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

[Harvey Friedman]

On 4/7/06 3:33 PM, "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.

He is one of the three people I contacted, and I haven't yet heard from any
three. I have to admit that I am becoming pessimistic about hearing from
them. 

My impression is that most of the leading senior number theorists have
published bounds either improving previous bounds or establishing a bound
where no bound previously existed. Also this particular leading senior
number theorist did publish on logical issues associated with number theory,
including, at least tangentially, bounds.
> 
>> 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.

The mere existence of a an effective bound for a Pi03 is of intrinsic
interest, and the interest to number theorists increases as the bounds get
lower and lower. 

>  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.

In this exchange, I was concentrating solely on statements of the form

(forall n)(therexists m)(forall r)(P(n,m,r))

where P(n,m,r) is innocent. And where there is a classical proof of this
statement, but where it is not known if it is recursively true. As a
consequence, it would not be known whether it is constructively provable
(under all proposed interpretations of constructivity).
> 
>  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?

When and if I get responses, I will ask.
> 
>> 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.

I'm all ears.
> 
>  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.
> 
I'm all ears.

Harvey Friedman