# How quickly did Goedel's Incompleteness Theorem become known and heeded throughout mathematics

Does anyone know how news of Goedel's incompleteness theorem spread? Did it do so little by little, or was it shouted in dramatic headlines throughout mathematical literature? If anyone can point me to the history of how people reacted to Goedel's result, I would be grateful.

I am asking this question because I have recently been reading a work that comes from a field very far from logic, namely Lie theory, written in 1940 (ten years after Über formal unentscheidbare...") wherein the author seems to be highly mindful of the different mathematical philosophies - he is at least paying lip service to intuitionism and how his work might make sense in that philosophy (see below). Now I should have thought that Goedel's incompleteness theorem would take much of the heat out of debates as to who had the "best" philosophy. Or did it put wind in the sails of the intuitionists, after Goedel had seemingly demolished the formalists, although I believe Goedel would not have seen his incompleteness theorem validating the intuitionists either, being as he was a strong Platonist. Anyhow, here is the quote: it is "Hauptsatz 1" in the paper and I was fascinated to read these words in the far-removed-from-logic field of Lie theory:

From H. Freudenthal "Die Topologie der Lieschen Gruppen Als Algebraisches Ph\"anomen" Annals of Mathematics vol 42. # 5 (1941) wherein he makes the following statement:

"Main Theorem 1: An isomorphism between two Lie groups, of which one is simple and of the second kind, is needfully continuous. Otherwise put: in the theory of Lie groups, the topology of simple groups of the second kind is a wholly algebraic phenomenon"

Lest you should think that the rewording "otherwise put ..." cannot be construed as a precise statement of a theorem (it does on the surface seem rather vague), Freudenthal goes on to explain:

"In the latter formulation the main theorem also makes sense for someone who outright refuses [the existence of] discontinuous mappings, such as [someone with] intuitionist leanings"

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Godel was hardly the last word on the philosophy of mathematics. Also, I think you are reading far too much into a single sentence in Freudenthal's paper. –  Andy Putman May 20 '11 at 5:35
@ Andy - certainly I agree with your statement about Goedel being the last word on philosophy, which is part of the point of my question - how did the different philosophies react to Godel, Turing, Post, Tarski etc? What I find interesting is the mention of different philophies in a paper far removed from logic. I know Freudenthal's Hilbert number was 2 (through van der Waerden) and he would have had much contact with the the latter, being a good friend. It's almost as though the 1920's debate between Brouwer and Hilbert is still to the fore in Freudenthal's mind when he makes the statement. –  WetSavannaAnimal aka Rod Vance May 20 '11 at 5:49
I don't see why this has to do with Goedel per se, still less the incompleteness theorems. Model theory, perhaps. –  Yemon Choi May 20 '11 at 5:53
@ Yemon I am only guessing (which is why I would like to {\it know} the history) that the incompleteness theorems might have taken some of the heat out of claims of various philosophies to be superior. I can't believe that adherents to various standpoints would have carried on heedless of such results as Goedel's, Tarski's, Turing's etc –  WetSavannaAnimal aka Rod Vance May 20 '11 at 6:00
Mathias's article "The ignorance of Bourbaki" (dpmms.cam.ac.uk/~ardm/bourbaki.pdf) seems to be quite relevant to your question. –  S. Carnahan May 20 '11 at 9:12

I believe the answer to the title of the question is "Quickly". The Wikipedia article on "Gödel's incompleteness theorems" has a nice discussion of the developments in the 1930's. The title of Gödel's paper on the incompleteness theorem is "Über formal unentscheidbare Sätze... I", and apparently he never needed to write part II.

It seems that the correctness of Gödel's paper was quickly realized, but rather than taking the heat out of any philosophical debate, it provided new and interesting fuel.

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A good supplement to the historical discussion in the Wikepedia article is the following paper. Paolo Mancosu (1999). Between Vienna and Berlin: The Immediate Reception of Gödel's Incompleteness Theorems. History and Philosophy of Logic 20 (1):33-45. The paper also appears in Mancosu's The Adventure of Reason, Oxford University Press, 2010. –  Ali Enayat May 20 '11 at 15:54
Thanks a bundle Ali. This seems EXACTLY what I am looking for. If you have not already done so, you might like to see S. Carnahan's article cited in the comments to my questions. –  WetSavannaAnimal aka Rod Vance May 21 '11 at 4:56

Another possibly relevant article:

Dawson, John W. Jr. (1984). "The Reception of Godel's Incompleteness Theorems". Proceedings of the Biennial Meeting of the Philosophy of Science Association, Vol. 1984,. Volume Two: Symposia and Invited Papers. The University of Chicago Press on behalf of the Philosophy of Science Association. pp. 253-271. JSTOR 192508.

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@none: fantastic! - just what I wanted, many thanks –  WetSavannaAnimal aka Rod Vance May 29 '11 at 2:41