Wittgenstein's philosophy of mathematics

Ludwig Wittgenstein considered his chief contribution to philosophy to be in the philosophy of mathematics, a topic to which he devoted much of his work between 1929 and 1944. As with his philosophy of language, Wittgenstein's views on mathematics evolved from the period of the Tractatus Logico-Philosophicus, changing from logicism which was endorsed by his mentor Bertrand Russell, to a general anti-foundationalism and constructivism that was not readily accepted by the mathematical community. The main source in Wittgenstein's thinking on mathematics is the text compiled as Remarks on the Foundations of Mathematics, which contains the late Wittgenstein's views, notably a controversial repudiation of Gödel's incompleteness theorems. The success of Wittgenstein's general philosophy has tended to displace the real debates on more technical issues.

Wittgenstein's initial conception of mathematics was logicist and even formalist. The Tractatus described the propositions of logic as a series of tautologies derived from syntactic manipulation, and without the pictorial force of elementary propositions depicting states of affairs obtaining in the world.

Wittgenstein asserted that “[t]he logic of the world, which is shown in tautologies by the propositions of logic, is shown in equations by mathematics” (6.22) and further that “Mathematics is a method of logic” (6.234).

During the 1920s Wittgenstein turned away from philosophical matters but his interest in mathematics was rekindled when he attended in Vienna a lecture by the intuitionist L. E. J. Brouwer. After 1929, his primary mathematical preoccupation entailed resolving the account of logical necessity he had articulated in the Tractatus Logico-Philosophicus—an issue which had been fiercely pressed by Frank P. Ramsey. Wittgenstein's initial response, Some Remarks on Logical Form, was the only academic paper he published during his lifetime, and marked the beginnings of a departure from the ideal language philosophy and correspondence theory of truth of the Tractatus.

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