''Remarks on the Foundations of Mathematics'' () is a book of

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 ...

's notes on 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 ...

. It has been translated from German to English by G.E.M. Anscombe, edited by G.H. von Wright and Rush Rhees, 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'' () is a work by the philosopher Ludwig Wittgenstein, published posthumously in 1953. ''Philosophical Investigations'' is divided into two parts, consisting of what Wittgenstein calls, in the preface, ''Bemer ...

'' which were composed later. When the book appeared it received many negative reviews mostly from working logicians and mathematicians, among them Michael Dummett, Paul Bernays, and Georg Kreisel. Kreisel's scathing review received particular attention although he later distanced himself from it. In later years however it received more positive reviews. 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 a deductive reasoning, deductive Argument-deduction-proof distinctions, argument for a Proposition, mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use othe ...

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 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 ...

. Multiple commentators read Wittgenstein as misunderstanding Gödel. In 2000 Juliet Floyd and

Hilary Putnam Hilary Whitehall Putnam (; July 31, 1926 – March 13, 2016) was an American philosopher, mathematician, computer scientist, and figure in analytic philosophy in the second half of the 20th century. He contributed to the studies of philosophy of ...

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 Kurt Friedrich Gödel ( ; ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel profoundly ...

who was a member of the Vienna Circle during the period in which Wittgenstein's early ideal language philosophy and ''

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 ...

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

Nachlass ''Nachlass'' (, older spelling ''Nachlaß'') is a German language, 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 word, compound in ...

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

{{Authority control 1953 non-fiction books Books by Ludwig Wittgenstein Philosophy of mathematics literature Logic literature

See also

References

External links