[FOM] Re Timothy Chow's "Re: Harvey's effective number theorists"

Gabriel Stolzenberg gstolzen at math.bu.edu
Sun Apr 16 20:59:52 EDT 2006

   In this message, I make a few comments about Timothy Chow's reply
to my "Harvey's effective number theorists." It consists in two quotes
from Timothy's message and my replies to them.

   Here is the first quote.

> but there are many other cases in number theory that are analogous
> to the simple group situation, where you push a bound low enough for
> explicit computations and thereby allow proofs of qualitatively new
> results.

   You say nothing in the simple group case about "pushing" a bound
low enough.  [I'm referring to a part of Timothy's message not quoted
here.]  We all understand the meaning of "getting" a bound that is
low enough.  But "pushing" suggests that use was made of "not low
enough" bounds (or at least techniques invented to get them) to get a
low enough one.  Is this what you mean?  It seems to me that you have
been silent about this.

> After enough experience with this sort of thing, one learns to
> respect the value of passing from no bound to some bound to a good
> bound just in general, knowing that this represents increased
> knowledge and power, as well as increased chances of crossing
> thresholds into new, uncharted territory.

   What does "just in general" mean here?  As I read you, you're
talking only about those happy cases in which one passes from no
bound to some bound to a good bound.  But my disagreement with
Harvey is about the value of pursuing any old bound, not taking into
account the time and effort that may be required.  (Skewes began
sometime between 1912 and 1933 and finished in 1955, three years
after Kreisel explained how to read a bound out of Littlewood's

   Thus, according to Harvey, as I understand him, it is a good in
itself to get ever lower upper bounds for no matter what, whether
or not we ever get a realistic one and whether or not, in the course
of trying, we invent techniques that help others to get one.  If you
agree, I hope you will give him your support.

   With best regards,

  Gabriel Stolzenberg

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