[FOM] Harvey's effective number theorists

[Timothy Y. Chow]

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