[FOM] Re Harvey Friedman's, "Number theorist's interest in bounds" (8 Apr)
Gabriel Stolzenberg
gstolzen at math.bu.edu
Sun Apr 9 19:54:11 EDT 2006
Harvey, I can't respond to this until I know what your number
theorist means by "an effective algorithm." If he means one thing,
he agrees with me. If he means what you say he does at the end of
your message, then I find his remarks puzzling and would need to ask
him a few more questions.
For now, I need to know whether he uses the word "effective"
according to its standard English meaning or the way that it
sometimes is used by classical mathematicians, as in an oxymoron
like "effectively computable."
Harvey's message follows.
_________________________________________________________________
> I just received a response from one of the three unnamed number
> theorists.
> "Some "effectivity" results are more interesting than others.
> Here's one that I consider interesting. I don't speak for anybody
> else, but I would expect that many other people would find it
> interesting too.
> An effective version of Faltings' theorem that contained a bound on
> the height (size of numerators and denominators) of the solutions
> would give us an effective algorithm for finding all rational
> points. That is a fundamental problem, and I would consider such a
> thing very worthwhile.
> On the other hand, an effective version of Faltings' theorem that
> only provided a bound on the number of rational points would be less
> interesting, because it would not (by itself) give us such an
> algorithm. But such an effective version might still be interesting:
> for example if that bound depended only on the genus of the curve,
> that would be new and important information."
> *************************************
> The situation in the second paragraph above is precisely the situation
> of a Pi03 sentence with a classical proof that is made constructive and
> in fact into a Pi01 sentence.
_________________________________________________________________________
Finally, the quotes from Alan Baker's book (in the second part of
your "Number theorists' interests in bounds") seem, on a superficial
reading, to mostly confirm what I have been saying. But I wouldn't
dare to draw any firm conclusion without talking with him about it.
Probably more than once. (I knew Alan a long time ago and have some
fond memories.)
Of course, I assume that, when you read these quotes, you see them
exactly the opposite of the way I do. Again, I think that, to resolve
this, we'll have to ask Baker some questions about his use of language.
With best regards,
Gabriel
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