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 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" 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. 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 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. 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 during the period in which Wittgenstein's early ideal language philosophy and "Tractatus Logico-Philosophicus" dominated the circle's thinking; multiple writings of Gödel in his Nachlass 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.