[FOM] afterthoughts re my reply to Tim Chow's "Re: Harvey's effective number theorists."

Gabriel Stolzenberg gstolzen at math.bu.edu
Mon Apr 17 14:06:10 EDT 2006

   After rereading my reply to Tim Chow's "Re: Harvey's effective number
theorists," I wish to add the following comments.

   Tim begins by quoting me inviting Harvey's unnamed number theorist
"to explain what makes the question of getting an 'effective' version
of Falting's theorem that yields an 'effective' algorithm for finding
all rational points a 'fundamental' problem?"

   But I don't see where Tim addresses my request for an explanation.
Notice that the number theorist didn't just say "problem."  He didn't
even say "homework problem."  He said "fundamental problem."  That's
what I would like to see explained.

   In reply to my, "Harvey's effective number theorists," Harvey said
that the number theorist's answer would be that it is obvious.  By
contrast, nothing in Tim's account seems to say this.

>   Sometimes if you keep pushing on a bound then you'll eventually
> cross some kind of threshold that suddenly opens up a qualitatively
> new realm of knowledge that you couldn't touch before.

   Tim, this sounds wonderful, almost magical.  But it's wholly
lacking in particulars.  Why, instead of providing some, do you offer
an analogy and then say the following?

> Similarly, in number theory, Tijdeman showed that Catalan's conjecture
> could have only finitely many exceptions, but until Mihailescu's work,
> we couldn't actually assert Catalan's conjecture as a theorem.

   Similarly, since 1934, we knew that there could be at most one
exception to Gauss's conjecture on quadratic imaginary fields of class
number 1 (a great result) but, until Stark, we couldn't assert the
conjecture as a theorem.  But what does this have to do with the
question at issue?  Did Mihailescu use Tijdeman's work?  If not, what
is its relevance?   Did Stark use the work of Heilbronn and Linfoot?
I don't think so.

> 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.  In some cases, of course,
> this optimistic viewpoint may turn out to be unfounded,

   Why do you talk about the value of passing "from no bound to some
bound to a good bound" when what is at issue is the value of passing
"from no bound to some bound"?

   Also, do you really think increased knowledge and power, no matter
what, is always desirable?  My head used to be filled with stuff I
couldn't forget, from old phone numbers to what was written on the
blackboard during last week's seminar talk.  I found it oppressive
and, on balance, unhelpful.

   Nevertheless, I think the value of increased knowledge is at the
heart of this discussion.  Harvey believes that, at least for bounds
and algorithms, increased knowledge, no matter what, is desirable.
But although some things you say also invite this reading, your praise
of "from no bound to some bound to a good bound" but not of "from no
bound to some bound" doesn't seem to fit with it.

   With best regards,


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