Ludwig Wittgenstein Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrian-British philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. He is con ...

considered his chief contribution to be in the

philosophy of mathematics The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in peop ...

, a topic to which he devoted much of his work between 1929 and 1944.Roydich V
''Wittgenstein's Philosophy of Mathematics''
The Stanford Encyclopedia of Philosophy As with his

philosophy of language In analytic philosophy, philosophy of language investigates the nature of language and the relations between language, language users, and the world. Investigations may include inquiry into the nature of meaning, intentionality, reference, ...

, Wittgenstein's views on mathematics evolved from the period of the

Tractatus Logico-Philosophicus The ''Tractatus Logico-Philosophicus'' (widely abbreviated and cited as TLP) is a book-length philosophical work by the Austrian philosopher Ludwig Wittgenstein which deals with the relationship between language and reality and aims to define th ...

: with him changing from

logicism In the philosophy of mathematics, logicism is a programme comprising one or more of the theses that — for some coherent meaning of 'logic' — mathematics is an extension of logic, some or all of mathematics is reducible to logic, or some or all ...

(which was endorsed by his mentor

Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ...

) towards a general anti-foundationalism and

constructivism Constructivism may refer to: Art and architecture * Constructivism (art), an early 20th-century artistic movement that extols art as a practice for social purposes * Constructivist architecture, an architectural movement in Russia in the 1920s a ...

that was not readily accepted by the mathematical community. The success of Wittgenstein's general philosophy has tended to displace the real debates on more technical issues. His '' Remarks on the Foundations of Mathematics'' contains his compiled views, notably a controversial repudiation of Gödel's incompleteness theorems.

Tractatus

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 An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...

force of elementary propositions depicting

states of affairs In philosophy, a state of affairs (german: Sachverhalt), also known as a situation, is a way the actual world must be in order to make some given ''proposition'' about the actual world true; in other words, a state of affairs is a ''truth-maker'', w ...

obtaining in the world. Wittgenstein asserted that “ e 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).

Philosophy of mathematics, post-1929

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 In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of f ...

L. E. J. Brouwer Luitzen Egbertus Jan Brouwer (; ; 27 February 1881 – 2 December 1966), usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, who worked in topology, set theory, measure theory and compl ...

. After 1929, his primary mathematical preoccupation entailed resolving the account of logical necessity he had articulated in the

—an issue which had been fiercely pressed by

Frank P. Ramsey Frank Plumpton Ramsey (; 22 February 1903 – 19 January 1930) was a British philosopher, mathematician, and economist who made major contributions to all three fields before his death at the age of 26. He was a close friend of Ludwig Wittgenste ...

. Wittgenstein's initial response,

Some Remarks on Logical Form "Some Remarks on Logical Form" (1929) was the only academic paper ever published by Ludwig Wittgenstein, and contained Wittgenstein's thinking on logic and the philosophy of mathematics immediately before the rupture that divided the early Wittgenst ...

, was the only academic paper he published during his lifetime, and marked the beginnings of a departure from the

ideal language philosophy Ideal language philosophy is contrasted with ordinary language philosophy. From about 1910 to 1930, analytic philosophers like Bertrand Russell and Ludwig Wittgenstein emphasized creating an ideal language for philosophical analysis, which would b ...

and correspondence theory of truth of the Tractatus.

The Lectures on the Foundations of mathematics

During the two terms of 1938/9 Wittgenstein lectured without any notes before students for two hours twice a week. From four sets of notes made during the lectures a text has been created, presenting Wittgenstein's views at that time.

The Remarks on the Foundations of mathematics (1937–44)

An editorial team prepared the edition of Wittgenstein's Remarks on the Foundations of mathematics from the manuscript notes he made during the years 1937–44. The material has been arranged in chronological order, allowing to observe some changes of emphasis or interest in Wittgenstein's views over the years.''Remarks on the Foundations of mathematics'' (1978) Revised Edition, Oxford: Basil Blackwell, G.H. von Wright, R. Rhees and G.E.M. Anscombe (eds.); translated by G.E.M Anscombe.

References

Crispin Wright Crispin James Garth Wright (; born 21 December 1942) is a British philosopher, who has written on neo-Fregean (neo-logicist) philosophy of mathematics, Wittgenstein's later philosophy, and on issues related to truth, realism, cognitivism, skep ...

, 1980, ''Wittgenstein on the Foundations of Mathematics'', Harvard University Press, {{ISBN, 0674953851 *Pasquale Frascolla, 1994, ''Wittgenstein's Philosophy of Mathematics'', Routledge

External links

*Sorin Bangu
''Ludwig Wittgenstein: Later Philosophy of Mathematics''
IEP *Victor Rodych
''Wittgenstein's Philosophy of Mathematics''
The Stanford Encyclopedia of Philosophy Mathematics Wittgenstein, Ludwig