Wittgenstein's philosophy of mathematics
Encyclopedia
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. 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.
force of elementary propositions depicting states of affairs obtaining in the world.
—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.
Philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of...
, a topic to which he devoted over half his work between 1929 and 1944. As with his philosophy of language
Philosophy of language
Philosophy of language is the reasoned inquiry into the nature, origins, and usage of language. As a topic, the philosophy of language for analytic philosophers is concerned with four central problems: the nature of meaning, language use, language cognition, and the relationship between language...
, Wittgenstein's views on mathematics evolved from the period of the Tractatus Logico-Philosophicus
Tractatus Logico-Philosophicus
The Tractatus Logico-Philosophicus is the only book-length philosophical work published by the Austrian philosopher Ludwig Wittgenstein in his lifetime. It was an ambitious project: to identify the relationship between language and reality and to define the limits of science...
to the ordinary language philosophy
Ordinary language philosophy
Ordinary language philosophy is a philosophical school that approaches traditional philosophical problems as rooted in misunderstandings philosophers develop by distorting or forgetting what words actually mean in everyday use....
of his later years, changing from an intense logicism
Logicism
Logicism is one of the schools of thought in the philosophy of mathematics, putting forth the theory that mathematics is an extension of logic and therefore some or all mathematics is reducible to logic. Bertrand Russell and Alfred North Whitehead championed this theory fathered by Richard Dedekind...
that placed him, alongside his mentor Bertrand Russell
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
, at the heart of the world of mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
, to a virulent anti-foundationalism
Anti-foundationalism
Anti-foundationalism as the name implies, is a term applied to any philosophy which rejects a foundationalist approach, i.e. an anti-foundationalist is one who does not believe that there is some fundamental belief or principle which is the basic ground or foundation of inquiry and knowledge...
and constructivism
Constructivism (mathematics)
In the philosophy of mathematics, constructivism asserts that it is necessary to find a mathematical object to prove that it exists. When one assumes that an object does not exist and derives a contradiction from that assumption, one still has not found the object and therefore not proved its...
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
Remarks on the Foundations of Mathematics
Remarks on the Foundations of Mathematics is a 1953 book of Ludwig Wittgenstein's notes on the philosophy of mathematics, translated from German to English by G.E.M. Anscombe, edited by G.H. von Wright and Rush Rhees, and published first in 1956...
, which contains the late Wittgenstein's deeply controversial repudiation of Gödel's incompleteness theorems
Gödel's incompleteness theorems
Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of...
- a line of thinking which is often described as unclear, and which led to harsh criticism from the logical and mathematical community.
Tractatus
Wittgenstein's initial conception of mathematics was formalist and described the propositions of logic as a series of tautologies derived from syntactic manipulation, and without the pictorialPicture theory of language
The picture theory of language, also known as the picture theory of meaning, is a theory of linguistic reference and meaning articulated by Ludwig Wittgenstein in the Tractatus Logico-Philosophicus. Wittgenstein suggested that a meaningful proposition pictured a state of affairs or empirical fact....
force of elementary propositions depicting states of affairs obtaining in the world.
Philosophy of mathematics, post-1929
After 1929, Wittgenstein's primary mathematical preoccupation entailed resolving the account of logical necessity he had articulated in the Tractatus Logico-PhilosophicusTractatus Logico-Philosophicus
The Tractatus Logico-Philosophicus is the only book-length philosophical work published by the Austrian philosopher Ludwig Wittgenstein in his lifetime. It was an ambitious project: to identify the relationship between language and reality and to define the limits of science...
—an issue which had been fiercely pressed by Frank P. Ramsey
Frank P. Ramsey
Frank Plumpton Ramsey was a British mathematician who, in addition to mathematics, made significant and precocious contributions in philosophy and economics before his death at the age of 26...
. Wittgenstein's initial response, Some Remarks on Logical Form
Some Remarks on Logical Form
Some Remarks on Logical Form 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 Wittgenstein of the Tractatus Logico-Philosophicus from the later...
, 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
Correspondence theory of truth
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 that world...
of the Tractatus.