[FOM] number theorists

[Harvey Friedman]

On 4/3/06 12:50 AM, "Gabriel Stolzenberg" <gstolzen at math.bu.edu> wrote:

Friedman wrote:

>> 1. Roth's theorem about approximability of irrational algebraic real
>> numbers by rationals.
> 
>> Roth was awarded the Fields Medal for his proof of this theorem. No
>> constructive proof is known. No effective bounds are known.
> 
>> I know that most leading number theorists are very much interested
>> in rectifying this situation.

Stolzenberg wrote:
 
>  I don't believe it.

Friedman wrote:
> 
>> ....Faltings was awarded the Fields Medal for his proof of this
>> theorem. No constructive proof is known. No effective bounds are
>> known.
> 
>> I know that most leading number theorists are very much interested
>> in rectifying this situation.

Stolzenberg wrote:
 
>  I don't believe it.
> 
>  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 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.
 
>   I don't know most leading number theorists.  But I know some.

Same for me.

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.

In fact, mathematicians will be much more satisfied with decent bounds, than
with enormous bounds - even if they are effective, and enough to get
constructivity. 

I have already email contacted three number theorists, with copies of the
materials, concerning this issue. I will report on what I find. I probably
will add to the list.

1. Very famous number theorist.
2. Well known number theorist.
3. Known number theorist.

All are Full Professors at major Math Depts.

Harvey Friedman