FOM: F.O.M./pure math; general intellectual interest

Lou van den Dries vddries at
Wed Dec 17 09:41:21 EST 1997

Okay, I had forgotten this old discussion where I mentioned number theory
in connection with complexity (but even there it was in reaction to
earlier claims by you concerning fom & complexity). Anyway, this is
not worth our energies to spell out further, we do not have a real
disagreement on this matter, I think. 

Otherwise I am not conceding anything. Sure, "suspect" is vague, so is
"general intellectual interest", and perhaps even more, the significance
of "general intellectual interest". I find P=NP a very interesting
problem, and it sure has to do with our basic intuitions and experience
with algorithms. As to how it would play for a fairly general audience
compared to abc, I wouldn't be able to predict. Both are the kind of
mathematical questions that can be discussed in a Scientific American
article (maybe this has already happened). Anyway, if P=NP is false
(as seems plausible), and even if it's true (with probably huge 
constants), and one of these is established, this would surely
increase our theoretical understanding of algorithms in a big way,
but there is no reason to think it would have direct practical
(technological) consequences. In discussing P=NP for a general
audience it seems to me this should be pointed out. 
  As to abc, it seems to express a very basic and at the same time
deep property of another kind of basic objects, namely natural
numbers (their behavior under addition and prime factorization).
(A special case says: there is a positive constant C such that if
a+b=c, where a and b are positive integers with gcd=1, then
c < C.P^2, where P is the product of the distinct prime factors of abc.
Reference: S. Lang, Old and new conjectured diophantine inequalities,
Bull. AMS 23, 37-74, 1990.) 

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