GÖDEL'S INCOMPLETENESS THEOREMS AND PHILOSOPHY OF MATH.

WITTGENSTEIN, LUDWIG. 1889-1951.

Autograph Letter Signed ("Ludwig Wittgenstein") to Moritz Schlick discussing Gödel's incompleteness theorems,

WITTGENSTEIN, LUDWIG. 1889-1951. Autograph Letter Signed ("Ludwig Wittgenstein") to Moritz Schlick discussing Gödel's incompleteness theorems, 13 pages, in German (English translation provided), Trinity College, Cambridge, July 31, 1935, folds, pages numbered in pencil.

WITTGENSTEIN ON GÖDEL'S THEORY IN GREAT DETAIL, a central document, often cited, in the the philosophy of mathematics and the Wittgenstein-Gödel debate. Written in response to a letter by Moritz Schlick, Wittgenstein advises an apparently astonished Schlick: "If someone says to you ... that there have to be unprovable theorems in mathematics, there is nothing at all astounding [Erstaunliches] in this" [emphasis Wittgenstein], because you don't yet have any idea what this apparently clear prose sentence says." Wittgenstein fundamentally considered himself a philosopher of mathematics, as he declared in his 1944 amendment to a short biography by John Wisdom, "[my] chief contribution has been in the philosophy of mathematics" (Monk, p 466). This exceptional letter, much debated, on unprovable theorems reveals essential Wittgenstein at the height of his philosophic genius.

Wittgenstein's middle period is largely defined by his investigations in mathematics – between 1929 and 1944, over half his writings are devoted to the subject. Investigating from a number of perspectives, including the examination of mathematical truth, the nature of propositions, the consistency of formal systems, he broke new ground, in particular, with an iconoclastic theory of Proof. Wittgenstein held that a theorem's meaning was dependent on the proof used to construct it, "If you want to know what is proved, look at the proof" (Philosophical Grammar, p 369), acknowledging more than one proof path or proof logic leading to a theorem. Wittgenstein's philosophy of mathematics is his most challenging work, and it remains an under-explored area of study.

Building on ideas expressed in his Remarks on the Foundation of Mathematics (written 1937-44), Wittgenstein in this letter expands on his post-Tractatus views:
"All that you can learn from 'my guidance' is that nothing can be said about such a proof before you have examined this particular proof. That is: the philosopher is always wrong, if he somehow wants to prophesy something in mathematics and says 'that is impossible,' 'that can never be proven.' Why not? That which was proved is only an expression of language [Wortausdruck], and the proof gives its particular meaning. And 'how much justification' we have for calling this proof the proof of this prose sentence is, to some extent, a matter of taste. That is: it's a matter of our judgment and our preference ... The funny thing is, that one cannot say anything 'formally' concerning a proof, that one hasn't in fact investigated, except that one must investigate it oneself in order to see how appropriate or inappropriate it is to call it a proof of this prose sentence. Philosophy can say nothing about a proof."

The letter, once in the possession of Schlick's daughter, is of particular relevance to Appendix 3 (of RFM), which contains a particularly controversial section now known as "the notorious paragraph." Wittgenstein faced a great deal of criticism for his critique of Gödel, including from Gödel himself, but recent debate and scholarship has given rise to myriad alternative interpretations, and this letter offers an important window to those arguments.

Physicist Moritz Schlick tagged Wittgenstein "the decisive turning point of philosophy," and made the Tractatus the philosophic foundation of "The Vienna Circle," the 20th-century's most prominent association of scientific philosophers (of which a young Gödel was a member). Wittgenstein and Schlick corresponded for some 10 years and the present is Wittgenstein's final letter to Schlick, who was suddenly murdered in 1936. Beyond its significance for the philosophy of mathematics, this lengthy letter is of considerable biographical importance. It records Wittgenstein's thoughts on two of his main "disciples," Alice Ambrose and Friedrich Waismann (respectively denigrating and praising), and it further contains a number of significant statements about his own personal relationship to philosophy, including his statements, "I probably won't practice philosophy" [after his trip to Russia], and "Who knows if I'll ever write anything more." Wittgenstein however does indicate at the end of this letter the manner in which he would like his philosophy transmitted to posterity.

Bertrand Russell described Wittgenstein as "the most perfect example I have ever known of genius traditionally conceived: passionate, profound, intense, and dominating." Wittgenstein letters of philosophical import are exceedingly rare in commerce. Evidencing Wittgenstein at his most essential, grappling extemporaneously with the most "astonishing" logical puzzle of modern times: Godel's incompleteness theorem. The present letter is the most spectacular Wittgenstein autograph to appear at auction in the past 25 years.

References:
Monk, Ray. Ludwig Wittgenstein: The Duty of Genius. New York: The Free Press, 1999.
Wittgenstein, Ludwig. Philosophical Investigations. Oxford: Blackwell, 1953.
Wittgenstein, Ludwig. Remarks on the Foundation of Mathematics. Cambridge: MIT Press, 1956.