[FOM] on harvey friedman's "number theorists" (4 Apr).

Gabriel Stolzenberg gstolzen at math.bu.edu
Fri Apr 7 15:33:47 EDT 2006

   This is a reply to Harvey Friedman's "number theorists" (4 Apr).

   All > > passages are statements in my message of April 3 that Harvey

> >  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 don't believe it. I personally discussed this matter over the years
> with a particularly famous and revered number theorist who is deeply
> interested in these issues.

   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.

> > 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.

> Easy. It is well accepted among many, perhaps most, leading
> mathematicians in the world that bounds are intrinsically
> interesting, and an important consideration in classical
> mathematics.  Period.

   If being fascinated with something is a way of being intrinsically
interested in it (and maybe even the same) then it seems that you now
agree with what I said near the beginning of this exchange.

   However, unless you mean to claim that EVERY bound is intrinsically
interesting/fascinating (and "an important consideration in classical
math"), which I'm sure you don't, UNQUALIFIED statements about bounds,
like yours above, are out of order.

   As I understand it, we're supposed to be talking about a PARTICULAR
KIND of case, the one that you formulated: a classical existence proof
but either no constructive one or they're all are grotesque.

   Is your number theorist intrinsically interested in this kind of
case?  If he is, then, to restate what I asked above, how does he
pursue this interest?

> In fact, we both should hope that I am correct. For the only
> detectable interest I sense in the mathematics community in
> constructivity is EXACTLY where it mathematically amounts to bounds.

   Harvey, the last thing I want is for the classical mathematical
community to have an interest in constructivity.

   If you want to know why, I'll try to explain.  But to understand,
you'll have to be willing to bracket some of your beliefs about the
constructivist project in order to give what I say a proper hearing.

   Finally, it may surprise you to hear that there is a substantial
body of constructive math that the math community admires and builds
upon for reasons that have almost nothing to do with bounds.  However,
classical mathematicians can't tell that it's constructive math, so
this is not an expression of an interest in constructivity.

   With best regards,


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