Hilbert and (especially) Whitehead and Russell.
Certain developments in mathematics proper also exerted an influence. Chief among these were the arithmetization of analysis
by Weierstrass, Dedekind and others, and the axiomatization of arithmetic by Peano and Dedekind. Of somewhat lesser
importance, though still significant for their effects on Hilbert’s thinking, were Einstein’s relativistic ideas in physics.
2 Intuitionism
A variety of views concerning the asymmetry of geometry and arithmetic emerged in the late nineteenth and early twentieth
centuries. That of the early intuitionists Brouwer and Weyl retained Kant’s synthetic a priori conception of arithmetic.
They responded to the discovery of non-Euclidean geometries, however, by denying the a priori status of that part of geometry
that could not be reduced to arithmetic by such means as Descartes’ calculus of coordinates. They retained, none the less, a
type of a priori intuition of time as the basis for arithmetical knowledge (see Brouwer 1913: 127-8). They also emphasized the
synthetic character of arithmetical judgment and inference, and sharply distinguished them from logical judgment and inference.
Brouwer described his intuition of time as consciousness of change per se - the human subject’s primordial inner awareness of
the ‘falling apart’ of a life-moment into a part that is passing away and a part that is becoming. He believed that, via a process
of abstraction, one could pass from this basal intuition of time to a concept of ‘bare two-oneness’, and from this concept to,
first, the finite ordinals, then the transfinite ordinals and, finally, the linear continuum. In this way, parts of classical arithmetic,
analysis and set theory could be recaptured intuitionistically. (See Brouwer 1907: 61, 97; 1913: 127, 131-2.)
Brouwer thus modified Kant’s intuitional basis for mathematics. He also modified his conception of knowledge of existence.
Kant believed that humans could obtain knowledge of existence only through sensible intuition. Only this, he believed, had the
type of involuntariness and objectivity that assures us that belief in an object is not a mere compulsion or idiosyncrasy of our
subjective selves. Like the post-Kantian romantic idealists, however, Brouwer (and Weyl, too) believed as well in knowledge of
existence via a kind of ‘intellectual intuition’ - an intuition carried by a purely internal type of mental construction (1907: 96-7).
The early intuitionists (especially Brouwer and Poincaré) remained Kantian in their conception of mathematical reasoning and
took it to be essentially different in character from ‘discursive’ or logical reasoning. Brouwer believed logical reasoning to mark
not patterns in mathematical thinking itself but only patterns in its linguistic representation. It was therefore not indicative of the
inferential structure of mathematical thinking itself and had no place within genuine mathematical reasoning per se. This was
essentially the idea expressed in Brouwer’s so-called ‘First Act of Intuitionism’ (1905: 2, 1981: 4-5).
Thus the early intuitionists (especially Brouwer and Weyl and, to some extent, Poincaré) discarded Kant’s view of geometry,
revised his conception of arithmetic and existence claims, and preserved his basic stance on the nature of mathematical
reasoning and its relationship to logical reasoning. Later intuitionists (for example, Heyting and Dummett) did not keep to this
plan. They rejected Brouwer’s view of the divide between logical and mathematical reasoning and made a significant place for
logic in their accounts of mathematical reasoning. Some of them (Dummett and his ‘anti-realist’ followers) even went so far as
to make the question ‘What is the logic of mathematical reasoning?’ central to their philosophy of mathematics (see §5 below).
3 Logicism
The view of the logicist Frege (and, to some extent, of Dedekind) accepted Kant’s synthetic a priori conception of geometry
but maintained arithmetic to be analytic. Russell, another logicist, rejected Kant’s views of both geometry and arithmetic (and
also of pure mechanics) and maintained the analyticity of both. (See Logicism.)
Frege’s logicism differed sharply from intuitionism. First, it differed in the place in mathematical reasoning it assigned to logic.
Frege (1884: preface, III-IV) maintained that reasoning is essentially the same everywhere and that even an inference pattern
such as that of mathematical induction, which appears to be peculiar to mathematics, is, at bottom, purely logical. Second, it
differed in its conception of geometry. Like the early intuitionists, Frege regarded the discovery of non-Euclidean geometries as
revealing an important asymmetry between arithmetic and geometry. Unlike them, however, he did not see this as grounds for
rejecting Kant’s synthetic a priori conception of geometry (1873: 3; 1884: §89), but rather as indicating a fundamental
difference between geometry and arithmetic. Frege believed the fundamental concept of arithmetic - magnitude - to be both
too pervasive and too abstract to be the product of Kantian intuition (1874: 50). It figured in every kind of thinking and so must,
he reasoned, have a basis in thought deeper than that of intuition. It must have its basis in the very core of rational thought
itself; the laws of logic.
The problem was to account for the cognitive richness of arithmetic on such a view. How could the ‘great tree of the science
of number’ (1884: §16) have its roots in bare logical or analytical ‘identities’? Frege responded by offering new accounts of
both the objectivity and the informativeness of arithmetic. The former he attributed to its subject matter - the so-called
‘logical objects’ (§§26, 27, 105). The latter he derived from a new theory of content which allowed concepts to contain (tacit)
content that was not needed for their grasp. On this view, analytic judgments could have content that was not required for the
mere understanding of the concepts contained in them. Consequently, they could yield more than knowledge of transparent
logical identities (§§64-66, 70, 88, 91).
Unlike Kant, then, Frege maintained an important epistemic asymmetry between geometry and arithmetic - an asymmetry
based upon his belief that arithmetic is more basic to human rational thought than is geometry. In addition, he departed from
Kant in maintaining a realistic conception of arithmetic knowledge despite its analytic character. He saw it as being about a
class of objects - so-called ‘logical objects’ - that are external but intimately related to the mind and therefore not the mere
expression of standing traits of human cognition. The differences between arithmetical and geometric necessity were to be
accounted for by separating the relationship the mind has to the objects of arithmetic from that which it has to the objects of
geometry.
Russell’s logicism differed from Frege’s. Perhaps most importantly, Russell did not regard the existence of non-Euclidean
ggeometries as evidence of an epistemological asymmetry between geometry and arithmetic. Rather, he saw the
‘arithmetization’ of geometry and other areas of mathematics as evidence of an epistemological symmetry between arithmetic
and the rest of mathematics. Russell thus extended his logicism to the whole of mathematics. The basic components of his
logicism were a general methodological ideal of pursuing each science to its greatest level of generality, and a conception of
the greatest level of generality in mathematics as lying at that point where all its theorems are of the form ‘p implies q’, all their
constants are logical constants and all their variables of unrestricted range. Theorems of this sort, Russell maintained, would
rightly be regarded as logical truths.
Russell’s logicism was thus motivated by a view of mathematics which saw it as the science of the most general formal truths;
a science whose indefinables are those constants of rational thought (the so-called logical constants) that have the most
ubiquitous application, and whose indemonstrables are those propositions that set out the basic properties of these indefinables
(Russell 1903: 8). In his opinion, such a view provided the only precise description of what philosophers have had in mind when
they have described mathematics as a necessary or a priori science.
Russell thus accounted for the necessity of mathematics by pointing to its logical character. He accounted for its richness
principally by invoking a new definition of syntheticity that allowed all but the most trivial logical truths and inferences to be
counted as informative or synthetic. Mathematical truths would thus be logical truths, but they would not, for all that, be
analytic truths. Similarly for inferences. An inference would count as synthetic so long as its conclusion was a different
proposition from its premises. Cognitive richness was conceived primarily as the production of new propositions from old, and,
on Russell’s conception (supposing the criterion of propositional identity to be sufficiently strict), even purely logical inference
could produce a bounty of new knowledge from old.
Russell was thus able to account for both the necessity and the cognitive richness of mathematics while making mathematics
part of logic. What had kept previous generations of thinkers and, in particular, Kant from recognizing the possibilities of such a
view was the relatively impoverished state of logic before the end of the nineteenth century. The new logic, with its robust
stock of new forms, its functional conception of the proposition and the ensuing fuller axiomatization of mathematics which it
made possible, changed all this forever and provided for the final refutation of Kant. Such, at any rate, was Russell’s position.
4 Hilbert’s formalism
Hilbert accepted the synthetic a priori character of (much of) arithmetic and geometry, but rejected Kant’s account of the
supposed intuitions upon which they rest. Overall, Hilbert’s position was more complicated in its relationship to Kant’s
epistemology than were those of the intuitionists and logicists. Like Russell, he rejected Kant’s specifically mathematical
epistemology - in particular, his conception of the nature and origins of its a priori character. Like Russell, too, he rejected the
common post-Kantian belief in the epistemological asymmetry of arithmetic and geometry. Hilbert was, however, unique
among those mentioned here in endorsing the framework of Kant’s general critical epistemology and making it a central
feature of his mathematical epistemology. Specifically, he adopted Kant’s distinction between the faculty of the understanding
and the faculty of reason as the guide for his pivotal distinction between the so-called ‘real’ and ‘ideal’ portions of classical
mathematics (Hilbert 1926: 376-7, 392).
Hilbert took ‘real’ mathematics to be ultimately concerned with the shapes or forms (Gestalten) of concrete signs or figures,
given in intuition and comprising a type of ‘immediate experience prior to all thought’ (1926: 376-7; [1928] 1967: 464-5). Hilbert
proposed this basic intuition of shape as a replacement for Kant’s two a priori intuitions of space and time. Like Kant’s a priori
intuitions, however, Hilbert, too regarded his finitary intuition as an ‘irremissible pre-condition’ of all mathematical (indeed, all
scientific) judgment and the ultimate source of all genuine a priori knowledge (1930: 383, 385).
The genuine judgments of real mathematics were the judgments of which our mathematical knowledge was constituted. The
pseudo-judgments of ideal mathematics, on the other hand, functioned like Kant’s ideas of reason. They neither described
things present in the world nor constituted a foundation for our judgments concerning such things. Rather, they played a purely
regulative role of guiding the efficient and orderly development of our real knowledge.
Hilbert did not, therefore, affirm the necessity of either arithmetic or geometry in any simple, straightforward way. Rather, he
distinguished two types of necessity operating within both. One, pertaining to the judgments of real mathematics, consisted in
the (presumed) fact that the apprehension of certain elementary spatial and combinatorial features of simple concrete objects is
a pre-condition of all scientific thought. The other, pertaining to the ideal parts of mathematics, had a kind of psychological
necessity, a necessity borne of the manner in which our minds inevitably or best regulate the development of our real
knowledge.
This conception of the necessity of mathematics was different from both Kant’s and the logicists’ and intuitionists’. So, too,
was Hilbert’s view of the cognitive richness of mathematics, which he attributed both to the objective richness of the shapes
and combinatorial features of concrete signs and to the richness of our imaginations in ‘creating’ complementary ideal objects.
In its overall structure, Hilbert’s mathematical epistemology thus resembled Kant’s general critical epistemology. This included
his so-called ‘consistency’ requirement (that is, the requirement that ideal reasoning not prove anything contrary to that which
may be established by real means), which resembled Kant’s demand that the faculty of reason not produce any judgment of
the understanding that could not in principle be obtained solely from the understanding (1781/1787: A328/B385).
5 Modifications
During the first four decades of the twentieth century, each of the post-Kantian programmes outlined above came under
attack. Frege’s logicism was challenged by Russell’s paradox. Russell’s logicism encountered difficulties concerning its use of
certain existence axioms (namely his axioms of reducibility and infinity) which did not appear to be laws of logic. Both were
challenged by Gِdel’s incompleteness theorems, as was Hilbert’s formalist programme. Finally, the intuitionists were criticized
both philosophically, where their idealism was called into question, and mathematically, where their ability to support a
significant body of mathematics remained in doubt. Various modifications have been proposed.
Modifications of logicism. On the mathematical side, a chastened successor to logicism can perhaps be seen in the
model-theoretic work of Abraham Robinson and his followers. They are interested in determining the mathematical content
latent in purely ‘logical’ features of various mathematical structures or the extent to which genuinely mathematical problems
concerning these structures can be solved by purely logical (that is, model-theoretic) means. They have been particularly
successful in their treatment of various algebraic structures (see Macintyre 1977; Robinson 1979; Hodges 1993).
Philosophically, too, there have been attempts to renew logicism. It re-emerged in the 1930s and 1940s as the favoured
philosophy of mathematics of the logical empiricists (see Carnap 1931; Hahn 1933). They did not, however, develop a logicism
of their own in the way that Dedekind, Frege and Russell did, but, rather, simply appropriated the technical work of Russell and
Whitehead (modulo the usual reservations concerning the axioms of infinity and reducibility) and attempted to embed it in an
overall empiricist epistemology.
This empiricist turn was a novel development in the history of logicism and represented a serious departure from both the
original logicism of Leibniz (§10) and the more recent logicism of Frege (and Dedekind). It was less at odds with Russell’s
logicism which saw mathematics and the empirical sciences as both making use of an essentially inductive method (the
so-called ‘regressive’ method - see Russell 1906, 1907).
Like all empiricists, the logical empiricists struggled with the Kantian problem of how to account for the apparent necessity of
mathematics while at the same time being able to explain its cognitive richness. Their strategy was to empty mathematics of all
non-analytic content while, at the same time, arguing that analytic truth and inference can be substantial and
non-self-evident.
Their ideas came under heavy attack by W.V. Quine, who challenged their pivotal distinction between analytic and synthetic
truths (1951, 1954). He argued that the basic unit of knowledge - the basic item of our thought that is tested against experience
- is science as a whole and that this depends upon empirical evidence for its justification. Mathematics and logic are used to
relate empirical evidence to the rest of science and, so, are inextricably interwoven into the whole fabric of science. They are
thus part of the total body of science that is tested against experience and there is no clean way of dividing between truths of
meaning (analytic truths) and truths of fact (synthetic truths).
Within a relatively brief period of time, Quine’s argument became a major influence in the philosophy of mathematics and the
logicism of the logical empiricists was largely abandoned. Newer conceptions of logicism have, however, continued to appear
from time to time. For example, Putnam (1967) addressed the difficult (for a logicist) question of existence claims, arguing that
such statements are to be seen as asserting the possible (as opposed to the actual) existence of structures. They are therefore,
at bottom, logical claims, and can be established by logical (or metalogical) means. Hodes (1984) takes a somewhat different
approach, arguing that arithmetic claims can be translated into a second-order logic in which the second-order variables range
over functions and concepts (as opposed to objects). In this way, commitment to sets and other specifically mathematical
objects can be eliminated and, this done, arithmetic can be considered a part of logic.
Field (1980, 1984) also presents a kind of logicist view - namely, that mathematical knowledge is (at least largely) logical
knowledge. Mathematical knowledge is defined as that knowledge which separates a person who knows a lot of mathematics
from a person who knows only a little, and it is then argued that what separates these two kinds of knowers is mainly logical
knowledge; that is, knowledge of what follows from what.
Modifications of Hilbert’s programme. Hilbert’s programme too has its latter-day adherents. For the most part, these have
adopted one of two basic stances: that of extending the methods available for proving the consistency of classical ideal
mathematics; or that of diminishing the scope and strength of classical ideal mathematics so that its consistency (or the
consistency of important parts of it) can more nearly be proved by the kinds of elementary means that Hilbert originally
envisaged.
Some in the first group (for example, Gentzen, Ackermann and Gِdel) have argued that there are types of evidence that
exceed finitary evidence in strength but which have the same basic epistemic virtues as it. Others (for example, Kreisel 1958;
Feferman 1988; Sieg 1988) argue for a change in our conception of what a consistency proof ought to do. They maintain that
its essential obligation is to realize an epistemic gain, and that finitary methods are not the only epistemically gainful methods for
proving consistency.
Those in the second group - the so-called ‘reverse mathematics’ school of Friedman, Simpson and others - try to isolate the
mathematical ‘cores’ of the various areas of classical mathematics and prove the consistency of these ‘reduced’ theories by
finitary or related means. So far, significant success has been achieved along these lines. (See Hilbert’s programme and
formalism §4.)
Modifications of intuitionism. Regarding intuitionism, Heyting’s work in the 1930s to formalize intuitionism and to identify its
logic has led to a vigorous programme of logical and mathematical investigation (see Heyting 1956; Troelstra 1973, 1977;
Beeson 1985 for descriptions of some of this work). In addition to Heyting and his students, Errett Bishop and his followers
have extended a constructivist approach to areas of classical mathematics to which such an approach had previously not been
extended (see Bridges 1987 for a survey).
On the more philosophical side, the most important development is the construction by Michael Dummett and his anti-realist
followers of a defence of intuitionism based upon - in their view - the best answer to the question ‘What is the logic of
mathematics?’. Their answer is based upon what they take to be a proper theory of meaning - a theory which, following
certain ideas set out by Wittgenstein in his Philosophical Investigations, equates the meaning of an expression with its
canonical use in the practice to which it belongs. They then identify the canonical use of an expression in mathematics with the
role it plays in the central activity of proof, and from this they infer an intuitionist treatment of the logical operators (Dummett
1973, 1977).
6 Later developments
Along with the modifications of the major post-Kantian viewpoints noted above, two other developments in the second half of
the twentieth century are important to note. One of these is the shift towards empiricism that was brought about by Quine’s
(following Duhem’s) merging of mathematics and the empirical sciences into a single justificatory unit governed by a basically
inductive-empirical method. On this view, mathematics may on the whole be less susceptible to falsification by sensory
evidence than is natural science, but this is a difference of degree, not kind.
This conception of mathematics dispenses with a ‘datum’ of mathematical epistemology that philosophers of mathematics from
Kant on down had struggled to accommodate: namely, the presumed necessity of mathematics. It puts in its place a general
empiricist epistemology in which all judgments - those of mathematics and logic as well as those of the natural sciences - are
seen as evidentially connected to sensory phenomena and, so, subject to empirical revision.
To accommodate the lingering conviction that mathematics is independent of empirical evidence in a way that natural science
is not, Quine introduced a pragmatic distinction between them. Rational belief-revision, he said, is governed by a pragmatic
concern to maximize the overall predictive and explanatory power of one’s total system of beliefs. Furthermore, predictive and
explanatory power are generally aided by policies of revision which minimize, both in scope and severity, the changes that are
made to a previously successful belief-system in response to recalcitrant experience.
Because of this, beliefs of mathematics and logic are typically less subject to empirical revision than beliefs of natural science
and common sense since revising them generally (albeit, in Quine’s view, not inevitably) does more damage to a belief-system
than does revising its common sense and natural scientific beliefs. The necessity of mathematics is thus accommodated in
Quine’s epistemology by moving mathematics closer to the centre of a ‘web’ of human beliefs where beliefs are less
susceptible to empirical revision than are the beliefs of natural science and common sense that lie closer to the edge of the
web.
In Quine’s view, merging mathematics and science into a single belief-system also induces a realist conception of mathematics.
Mathematical sentences must be treated as true in order to play their role in this system, and the world is to be seen as being
populated by those entities that are among the values of the variables of true sentences. Mathematical entities are thus real
bbecause mathematical sentences play an integral part in our best total theory of experience (see Quine 1948, 1951; Putnam
1971, 1975).
Quine’s views have been challenged on various grounds. For example, Parsons (1980) argues that treating the elementary
arithmetical parts of mathematics as being on an epistemological par with the hypotheses of theoretical physics fails to capture
an epistemologically important distinction between the different kinds of evidentness displayed by the two. Even highly
confirmed physical hypotheses such as ‘The earth moves around the sun’ are more ‘derivative’ (that is, roughly, more
theory-laden) than is an elementary arithmetic proposition such as ‘ ’. It is therefore not plausible to regard the
two claims as based on essentially the same type of evidence.
Others have challenged different aspects of Quine’s position. Field (1980) and Maddy (1980), for example, both question his
merging of mathematics and natural science, though in different ways. Field argues that natural science that utilizes
mathematics is a conservative extension of it and, so, has no need of its entities. The mathematical part of natural science can
thus, in an important sense, be separated from the rest of it. (See Shapiro (1983) for an apt criticism of Field’s arguments.).
Maddy investigates the possibility that our knowledge of at least certain mathematical objects might not be so diffuse and
inextricable from the whole scheme of our natural scientific knowledge as Quine suggests. She argues that perceptual
experience can be tied closely and specifically to certain mathematical objects (in particular, to certain sets) in a way that
seems out of keeping with Quine’s holism.
In addition to Quine, others have suggested different mergings of mathematics and natural sciences. Kitcher (1983), for
instance, presents a generally empiricist epistemology for mathematics in which history and community are important
epistemological forces. Gِdel, on the other hand, argued that mathematics, like the natural sciences, makes use of what is
essentially inductive justification ([1947] 1964: 477, 485) when it justifies higher-level mathematical hypotheses on the grounds
of their explanatory or simplificatory effects on lower-level mathematical truths and on physics. He allowed, however, that only
some of our mathematical knowledge arises from empirical sources and regarded as absurd the idea that all of it might do
(1951: 311-12).
Another important influence on recent philosophy of mathematics is Benacerraf’s ‘Mathematical Truth’ (1973), in which he
argues that the philosophy of mathematics faces a general dilemma. It must give an account of both mathematical truth and
mathematical knowledge. The former seems to demand abstract objects as the referents of singular terms in mathematical
discourse. The latter, on the other hand, seems to demand that we avoid such referents. There are mathematical
epistemologies (for example, various Platonist ones) that allow for a plausible account of the truth of mathematical sentences.
Likewise, there are those (for example, various formalist ones) that allow for a plausible account of how we might come to
know mathematical sentences. However, no known epistemology does both. Towards the end of the twentieth century a great
deal of work has been devoted to resolving this dilemma. Field (1980, 1984), Hellman (1989) and Chihara (1990) attempt
anti-Platonist resolutions. Maddy (1990), on the other hand, attempts a resolution at once Platonist and naturalistic. To date
there is no general consensus on which approaches are the more plausible.
An earlier argument of Benacerraf’s (see Benacerraf 1965, but also Dedekind 1888: §73; Hilbert 1900; Weyl 1927; Bernays
1950) was similarly influential in shaping later work. It is the chief inspiration of the position known as ‘structuralism’ - the
view that mathematical objects are essentially positions in structures and have no important additional internal composition or
nature (see Resnik 1981, 1982; Shapiro 1983). Apart from the desire for a descriptively more adequate account of
mathematics, the chief motivation of structuralism is epistemological. Knowledge of the characteristics of individual abstract
objects would seem to require naturalistically inexplicable powers of cognition. Knowledge of at least some structures, on the
other hand, would appear to be explicable as the result of applying such classically empiricist means of cognition as abstraction
to observable physical complexes. Structures identified via abstraction become part of the general framework of our thinking
and can be extended and generalized in a variety of ways as the search for the simplest and most effective overall conceptual
scheme is pursued.
Structuralism as a general philosophy of mathematics has been criticized by Parsons (1990) who argues that there are
important mathematical objects for which structuralism is not an adequate account. These are the ‘quasi-concrete’ objects of
mathematics - objects that are directly ‘instantiated’ or ‘represented’ by concrete objects (for example, geometric figures and
symbols such as the so-called ‘stroke numerals’ of Hilbert’s finitary arithmetic, where these are construed as types whose
instances are written marks or symbols or uttered sounds). Such objects cannot be treated in a purely structuralist way
because their ‘representational’ function cannot be reduced to the purely intrastructural relationships they bear to other objects
within a given system. At the same time, however, they are among the most elementary and important mathematical entities
there are.
Routledge Encyclopedia of Philosophy, Version 1.0, London: Routledge |
