FOM: fom & matiyasevich's theorem
Dave Marker
marker at math.uic.edu
Tue Oct 21 18:57:29 EDT 1997
Steve writes:
>For example, most number theorists are completely
>uninterested in the foundational issues raised by Matiyasevic's
>theorem. If they took an interest in those issues, number theory
>might have a different shape.
While I won't take the bait offered in the second sentence, I thought it
worth pointing out that there are some number theorists interested in
Matiyasevich's theorem and the questions it raises.
-In "Number Theory I", which appears in the Encyclopedia of Mathematical
Sciences, Manin and Panchiskian devote a chapter to a discussion of
Matiyasvich's theorem. Indeed in the introduction they offer this as a
possible justification for Gauss' characterization of number theory as the
queen of science. (For those of you who haven't seen this book I strongly
recommend it as an outline of the main lines of research themes in number
theory).
-Mazur has a very interesting article on number theoretic issues comming
out of Matiyasevich's theorem which appeared a couple of years ago in the
JSL. The most interesting question perhaps being
whether it can be extended to rational solutions to diophantine equations.
My guess is that the first reaction of most logicians is that there should
be little difference, but Mazur offered some interesting conjectures
(since refuted) that would lead to decidability. (I believe I have also
heard speculation that decidability might follow from the
Birch-Swinnerton-Dyer conjecture). Mazur also mentions an observation of
Vojta that the undecidability of a problem of Buchi's would follow from
the Lang conjecture.
Dave Marker
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