Friday, September 23, 2005

75 Years of Incompleteness

On October 7th, we'll see the 75th anniversary of Godel's unveiling of his (first) incompleteness. I'm now reading Rebecca Goldstein's excellent "Incompleteness", which not only explains Godel's theorems in simple language, but also provides the context both of the history of mathematics at that point, and of Godel's personality.

The book raises some thoughts:

* If there were no WWI, there would be no Vienna Circle (or Schlick's Circle, as it was originally called; there were many discussion circles in post-war Vienna). The modernist movement in Vienna, attempting to rebuild all thought on clean, solid foundations, was in part a reaction to everything in past thinking habits and assumptions, habits that to some had seemed to lead to the war.

* The stark constrast between Wittgenstein's classic, tortured brand of genius, and Godel's quiet, hermetic brand. Goldstein suspects, but cannot prove, that Godel was spurred on in his work, subconsciously at least, by the idea of shutting up the arrogant Wittgenstein.

(Goldstein notes that the fact that Godel could use only mathematics to say something metamathematical, even metaphysical, is in complete opposition to what Wittgenstein maintained about the limits of sayable things in a language.)

* Wittgenstein and Russell were both aristocrats. I knew this of each of them, but never really noticed that it was true of both of them at the time they were interacting at Cambridge. I'm not sure if it means anything, or if it's just an interesting coincidence.

* When Godel unveiled his idea at a conference (Oct. 7, 1930, in Konigsberg), it went almost completely unnoticed. He spoke for perhaps 30 seconds, in a quiet, casual voice (judging by everything else in his life), and was only 24 at the time. Had not John von Neumann took interest and buttonholed him later, it might have taken years longer for Godel's theorem to take hold. (He had been sitting on the theorem since the previous year, and throughout his life his unsent letters and papers far outnumbered his actual publications.)