[FOM] re "Number theorist's interest in bounds (10 Apr)"

Gabriel Stolzenberg gstolzen at math.bu.edu
Fri Apr 14 15:00:37 EDT 2006

   In this message, I comment on the 3rd part of Harvey Friedman's
message of April 10, "Number theorist's interest in bounds."

   I begin by quoting from that message.

> Evidence from the number theorist Alan Baker, an unnamed well known
> number theorist, and postings of Chow, and an expected posting about
> the situation with Ax, Kochen, and Cohen, are pretty decisive
> concerning the general character of number theorist's interest in
> bounds.

   I can't in good conscience call this "pretty decisive."  How about
"plausible"?  For me, so far, the evidence is too superficial to make
a definitive judgment.  E.g., I haven't yet seen one example where an
unrealistic bound is used to get a realistic one.  (I don't want to
just hear about it.  I want to see it.)  Yet, once we have agreed that
it is not a matter of an "intrinsic" interest in bounds, this seems to
be the main reason for wanting them.

> However, there is still considerable interest in even a very high
> bound, because it suggests the possibility that that upper bound
> can be improved to a much lower upper bound.

   How does having an unrealistic upper bound suggest the possibility
of getting a realistic one?  Past experience?  How often does one go
from a classical existence proof to a realistic bound, skipping an
unrealistic one?  Is there a higher success rate if we stop to get an
unrealistic one?

> Note that this follows the pattern indicated above for Pi02 sentences,
> exactly. Unreasonable bounds came first. Then a lot of activity to try
> to make them better. They finally became "somewhat reasonable" but not
"reasonable" yet. Also serious interest in lower bounds. The matter is
surely quite ongoing...

   Right, but this is not what one would ordinarily call an "intrinsic
interest" in unrealistic (unreasonable) bounds, which is more or less
where I came in.

   Gabriel Stolzenberg


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