[FOM] re Timothy Chow's Re: contra harvey on what number theorists want (4 Apr)

Gabriel Stolzenberg gstolzen at math.bu.edu
Sat Apr 8 10:28:16 EDT 2006

   This is a reply to Timothy Chow's April 4 reply to my message of
April 3.

   I begin with a sentence from my message that Timothy quotes in his,
followed by his reply to it.

> >  A list of number theorists making pronouncements about this
> > is surely no substitute for sound mathematical reasons.

> I can sort of understand what it might mean to have "sound
> mathematical reasons" for, say, *conjecturing* that the Riemann
> hypothesis is true.  This would mean partial results, analogous
> theorems, empirical results,  etc.  But what on earth does it
> mean to have "sound mathematical reasons" for *wanting* a certain
> result?

  In the case of the Riemann hypothesis, it usually means that you
want it in order to make use of certain known consequences of it.

>  Furthermore, you at least agree that number theorists sometimes do
> *say* they are interested in effectivizing ineffective theorems, and
> spend time proving effective versions of ineffective results.

   Where do you see me saying that they spend time proving such things?

   I know only one case of this kind.  Moreover, several years before
some number theorist hacked out the required bound, Kreisel had pointed
out that one can be read off, in a routine way, from the classical proof.
Yet, in the one account I've seen of this alleged case of "effectiving
an ineffective result," the author, who is a number theorist, makes no
reference to Kreisel's observation.

   I find it difficult to square this failure to master the relevant
literature with a belief in the keen interest of number theorists in
getting proofs of this kind.

>   Why do you think they are lying, and that they are *not* in
> fact interested in these effective results, and are spending their
> careers on things they're not interested in?

   I don't think they are lying.  God forbid.  What did I say that
could have made you believe that I think such a thing?

   Nor do I think they are "spending their careers on things they're
not interested in."  Again, what did I say that you read this way?

   Nor, finally, do I think number theorists are spending their time
"proving effective versions of ineffective results."  More precisely,
I suspect that attempts at proofs of this kind are extremely rare.

   Gabriel Stolzenberg

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