A Dictionary of Philosophy, Third Edition
Mathematics (philosophy of)
. The study of concepts and systems appearing in mathematics, of the nature of mathematical knowledge, and of the justification of mathematical statements.
The basic objects of mathematics are NUMBERS, of which there are various kinds (cardinal, ordinal, natural, real, etc.; see Russell). Whether numbers are real entities, and how we decide this, are questions shared with metaphysics (cf. BEING). Platonists, or realists, think that numbers are ABSTRACT entities in the sense of being outside space and time, and that mathematical truths, including those about infinite numbers, exist independently of our researches. Constructivists emphasize the dependence of mathematics on the activity of mathematicians (see INTUITIONISM, FINITISM). Numbers have been defined in terms of CLASSES or sets which are themselves studied by set theory (cf. CALCULUS). In fact the definition of ‘equinumerous class’ helped G.Cantor (1845–1918) to elaborate the study of infinite numbers. One class is equinumerous (‘equivalent’ and ‘equipollent’ are other terms used here) to another if for each member of one there can be found exactly one corresponding member of the other. These definitions enable the calculus of classes, and set theory, to be used to axiomatize mathematics (cf. AXIOM SYSTEM, LOGIC). One difficulty about classes stems from RUSSELL’S PARADOX. Logicists, however (notably Frege and Russell), who claim to reduce mathematics entirely to logic, have used classes for this purpose. They claim that mathematical objects can be defined in logical terms, via classes, and also that mathematical proofs can be reduced to logical proofs. But there are difficulties in proving the existence of classes in general from axioms which can be reasonably regarded as purely logical.
Geometry was originally regarded as fundamentally different from arithmetic, and as dealing with space. Kant’s views on space provoked questions about how different geometries relate to each other and to real SPACE.
Arithmetic suggests the question how different algebras relate to each other and to the world (cf. AXIOM SYSTEM). These questions are connected with the questions whether our knowledge of mathematical concepts and propositions is empirical as, in particular, J.S.Mill thought, or A PRIORI. Modern developments on all these questions have tended to unite arithmetic and geometry.
Other questions include: what is truth in mathematics, and how is it related to provability? Is every mathematical truth provable (cf. GÖDEL’S THEOREMS), and can we know mathematical truths by direct insight as well as by proof?
*S.F.Barker, Philosophy of Mathematics, Prentice-Hall, 1964. (Elementary.)
P.Benacerraf and H.Putnam (eds), Philosophy of Mathematics, Blackwell, 1964. (Important collection of readings, of varying difficulty.)
*G.Frege, Foundations of Arithmetic, Blackwell, 1884, trans. by J. L.Austin, 1950. (Fairly elementary discussion of numbers, etc. from Platonist and logicist viewpoint by a pioneer of modern logic. Includes more technical appendix on Russell’s paradox.)
L.Goddard, ‘“True” and “provable”’, Mind, 1958. (Cf. discussions in Mind, 1960, 1962.)
S.Körner, Philosophy of Mathematics, Hutchinson, 1960. (Rather more advanced.)
I.Lakatos, Proofs and Refutations, Cambridge UP, 1976, original version 1963–4. (Elaborate discussion, in dialogue form with historical notes, of genesis of some problems about proof.)
H.Lehman [see bibliography to A PRIORI.]
B.Russell, Introduction to Mathematical Philosophy, Allen and Unwin, 1919. (General.)
G.Ryle, C.Lewy, K.R.Popper, ‘Why are the calculuses of logic and arithmetic applicable to reality?’, Proceedings of the Aristotelian Society , supplementary vol., 1946.
*H.Wang, ‘Process and existence in mathematics’, in Y.Bar-Hillel, et al. (eds), Essays on the Foundations of Mathematics, Magnes Press, Hebrew University, Jerusalem, 1962. (Brings out some of the issues in mainly simple language.)
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