[FOM] Re Timothy Chow's Re on harvey friedman's number theorists (8 Apr)

Gabriel Stolzenberg gstolzen at math.bu.edu
Sun Apr 9 18:50:52 EDT 2006

   In his reply to my "on harvey friedman's number theorists" of 8 April,
Timothy Chow begins by quoting something I say in it.

Gabriel Stolzenberg <gstolzen at math.bu.edu> wrote:

> >   I'm surprised that you don't tell us how this interest is manifested
> > mathematically.  Isn't that important?  I'd like to see some of the
> > work that he did on questions of this kind.

> Oh, is that all you mean?  This request is easy to satisfy.  At one
> time I had an officemate, Mike Bennett, whose career was and is devoted
> to this kind of thing.  Browse his publications at

> and you'll quickly find that even the papers that don't have the
> words "effective" or "explicit" in the title are deeply concerned
> with effective methods in Diophantine approximation.  For him, a
> nonconstructive theorem such as Roth's theorem is little more than
> a starting point that guides his search for effective (and better,
> computationally efficient) methods.

   Tim, you're talking about "effective" and "computationally efficient"
methods.  That's great stuff.  I'm with you.  But, as I understand it,
Harvey is insisting that every bound is, as he puts it, "intrinsically"
interesting.  Every bound.

   And even if he doesn't really mean this, he's made it clear that he
does mean it for every bound for a "pure existence" theorem in number
theory.  Every one.  (I don't know how he feels about this outside of
number theory.  I've proved more results of this kind than I care to
remember, including at least one in number theory, but the earth never
moved.  My reasons for proving them were *not* what I called "sound
mathematical reasons.")

   Gabriel Stolzenberg

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