[FOM] Re the future of history

Gabriel Stolzenberg gstolzen at math.bu.edu
Sat Nov 10 23:43:45 EST 2007

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

> On November 6 Gabriel Stolzenberg wrote:

>  > Question.  What is the name of the famous conjecture in analysis
>  > whose proof by Abraham Robinson is sometimes offered as a
>  > demonstration of the power of nonstandard analysis?  (Jim Holt
>  > did this in the NYR >but, as I recall, without identifying the
>  > conjecture.)
> The reference is likely to the Bernstein-Robinson theorem:
> Let T be a linear operator on Hilbert space H such that for some
> polynomial p, p(T) is compact. Then H has a non-trivial closed
> linear subspace E such that T maps E into itself.
> This answered a problem of Paul Halmos that had been open for a long
> time. The proof used non-standard methods in a particularly beautiful
> way, approximating an infinite dimensional space from above by a
> space with hyperfinite dimension, so the theorems of finite
> dimensional linear algebra could be brought to bear. Soon afterwards
> generalizations of the theorem were proved by standard methods.
> Martin

   Below I include a reply to Robert Knighten that I initially sent
to him off line.  It is, in a somewhat different way, also a reply
to Martin Davis.  If we contrast the plausible story Davis tells in
his second paragraph above with the one I tell below about how experts,
especially Errett Bishop, assessed the importance and difficulty of
the result, we have a nice example of what concerns me about what,
near the end of my reply to Knighten, I call "the future of history."

   Will there continue to be two stories, Davis said this, Stolzenberg
said that?  Or will one eventually become the received wisdom?  If so,
will it be as a result of painfully careful scholarship?  Or will it
be something that is stated as if the author knows what he is talking
about and, because of his credentials, etc., his readers see no reason
to doubt.  (E.g., a physicist at a leading university mocks comments
about elementary physics by a layman.  Even if none of it is intelligible
to us, we will take for granted that he knows what he is talking about.
Unless we have good reason to think otherwise.)

   Knighten's reply (6 Nov 2007) to my question was:
> I suspect you are refering to the "invariant subspace problem" as in
> A. R. Bernstein and A. Robinson, "Solution of an invariant subspace
> problem of K. T. Smith and P. R. Halmos", Pacific J. Math., 16 (1966),
> 421-431.  MR33 #1724.

   My slightly cropped reply follows.

   Not really but your answer pleases me.  Let me explain why.  I
was referring to the big invariant subspace conjecture, which, so far
as I know, is still unproven.  Whereas you are referring to the very
special case of it that Robinson actually proved.

   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.  E.g., when Jim Holt said it (at least, as I read him)
in the New York Review of Books.  I might have written a letter
pointing this out but I feared that I would be seen as hostile to
nonstandard analysis, which I am not; it would have been much better
for such a letter to be written by a nonstandard analyst.

   In my view, the rumor (is this the right word?) that Robinson
proved the big conjecture played a role in how the nonstandard
analysis program was seen, at least initially.  I have no idea who
started it or how.  Nonstandard analysts would surely have known
that this not what Robinson proved.  But then who?

   Bernstein and Robinson proved that if T^n is compact for some
positive integer n, then T has an invariant subspace.  I think
that Halmos may have proved this for T^n replaced by an arbitrary
polynomial in T.  (But I don't know if this was before or after
Bernstein and Robinson.)

            [Compare what follows with Davis's account.]

   Although I'm an analyst, I'm not at all an expert in this stuff.
But what I heard from experts is that these were basically routine
generalizations of the case of a compact operator and didn't seem
to bring us any closer to proving the general conjecture.  I have
no idea whether Halmos saw it this way.  However, a former colleague
of Errett Bishop told me about 20 years ago that, when he heard
about the proof (of the case when T^n is compact) using nonstandard
analysis, he hurried to Bishop's office to tell him the exciting news.
When he did, Bishop's reply was something like, "If I prove this for
you in half an hour, will you promise never to think about nonstandard
analysis again?"  When I asked whether Bishop had succeeded, he said
that he did it in 20 minutes.

   I offer this only as an anecdote.  I think it's interesting.
But what it says about the difficulty of proving Robinson's result
is not so clear.  (After all, Bishop was a very powerful analyst.
If nonstandard analysis were to contribute as much as he did, that
would be quite impressive.)

   Finally, I should explain that my basic concern is with history,
or, rather, with the future of history, and not only with this
minor episode concerning nonstandard analysis.  I have found that
many people, including scientists, prefer a history that consists
of relatively neat, plausible (if you don't know too much) stories.
Plausible as seen from our current perspective.  So my concern re
nonstandard analysis and the invariant subspace conjecture is that
eventually this exciting but false account may become the accepted
truth.  As it did for Jim Holt.

   Soon after Andrew Wiles announced his proof of the Fermat Conjecture,
Andre Weil was asked what he thought about it.  (About Wiles proving
it and not he.)  He allegedly replied, "Andrew Wiles, Andre Weil, in
a few hundred years nobody will know the difference."

   This completes my reply to Knighten.  I will complete my reply to
Davis by noting that the disagreement, such as it is, now seems to be
about the difficulty and interest of the result that Robinson proved.

   Gabriel Stolzenberg

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