Ludwig Wittgenstein Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrian philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. From 1929 to 1947, Witt ...

considered his chief contribution to be in the

philosophy of mathematics Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship to other areas of philosophy, particularly epistemology and metaphysics. Central questions posed include whether or not mathem ...

, 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 The ''Stanford Encyclopedia of Philosophy'' (''SEP'') is a freely available online philosophy resource published and maintained by Stanford University, encompassing both an online encyclopedia of philosophy and peer-reviewed original publication ...

As with his

philosophy of language Philosophy of language refers to the philosophical study of the nature of language. It investigates the relationship between language, language users, and the world. Investigations may include inquiry into the nature of Meaning (philosophy), me ...

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

Tractatus Logico-Philosophicus The ''Tractatus Logico-Philosophicus'' (widely abbreviated and Citation, cited as TLP) is the only book-length philosophical work by the Austrian philosopher Ludwig Wittgenstein that was published during his lifetime. The project had a broad goal ...

'': 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 al ...

(which was endorsed by his mentor

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

) towards a general

anti-foundationalism Anti-foundationalism (also called nonfoundationalism) is any philosophy which rejects a foundationalist approach. An anti-foundationalist is one who does not believe that there is some fundamental belief or principle which is the basic ground or ...

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 the Soviet Union in t ...

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 ''Remarks on the Foundations of Mathematics'' () is a book of Ludwig Wittgenstein's notes on the philosophy of mathematics. It has been translated from German to English by G.E.M. Anscombe, edited by G.H. von Wright and Rush Rhees, and publis ...

'' contains his compiled views, notably a controversial repudiation of

Gödel's incompleteness theorems Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the phi ...

Tractatus

Wittgenstein's initial conception of mathematics was logicist and even

formalist Formalism may refer to: * Legal formalism, legal positivist view that the substantive justice of a law is a question for the legislature rather than the judiciary * Formalism (linguistics) * Scientific formalism * A rough synonym to the Formal sys ...

. The ''Tractatus'' described the propositions of logic as a series of tautologies derived from syntactic manipulation, and without the

pictorial An image or picture is a visual representation. An image can be two-dimensional, such as a drawing, painting, or photograph, or three-dimensional, such as a carving or sculpture. Images may be displayed through other media, including a project ...

force of

elementary propositions In logic and analytic philosophy, an atomic sentence is a type of declarative sentence which is either true or false (may also be referred to as a proposition, statement or truthbearer) and which cannot be broken down into other simpler sentences ...

depicting

states of affairs In philosophy, a state of affairs (), 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'', whereas a proposit ...

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 ...

L. E. J. Brouwer Luitzen Egbertus Jan "Bertus" Brouwer (27 February 1881 – 2 December 1966) was a Dutch mathematician and philosopher who worked in topology, set theory, measure theory and complex analysis. Regarded as one of the greatest mathematicians of the ...

. 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 people, 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 ...

. 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 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 In metaphysics and philosophy of language, the correspondence theory of truth states that the truth or falsity of a statement is determined only by how it relates to the world and whether it accurately describes (i.e., corresponds with) that worl ...

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, ske ...

, 1980, ''Wittgenstein on the Foundations of Mathematics'', Harvard University Press, *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

{{Ludwig Wittgenstein

Mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

Wittgenstein, Ludwig