[FOM] Harvey's effective number theorists

Timothy Y. Chow tchow at alum.mit.edu
Thu Apr 13 16:49:11 EDT 2006

```Gabriel Stolzenberg writes:
>    Finally, I'd like to thank Harvey's first unnamed number theorist for
> his comments and invite him 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?

Mathematicians whose own research doesn't involve hard analysis or other
techniques requiring explicit quantitative calculations on a daily basis
sometimes find it hard to understand why (for example) analysts sometimes
get so excited about reducing an exponent from 3/4 to 2/3 or something
like that.  Part of it, of course, is that numbers like that provide a
handy, quotable benchmark for progress, and what people are really excited
about are the new techniques that enable one to break through what seemed
to be a tough barrier.

However, things also work in the opposite direction.  That is, 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.  For an analogy from a different area of
math, consider the classification of finite simple groups.  The
classification theorem would be significantly weaker if we could only say,
"there are finitely many sporadic groups," while having no idea *how*
many.  If you have an explicit list then you can prove all kinds of
previously inaccessible theorems just by checking all the groups.  This
isn't usually the most satisfying type of proof, but at least it's a
proof, and you might have no other proof available.

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.  I don't know
of any applications of Catalan's conjecture, 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.

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, just as any kind of study "for its own sake" may not
yield the results that a hard-headed applications-oriented person wants to
see.  But I'm sure I don't need to teach FOM readers how to respond to
someone who asks, "But is this work going to lead to applications?"

Tim
```