[FOM] re re re the future of history

Gabriel Stolzenberg gstolzen at math.bu.edu
Wed Nov 14 19:51:15 EST 2007

   This is a reply to S.S. Kutateladze's "Re [FOM] Re the future
of history" (12 nov).

   He begins by quoting me from a message to which his is a reply.

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

    Nevertheless, it is the problem I raised, using the expression
"the future of history" and referring to an essay published in the
New York Review of Books."

> 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.

    I agree thast this is a problem.  But to me the two problems
should be dealt with together..

> > "... the disagreement, such as it is, now seems to be 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.

   I accept this but I was talking only about the reputation of
nonstandard analysis, in particular, among educated laymen, not the
truth about it.

> 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.

   Well, that's one reading of it.  But don't we have to understand
this ancient tradition of dichotomy of points and monads in order to
be able to assess this claim of resurrection?  (Does everyone except
me know what a monad is?  I mean really know, not just kinda sorta.)
I also need a precise explanation of just what "in concordance with"
means here.  This is crucial.

> 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."

   Why must we think of Nelson's observation in order to understand
Goedel's prediction?  And why aren't we told what these allegedly
good reasons are so we can decide for ourselves?  (In my experience,
Goedel was not always a reliable source.)

   Gabriel Stolzenberg

More information about the FOM mailing list