[FOM] Deep Thought on number theory

Gabriel Stolzenberg gstolzen at math.bu.edu
Thu Apr 13 12:52:57 EDT 2006

   This morning I talked with a number theorist, who will be known
here as "Deep Thought," who disagrees with the idea that bounds
are intrinsically interesting and said that number theorists are
sometimes able---often enough to make it interesting---to go from
an "effective" bound to a realistic one.

   I find this fascinating and wonder why number theorists don't
advertise it.  E.g., with simple examples in number theory courses.
(I know about bootstrap and speedup methods in analysis but I have
no idea how to go from "effective" to realistic in number theory.)

   As for the role of the classical existence proof with which number
theorists allegedly start, Deep Thought said that his colleagues go
directly for a constructive proof---except that (1) they really don't
know what this means and (2) as we know, they sometimes end up with
a classical proof.

   Finally, Deep Thought agrees that "effective" algorithm is an
oxymoron.  Maybe this is a sign that he thinks constructively rather
than classically.  There is other evidence of this.  Yet he claims
to be a Platonist, as in "Aristotle no, Plato yes."

   Gabriel Stolzenberg

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