''Remarks on the Foundations of Mathematics'' (german: Bemerkungen über die Grundlagen der Mathematik) is a book of

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 ...

's notes on 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 ...

. It has been translated from German to English by G.E.M. Anscombe, edited by

G.H. von Wright Georg Henrik von Wright (; 14 June 1916 – 16 June 2003) was a Finnish philosopher. Biography G. H. von Wright was born in Helsinki on 14 June 1916 to Tor von Wright and his wife Ragni Elisabeth Alfthan. On the retirement of Ludwig Wittgenste ...

and

Rush Rhees Rush Rhees (; 19 March 1905 – 22 May 1989) was an American philosopher. He is principally known as a student, friend, and literary executor of the philosopher Ludwig Wittgenstein. With G. E. M. Anscombe he was co-editor of Wittgenstein's pos ...

, and published first in 1956. The text has been produced from passages in various sources by selection and editing. The notes have been written during the years 1937–1944 and a few passages are incorporated in the ''

Philosophical Investigations ''Philosophical Investigations'' (german: Philosophische Untersuchungen) is a work by the philosopher Ludwig Wittgenstein, published posthumously in 1953. ''Philosophical Investigations'' is divided into two parts, consisting of what Wittgens ...

'' which were composed later. When the book appeared it received many negative reviews mostly from working logicians and mathematicians, among them

Michael Dummett Sir Michael Anthony Eardley Dummett (27 June 1925 – 27 December 2011) was an English academic described as "among the most significant British philosophers of the last century and a leading campaigner for racial tolerance and equality." He w ...

, Paul Bernays, and

Georg Kreisel Georg Kreisel FRS (September 15, 1923 – March 1, 2015) was an Austrian-born mathematical logician who studied and worked in the United Kingdom and America. Biography Kreisel was born in Graz and came from a Jewish background; his family ...

. Today ''Remarks on the Foundations of Mathematics'' is read mostly by philosophers sympathetic to Wittgenstein and they tend to adopt a more positive stance. Wittgenstein's philosophy of mathematics is exposed chiefly by simple examples on which further skeptical comments are made. The text offers an extended analysis of the concept of

mathematical proof A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proo ...

and an exploration of Wittgenstein's contention that philosophical considerations introduce false problems in mathematics. Wittgenstein in the Remarks adopts an attitude of doubt in opposition to much orthodoxy in the philosophy of mathematics. Particularly controversial in the Remarks was Wittgenstein's "notorious paragraph", which contained an unusual commentary on Gödel's incompleteness theorems. Multiple commentators read Wittgenstein as misunderstanding Gödel. In 2000

Juliet Floyd Juliet Floyd is professor of philosophy at Boston University. Her strongest research interests lie in early analytic philosophy (on which she has edited a volume) and she has used early analytic philosophy as a lens to examine a diverse array of ...

and Hilary Putnam suggested that the majority of commentary misunderstands Wittgenstein but their interpretation has not been met with approval. Wittgenstein wrote The debate has been running around the so-called ''Key Claim'': If one assumes that P is provable in PM, then one should give up the "translation" of P by the English sentence "P is not provable". Wittgenstein does not mention the name of Kurt Gödel who was a member of the

Vienna Circle The Vienna Circle (german: Wiener Kreis) of Logical Empiricism was a group of elite philosophers and scientists drawn from the natural and social sciences, logic and mathematics who met regularly from 1924 to 1936 at the University of Vienna, ch ...

during the period in which Wittgenstein's early

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 ''

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 ...

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

Nachlass ''Nachlass'' (, older spelling ''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 ...

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

Rebecca Goldstein Rebecca Newberger Goldstein (born February 23, 1950) is an American philosopher, novelist, and public intellectual. She has written ten books, both fiction and non-fiction. She holds a Ph.D. in philosophy of science from Princeton University, and ...

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

References

External links

*Sorin Bangu
''Ludwig Wittgenstein: Later Philosophy of Mathematics''
IEP *Victor Rodych
''Wittgenstein's Philosophy of Mathematics''
The Stanford Encyclopedia of Philosophy {{Authority control 1953 non-fiction books Ludwig Wittgenstein Books by Ludwig Wittgenstein Philosophy of mathematics literature Logic literature