[FOM] contra harvey on what number theorists want
Timothy Y. Chow
tchow at alum.mit.edu
Tue Apr 4 11:57:25 EDT 2006
Gabriel Stolzenberg <gstolzen at math.bu.edu> wrote:
[Re: obtaining effective versions of theorems of Roth, Faltings, etc.]
> More generally, I don't believe that any number theorist, leading
> or following, is "very much interested in rectifying this situation."
> Yes, they sometimes talk this way. But that has more to do with the
> grip of their metaphysics than with number theory.
> I would change my mind if I was shown, in at least one case, a
> sound mathematical reason for wanting such a constructive proof or
> bound. A list of number theorists making pronouncements about this
> is surely no substitute for sound mathematical reasons.
What exactly is a "sound mathematical reason for wanting ... a
constructive proof or bound"? More generally, what is a "sound
mathematical reason" for wanting *any* theorem?
For example, lots of mathematicians talk as if they are interested in
seeing a proof of the Riemann hypothesis. How are we supposed to tell
whether they have "sound mathematical reasons" for this, versus simply
being "metaphysical" and "making pronouncements"?
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?
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. 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? Note that saying that they're interested for *metaphysical reasons*
is different from saying that they're *not interested at all*.
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