[FOM] Re the future of history

Robert L Knighten RLK at knighten.org
Sun Nov 11 02:08:28 EST 2007

Gabriel Stolzenberg writes:
 > 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.

History is hard, and the history of mathematics is particularly so, but this
isn't an example of the difficulty.  Wikipedia has a fine brief account of the
history and status
(http://en.wikipedia.org/wiki/Invariant_subspace_conjecture) including a
reference to Halmos' note "Invariant Subspaces", American Mathematical Monthly,
Vol. 85, No. 3 (March 1978), pages 182-183. In his note Halmos comments:

"The Aronszajn-Smith technique seemed to be so sharply focused on its
particular purpose that for a dozen years it resisted even mild
generalizations; it was, for instance, not known whether the conclusion
remained true for operators whose square is compact.  Now that is known; the
extension to polynomially compact operators was obtained by Bernstein and
Robinson (1966).  The presented their result in the metamathematical language
called non-standard analysis, but, as it was realized very soon, that was a
matter of personal preference, not necessity."

[The Aronszajn-Smith technique proved compact operators have non-trivial
invariant subspaces.]

There may be those who think that Bernstein and Robinson proved something
more, or that the Bernstein and Robinson result was important to the
non-standard analysis program, but this is surely willful ignorance.

. . .

 >    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.)

Sometimes anecdote is the only history we have, so I will add my own.  I was a
graduate student at MIT when the Bernstein and Robinson paper appeared.  Only
a year or so before I had taken courses in non-standard analysis and
functional analysis (where the Aronszajn-Smith theorem was proved) so I too
was excited to hear the news.  I remember very well the common room tea where
it was first discussed, including Ken Hoffman declaring in his usual boisterous
manner that he was too old to learn about ultrafilters (which he compared to
his favorite floating flapdoodles).  There was a general sense of relief among
the analysts when the word came only a couple of weeks later that a standard
proof using the clever insight that Bernstein and Robinson had found was easy
and actually shorter.  The argument from the logicians in the crowd that it
was the non-standard methods that enabled them to get the insight were brushed
aside with the remark that it was probably "in the air".

 >    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.

Is there a disagreement?   The Aronszajn-Smith theorem certainly seems to have
been the first breakthrough in studying the Invariant Subspace Conjecture.  It
was 12 years with no progress until the Bernstein-Robinson theorem which was
the next breakthrough, then another seven years for Lomonsov's result.
Assuming that de Branges' "proof" deserves the treatment it is getting, that
still seems to be the status today.

(I didn't realize from your first posting that you were referring to the
Invariant Subspace Conjecture.  The analysts I knew in 1966 were all quite
impressed with solving the invariant subspace problem of Smith and Halmos.)

-- Bob

Robert L. Knighten
RLK at knighten.org

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