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The first question is a metaphysical question with close affinities to questions about the existence of other entities such as universals, properties and values. According to many philosophers, if such entities exist then they do so outside of space and time, and they lack causal powers; they are often termed abstract (as opposed to concrete) entities. If we accept the existence of abstract mathematical objects then an adequate epistemology of mathematics must explain how we can know about them. Of course, proofs seem to be our main source of justification for mathematical propositions but proofs depend on axioms and so the question of how we can know the truth of the axioms remains.
It is usually thought that mathematical truths are necessary truths; how then is it possible for finite, physical beings inhabiting a contingent world to have knowledge of such truths? Two broad views are possible: either mathematical truths are known by reason; or they are known by inference from sensory experience. The former rationalist view is adopted by Descartes and Leibniz who also thought that mathematical concepts are innate. Locke and Hume agreed that mathematical truths were known by reason but they thought all mathematical concepts were derived by abstraction from experience. Mill was a complete empiricist about mathematics and held both that mathematical concepts are derived from experience and also that mathematical truths like 2+2=4 are really inductive generalisations from experience. (N.B. Kant’s views on mathematics are complex and important; see Kant.)
The discovery in the mid-nineteenth century of non-Euclidean geometry meant that philosophers were forced to reassess the status of Euclidean geometry which had previously been regarded as the shinning example of certain knowledge of the world. Many took the existence of consistent non-Euclidean geometries to be a direct refutation of both Mill’s and Kant’s philosophies of mathematics. By the end of the nineteenth century Cantor had discovered various paradoxes in the theory of classes and there was something of a crisis in the foundations of mathematics. The early twentieth century saw great advances in mathematics and also in mathematical logic and the foundations of mathematics.
Most of the fundamental issues in the philosophy of mathematics are accessible to anyone who is familiar with geometry and arithmetic and who has had the experience of following a mathematical proof. However, some of the most important philosophical developments of the twentieth century were instigated by the profound developments that have taken place in mathematics and logic, and a proper appreciation of these issues is only available to someone who has an understanding of basic set theory and intermediate logic. To study philosophy of mathematics at an advanced level one ought really to have followed a course which includes proofs of Gödel’s incompleteness theorems. What follows is a brief guide to reading on various topics in the philosophy of mathematics.
Realism about mathematics, namely the view that mathematical theorems are true propositions about abstract entities, is often called platonism because it was Plato who first proposed the existence of a realm of abstract, unchanging forms wherein the numbers and other mathematical objects reside.
The most well-known modern platonist is Kurt Gödel:
A recent version of platonism is the neo-Fregean form advocated chiefly by Crispin Wright:
There is a tradition in philosophy called nominalism which rejects the existence of all abstract entities. Nominalists face perhaps their biggest difficulty in accounting for the successful application of mathematics. One form of nominalism is formalism which is the doctrine that mathematics is nothing more than the manipulation of symbols according to certain rules. A version of this view was first propounded by the great mathematician of the early twentieth century David Hilbert. Formalism avoids many problems faced by platonism but leaves us with the question of why anyone would bother to do mathematics or find it useful. Hilbert’s programme was devastated by Gödel’s incompleteness theorems but an answer to this question has recently been proposed by Hartry Field.
Intuitionism was proposed by Brouwer as a philosophy of mathematics without foundations. Whereas Kant had sort to ground arithmetic in the experience of time and geometry in the experience of space, Brouwer attempted to account for all of mathematics in terms of the intuition (conscious experience) of time. Intuitionism clashed with classical mathematics in so far as Brouwer held that there are no truths beyond experience, and hence that the law of the excluded middle could not be applied to all mathematical statements (in particular infinitary parts of mathematics are indeterminate with respect to some properties). The successor of intuitionism, namely constructivism, has abandoned Kantian metaphysics and epistemology but still maintains that some mathematical statements, namely those for which no proof has been constructed, have no truth value.
The modern analytic tradition begins with the work of Frege and Russell (among others) for both of whom mathematics was a central concern. As mentioned above, mathematical statements, if they are true at all, are true necessarily. The principles of logic are also usually thought to be necessary truths; perhaps then mathematical truths are really just complicated logical truths. Logicism is the name given to the research programme initiated by Frege and developed by Russell and Whitehead the aim of which was to show how mathematics is reducible to logic. Frege attempted to provide mathematics with a sound logical foundation; unfortunately Russell discovered that Frege’s system was inconsistent. Russell’s famous work on the theory of types was an attempt to avoid the paradoxes that beset Frege’s version of logicism.
Set theory was developed in the late nineteenth century and by the mid-1920s had reached maturity with the axiomatic formulation developed by Zermelo, Fraenkel, Skolem and others which is now known as ZF set theory.
Mathematical logic is the formal study of mathematical structures and systems; its subparts include proof theory and model theory. The most important development in mathematical logic for the philosophy of mathematics was Gödel’s proof that any axiomatic system powerful enough to formalise arithmetic will be incomplete in the sense that there will be truths which are not provable within the system. This result is widely perceived as having dealt a mortal blow to the foundational programmes of both Hilbert and Russell, and its philosophical importance is hard to overestimate.
Quine proposed an empiricist view of mathematical knowledge combined with a form of platonism. He also proposed the much discussed indispensibility argument for plaotnism about mathematics.
Structuralists argue that mathematics is not about some particular collection of abstract objects but rather mathematics is the science of patterns of structures, and particular objects are relevant to mathematics only in so far as they instantiate some pattern or structure. Various versions of structuralism have been proposed the most important of which can be found in the sources below.
The notion of proof and its role in mathematical practice has been the subject of increased scrutiny in recent years. Developments in computer science have allowed some previously intractable problems to be tackled with computers. The status of the resulting ‘computer proofs’ is of considerable philosophical interest.
Since Zeno’s paradoxes and the discovery of the irrationality of square root of 2 by ancient Greek mathematicians, the infinite has been a source of puzzlement and controversy in mathematics. Of course, we learn as children that for any natural number we can think of there is always a bigger one, and so there is a sense in which the infinite is part of our most basic mathematics. However, mathematicians have always been concerned about the actually as opposed to the merely potentially infinite in their subject. In the nineteenth century Cantor developed his theory of transfinite sets in which he proved the existence of infinite sets which were bigger than the set of natural numbers. Although Cantor’s theory was greeted with a fair amount of hostility at the time, it has proved to be so useful that it is an indispensible part of modern mathematics forcing mathematicians to grapple with the infinite. However, there is a long tradition from Hilbert and Brouwer onwards of attempting to do without the actually infinite in mathematics.