Ludwig Wittgenstein considered his chief contribution to philosophy to be in the philosophy of mathematics, a topic to which he devoted over half his work between 1929 and 1944.[1] As with his philosophy of language, Wittgenstein's views on mathematics evolved from the period of the Tractatus Logico-Philosophicus to the ordinary language philosophy of his later years, changing from an intense logicism that placed him, alongside his mentor Bertrand Russell, at the heart of the world of mathematical logic, to a virulent anti-foundationalism and constructivism that was widely repudiated by the mathematical community. Of particular importance in Wittgenstein's thinking on mathematics is the text Remarks on the Foundations of Mathematics, which contains the late Wittgenstein's deeply controversial repudiation of Gödel's incompleteness theorems - a line of thinking which is often described as unclear, and which led to harsh criticism from the logical and mathematical community.
Wittgenstein's initial conception of mathematics was formalist[1] and 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.
After 1929, Wittgenstein's 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.[2] 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.