From: | John Cowan <cowan@...> |
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Date: | Saturday, November 18, 2000, 23:31 |

On Sat, 18 Nov 2000, Yoon Ha Lee wrote:> I love how the history of mathematical prejudice is recorded in the names > of number-types: rational, irrational, imaginary, complex, nonstandard...<G>Don't forget negative.> > Wouldn't it be cool if there was a finite proof for G? Nobody actually > > knows if it's true -- but if it were, nonstandard numbers would be > > *hard-wired* into number theory, willy-nilly. > > G? <puzzled look> Clarify, please? I'm mightily curious, but also > rather ignorant. :-( (The only thing that comes to mind is "g" in > psychology, which I'm guessing is an entirely different animal.)Right. G is the Goedel sentence, the one whose meta-mathematical interpretation is "G has no proof". (Its purely *mathematical* interpretation is just a relationship between some rather large numbers, but when you map the numbers onto statements in proofs, you find that G asserts that it has no proof.) However, I blundered above by saying "finite proof for G"; it's precisely Goedel's Theorem that G has no proof. I meant "finite proof for not-G". This is an open question, and most number theory people believe it's false, but there is no proof either way. In standard number theory, we assume G is true, so there are no nonstandard numbers. In nonstandard number theory, we assume ~G is (i.e. "G has a proof"), but we still cannot construct a proof for G that is not infinitely long (and a good thing too, since it's false)! But if there were an *independent* proof of ~G, then nonstandard number theory (analysis, topology, etc. etc.) is the only kind there is, and nonstandard numbers must exist whether we want them or not.> YHL, who loves history of math but doesn't quite know enough math > (Actually, I'm not sure anyone does, what with the explosion of > sub-sub-fields and specialists who can't talk to each other. Some > professors at Cornell have scared me with tales of going to mathematical > conferences where someone would give a paper and only one person in the > audience would pay attention while the rest went out for coffee, because > everyone else had no clue what was going on!)Too right. There is an article somewhere (but apparently not on the net) about a professor applying for a grant to study "hypertwistoploppic pseudotheomorphisms" (fictional); one of the lines in the application is "Only four other people in the world understand anything about it." -- John Cowan cowan@ccil.org One art/there is/no less/no more/All things/to do/with sparks/galore --Douglas Hofstadter