[FOM] Re the future of history

S. S. Kutateladze sskut at math.nsc.ru
Mon Nov 12 11:34:30 EST 2007

Gabriel Stolzenberg> On November 6 in Re: [FOM] Q and A (nonstandard analysis),  Martin
Gabriel Stolzenberg> Davis wrote:

Gabriel Stolzenberg> Over the years, especially
Gabriel Stolzenberg>early on, I would hear it said that Robinson had
Gabriel Stolzenberg>proved the big conjecture.  And I've never seen
Gabriel Stolzenberg>this corrected...

I am sure that you would hear this, but this was not written in the
established  mathematical texts on the matter. The problem is not to correct
misleading remarks outside the "formal" mathematics. The problem is to
correct the prevalent but incorrect view that if Robinson  had managed to prove the "great"
conjecture  then this would become an ultimate demonstration of the relevance of
nonstandard analysis.

Gabriel Stolzenberg>   ... the disagreement, such as it is, now seems to be
Gabriel Stolzenberg> about the difficulty and interest of the result that Robinson proved.
The quality of the  contribution of Robinson to invariant subspaces is
completely inessential for positioning nonstandard analysis.
Since nonstandard analysis is conservative over the "standard" analysis and the "great"
conjecture is stated in standard terms, any nonstandard proof of the conjecture will
be immediately  translated into the standard language (there is a
formal algorithm  by Nelson to this end). The power nonstandard
analysis lies far from the fact that nonstandard analysis allows one to use
infinitesimals in inventing new proofs of standard theorems.
Nonstandard analysis is relevant since it resurrected  infinitesimals
and infinities, explaining that the body of "genuine" mathematics may proceed in
concordance with the ancient tradition of dichotomy between points and
monads, which was  forbidden in mathematics for a few decades of the twentieth century.

Nelson wrote that ``really new in nonstandard analysis are not theorems
but the notions, i.e., external predicates.'' We must think of this to
understand the prediction of Goedel: "...there are good reasons to believe
that nonstandard analysis, in some version or other, will be the analysis of the future."

Sobolev Institute of Mathematics
Novosibirsk State University
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