[FOM] Re the future of history

Gabriel Stolzenberg gstolzen at math.bu.edu
Mon Nov 12 20:13:45 EST 2007

   On 10 Nov, Robert L Knighten wrote a reply to my "Re the future
of history" (10 Nov).  He begins by quoting Martin Davis's reply of
nov 6 to the Q in my "Q and A" also, of nov 6.  Since Davis's reply
is available on the list, I will not requote it here.  Knighten then
quotes me:

>  >    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 [see Davis Nov 6] 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.

   Knighten continues:

> History is hard, and the history of mathematics is particularly so, but
> this isn't an example of the difficulty.

   I disagree.  You call the result (and Halmos's subsequent one) a
"breakthrough."  And, from what you say, there were plenty of other
analysts who agreed with this.  Others, like Bishop, definitely
did not.  If there is no difficulty, how would you fit Bishop's 20
minute proof (see my "Re the future of history" 10 nov) into your
story/history of breakthroughs?

>  Wikipedia has a fine brief account of the history and status.

   Does it include what Bishop allegedly did?  If you think that this
is unimportant, then that's an example of our disagreement.

>  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).

   What precisely is being referred to by "it resisted even mild
generalizations"?  Who made these attempts and what were they?  This
matters because what it suggests about the difficulty of the problem
has to be reconciled with Bishop allegedly proving it in 20 minutes.

> 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.
   My question is how important they were, not to the program, but
to the reputation that nonstandard analysis acquired at the outset
on the basis of this work.

> 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".

   I think I get what you're saying here.  "Cheap talk" is probably
too nice a way to put it.  In the MIT common room (where I spent many
hours avoiding working on my thesis---with Hoffman and Singer), there
was a great amount of it.  Understatement.  But unless the logicians
in the crowd could demonstrate that "it was the non-standard methods
that enabled them to get the insight," then this claim was part of
the cheap talk.  In my experience, it sometimes is very difficult to
demonstrate this kind of thing conclusively.

>  >    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,

   Breakthrough?  To me, your second use of this word above suggests
either that there was reason to think that the Bernstein-Robinson
theorem would be of help in proving either the big conjecture or in a
search for a counterexample.  If this is what you mean, what is your
evidence for either of these?  (Also, by itself, 12 years with no
progress doesn't say much.  We need to know about the attempts made
during that time and who made them.  Were there any?)

   Gabriel Stolzenberg

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