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11. Philosophy of Mathematics

MARY TILES


Subject Philosophy

Key-Topics mathematics

DOI: 10.1111/b.9780631219088.2002.00016.x


Extract

Introductions to the philosophy of mathematics often begin where Körner's (1960) influential introduction began, outlining three positions: logicism, formalism and intuitionism. These were the three contending schools to emerge from nineteenth-century mathematical moves to provide rigorous foundations for mathematical analysis (including infinitesimal calculus). The problem for the philosophy of mathematics has been that (1) these seemed to represent all the reasonable positions available, and (2) in the light of Gödel's incompleteness theorems and other results proved by Turing, Church, Skolem and Tarski in the 1930s, neither logicism nor formalism seemed a philosophically viable position. Intuitionism, whilst having its philosophical credentials intact, was unacceptable to most mathematicians because it involved discarding parts of classically accepted mathematics. This apparent impasse partly explains the decline of interest in the philosophy of mathematics since the first part of this century, when for a while, with the work of Russell and Whitehead, and the Vienna Circle logical positivists, it seemed to occupy centre stage. Hao Wang, in his perceptive retrospective analysis of the philosophy of mathematics of this period ( Wang 1968 ), explains this trajectory in terms of the wider movements of analytic philosophy, which were initially strongly empiricist and which consequently ... log in or subscribe to read full text

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