Remarks on the Foundations of Mathematics
Remarks on the Foundations of Mathematics (German: Bemerkungen über die Grundlagen der Mathematik) 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. The text, embodying much of Wittgenstein's philosophy of mathematics, contains an extended analysis of the concept of mathematical proof 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, Paul Bernays, and Georg Kreisel at publication.
Particularly controversial in the Remarks was Wittgenstein's "notorious paragraph", which contained an unusual commentary on Gödel's incompleteness theorems. Multiple commentators have read Wittgenstein as misunderstanding Gödel, although recently Hilary Putnam has suggested that the majority of commentary misunderstands Wittgenstein. Gödel 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 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.
- Wittgenstein, Ludwig; G. H. von Wright, R. Rhees, G. E. M. Anscombe. "Remarks on the Foundations of Mathematics, Revised Edition". MIT. p. 1. Retrieved 5 September 2010.
- Kreisel, G. (2010). "WITTGENSTEIN'S REMARKS ON THE FOUNDATIONS OF MATHEMATICS". The British Journal for the Philosophy of Science (British Society for the Philosophy of Science) IX (61 (3)): 135–158. doi:10.1093/bjps/IX.34.135.
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- "Gödel And The Nature Of Mathematical Truth". Edge. 2002-04-22. Retrieved 2013-10-17.