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                                   [log in to unmask]
One art/there is/no less/no more/All things/to do/with sparks/galore
        --Douglas Hofstadter