Remarks on the Foundations of Mathematics is a 1953 book of Ludwig Wittgenstein

Ludwig Wittgenstein

Ludwig Josef Johann Wittgenstein was an Austrian philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. He was professor in philosophy at the University of Cambridge from 1939 until 1947...

's notes on the philosophy of mathematics

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...

, translated from German to English by G.E.M. Anscombe, edited by G.H. von Wright and Rush Rhees

Rush Rhees

Rush Rhees was a philosopher at Swansea University from 1940 to 1966Rhees is principally known as a student, friend, and literary executor of the philosopher Ludwig Wittgenstein. With G. E. M. Anscombe, he edited Wittgenstein's posthumous Philosophical Investigations , a highly influential work...

, and published first in 1956. The text, embodying much of Wittgenstein's philosophy of mathematics

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 over half his work between 1929 and 1944...

, contains an extended analysis of the concept of mathematical proof

Mathematical proof

In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...

 and an exploration of Wittgenstein's contention that many philosophical problems arise in mathematics at the level of elementary computations. Wittgenstein in the Remarks adopts an attitude of doubt in opposition to much orthodoxy in the philosophy of mathematics, which led to skepticism and multiple negative reviews by such Wittgensteinian logicians as Michael Dummett

Michael Dummett

Sir Michael Anthony Eardley Dummett FBA D.Litt is a British philosopher. He was, until 1992, Wykeham Professor of Logic at the University of Oxford...

, Paul Bernays

Paul Bernays

Paul Isaac Bernays was a Swiss mathematician, who made significant contributions to mathematical logic, axiomatic set theory, and the philosophy of mathematics. He was an assistant to, and close collaborator of, David Hilbert.-Biography:Bernays spent his childhood in Berlin. Bernays attended the...

, and George Kreisel at publication.

Particularly controversial in the Remarks was Wittgenstein's "notorious paragraph", which contained an unusual commentary on 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...

. Multiple commentators have read Wittgenstein as misunderstanding Gödel

Godel

Godel or similar can mean:*Kurt Gödel , an Austrian logician, mathematician and philosopher*Gödel...

, although recent analysis by Hilary Putnam

Hilary Putnam

Hilary Whitehall Putnam is an American philosopher, mathematician and computer scientist, who has been a central figure in analytic philosophy since the 1960s, especially in philosophy of mind, philosophy of language, philosophy of mathematics, and philosophy of science...

 has suggested that the majority of commentary misunderstands Wittgenstein. Gödel was a member of the Vienna Circle

Vienna Circle

The Vienna Circle was an association of philosophers gathered around the University of Vienna in 1922, chaired by Moritz Schlick, also known as the Ernst Mach Society in honour of Ernst Mach...

 during the period in which Wittgenstein's early ideal language philosophy and 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...

 dominated the circle's thinking; multiple writings of Gödel in his Nachlass

Nachlass

Nachlass is a German word, used in academia to describe the collection of manuscripts, notes, correspondence, and so on left behind when a scholar dies. The word is a compound in German: nach means 'after', and the verb lassen means 'leave'. The plural can be either Nachlasse or Nachlässe...

 contain his antipathy for Wittgenstein, and belief that Wittgenstein wilfully misread the theorems. Some commentators, such as Rebecca Goldstein

Rebecca Goldstein

Rebecca Goldstein is an American novelist and professor of philosophy. She has written five novels, a number of short stories and essays, and biographical studies of mathematician Kurt Gödel and philosopher Baruch Spinoza....

, have hypothesized that Gödel developed his logical theorems in opposition to Wittgenstein.