FOM: Matiyasevic's theorem versus Lang conjectures

[Stephen G Simpson]

Anand says that Matiyasevic's theorem is about integral points on
varieties.  That's one way of looking at it, but the truth of the
matter is that Matiyasevic's theorem is about objects of a much more
basic kind: polynomial equations with integer coefficients.  Pure
mathematicians with no appreciation for foundational issues will reply
that these two ways of looking at Matiyasevic's theorem are
equivalent.  But in actual fact, there's a world of difference in
terms of understandability and accessibility to non-mathematicians.
And Matiyasevic's theorem says that the most immediately obvious
problem concerning such equations, namely the problem of which ones
have integer solutions, is unsolvable.  This was, literally, front
page news.

Contrast this with the Lang conjectures.  In a sense, the Lang
conjectures and Matiyasevic's theorem are about the same kinds of
objects.  But the Lang conjectures additionally require a lot of
high-level concepts of algebraic geometry, etc etc, and are therefore
much more technical and non-basic, to the point where nobody with less
than two or three years of graduate study in several branches of
mathematics can grasp them, and even then the interest for
nonspecialists is highly dubious.  Matiyasevic's theorem is much more
readily understandable, because no curves or surfaces are involved.

This explains why Matiyasevic's theorem may be called foundational,
while the Lang conjectures may not.  This is the answer to Anand's
implicit question.

-- Steve