FOM: Lang/Greece = Matijasevic/India
Stephen G Simpson
simpson at math.psu.edu
Thu Oct 23 04:01:00 EDT 1997
Anand Pillay writes:
> 2) As the discussion proceeds it is interesting to see how
> Simpson's (I hope not Harvey's) foundational view unfolds. As in
> many cases, the original "rhetoric" or "manifesto" turns out to be
> in direct contradiction with the reality. Steve's "absolute
> objectivism" turns out to be "absolute subjectivism":
Anand, I truly have no idea what you are going on about. Where has my
"rhetoric" or "manifesto" contradicted reality? Actually, maybe it's
better if you don't answer that. I think we are simply not connecting
at some level and there's no point in trying to fix it.
> what is "basic", or of "general intellectual interest" etc.
> amounts just to what Steve happens to understand.
Well, if I personally am not interested in something or don't
understand something, then obviously that thing can't be of *general*
interest, right???? (Just kidding ....)
But seriously Anand, I think you are missing the point of what Harvey
and I have been saying. Let me once again try to explain it (at the
risk of being insulted yet again by you and/or Dave).
1. We all know that many pure mathematicians routinely identify
"rational solutions of systems of polynomial equations" with "rational
points on algebraic curves and surfaces". And the pure mathematicians
are justified in this; there is a formal mathematical equivalence
between the two concepts. But notice: From a general intellectual
asoect, these two concepts are very, very different. Perhaps the pure
mathematician could most readily appreciate this distinction as an
intuitive, nonrigorous matter of cultural/historical emphasis. The
"curves and surfaces" aspect emphasizes geometrical intuition (ancient
Greece). The "polynomial equations" aspect emphasizes algebraic
calculation (ancient India). These are the two souls of mathematics.
They are to a large extent effectively merged in 20th century pure
mathematics, but in ancient times there was a world of difference, and
that world of difference is what I want to emphasize now.
2. I think most mathematicians would agree that Lang's conjectures can
only be grasped from the geometrical or "curves and surfaces"
perspective (needed in order to make sense out of genus, etc.). And
conversely, Matijasevic's theorem can only be grasped from the
algebraic or "polynomial equations" perspective; geometrical intuition
is of no help here.
3. Now, once again, let's look at this from the perspective of the
educated non-mathematician. On the one hand, it just so happens that
the concepts needed to grasp Matijasevic's theorem (integers,
polynomial equations, algorithms) are readily intelligible to a wide
audience of educated people beyond the pale of pure mathematics. On
the other hand, it just so happens that the concepts needed to grasp
Lang's conjectures (complex algebraic varieties, absolute
irreducibility, genus, cohomology, etc) are relatively arcane. This
may be particularly galling to the pure mathematician, because from
his vantage point, Lang's conjectures may seem much more vital.
However, the man in the street does not share that vantage point. The
difference is one of intellectual or cultural perspective.
4. When you look at this from the viewpoint mentioned above, you can
perhaps see why it might be reasonable to say that Matijasevic's
theorem is more "foundational" than Lang's conjectures. But let's not
quibble over terminology. If you don't like the word "foundational",
use another word of your choice. The point is that Matijasevic's
theorem is more accessible from the general intellectual perspective.
This is simply a cultural fact. I don't think anyone can deny this or
ignore it.
5. Somebody like me who points these things out runs the risk of
angering pure mathematicians, incurring insults and abuse, appearing
boorish, stupid, uninformed, ignorant, inconsistent, etc., and perhaps
even treasonous, because he may appear to be selling out his academic
friends and taking up the cause of the ignorant hoards. Nevertheless,
I'm willing to risk these barbs and arrows, because f.o.m. is so
important to me. Actually, I deserve a gold medal for my courage.
Some day Anand and Dave will agree and give me one. :-)
Sincerely,
-- Steve
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