FOM: Thinking and faking with universes
Colin McLarty
cxm7 at po.cwru.edu
Wed Apr 14 14:03:31 EDT 1999
reply to friedman at math.ohio-state.edu (Harvey Friedman)
>Reply to McLarty 5:34PM 4/9/99 and 12:32AM 4/12/99. The disappointing thing
>about McLarty's postings is that he has been asked to explain what the
>point is of considering, say, maps from gigantic objects into the small
>(relatively) concrete objects that people are really interested in. His
>responses contain virtually no substantial mathematical information - just
>a huge pile of names and references to books (and some papers)!
Suppose you want to know about analytic functions on a Riemann surface S
with certain poles. That is, the function should go to infinity, in a
specified way, at each of some specified finite number of points of S, and
it should be analytic away from those points.
Does such a function always exist? Not such an easy problem. It is very
easy if there are no poles, any constant function will do. And easy around a
single pole--the specification given for that pole will work on some
neighborhood around it. But can we find solutions around each pole that
patch together? (Would be trivial for differentiable functions, but not for
analytic.)
So, define a "local solution" <U,f> to be an open subset U of S and a
function f solving the problem on U--it has appropriate poles at any of the
selected points that is in U and it is analytic in the rest of U. We want
to look at all the local solutions and see if we can paste some of them into
a function f with those properties all over S.
The local solutions form a sheaf on S. Let T be a disjoint union of copies
of open subsets U, one for each local solution <U,f>, except that whenever a
local solution <V,g> is a restriction of <U,f> and also <U',f'> we paste
together those copies of U and of U' along that copy of V. Let h:T-->S map
each copy of U onto U in the obvious way.
On one hand T is much larger than S. Over any small enough subset U with at
most one pole, T obviously stacks uncountably many copies. Yet h:T-->S
merely organizes information we would use anyway to solve the problem. When
Mittag-Leffler solved the problem he did not have this organization, but now
that we do it simplifies things and leads to new theorems.
I'll cite some named experts: sheaves and much more are needed in
order to "develop only that general machinery necessary to study the
concrete geometric questions [about curves and surfaces in classical two and
three dimensional complex projective space] and special classes of algebraic
varieties [in those same spaces]". Griffiths and Harris PRINCIPLES OF
ALGEBRAIC GEOMETRY, page v.
> McLarty 5:34PM 4/9/99 writes:
>
>> Historical nuance: universes were a clear and present issue to
>>Grothendieck, Deligne as they first created the cohomology of schemes, and
>>they thought about large structures as part of their serious number theory.
>>You can see this in the SGAs.
>
>What does "issue" mean? Are you trying to suggest that these two people
>thought they needed large structures as part of their serious number
>theory, or just that they liked the extra generality in Lemmas?
Yes, they needed these structures. But not in the sense you use when you
talk about needing an assumption.
When you say you "need" a certain assumption to prove a theorem, you mean
that the theorem is formally derivable in ZFC with that assumption (or a
technical surrogate, such as consistency of that assumption) and not
without. This presents very interesting problems, and I like seeing your
results. But this is not the only place mathematical problems come from.
Grothendieck "needed" whatever methods he could find to solve an
outstanding problem: the Weil conjectures.
Did he use "excess generality"? Well, "excess" to what? Obviously he had no
idea at the time what generality he would need. In hindsight his method has
"excessive" proof-theoretic strength. Some aspects (famously toposes) are
considered excessive in practice by many number theorists. Universes are
apparently not excessive in practice since number theorists continue using
them.
Try posing these two questions to your experts: "Can you name a textbook on
cohomological number theory that does not refer to large structures?" And
"Are the papers in MODULAR FORMS AND FERMAT'S LAST THEOREM written with
excess generality?"
>McLarty writes:
>> Certainly the very small structures are "where the action is" in the
>>number theoretic proofs.
>
>The issue is: what is then the nature of the "uses" of universes - since
>they are not "where the action is"?
I already answered, in the sentence after the one you quote. The
universes function in the (now) routine parts. The theorist giving the proof
never proves that a given field extension has a Galois cohomology. He or she
simply uses that fact to justify particular calculations. If the theorist
even cites a source to show these cohomologies exist, that source is likely
to define them as derived functors on a certain category, thus referring to
and even quantifying over universe-sized structures. The theorist will not
think about that.
In response to my claim that people do not introduce algebraic
numbers over the rationals by first proving every field has an
algebraic closure, Friedman writes:
>Why not see "Since every field has an algebraic closure...", or "take the
>algebraic closure of..."? Why doesn't that appear and why not teach it? In
>Lang's Algebra textbook, "every field has an algebraic closure" is proved
>on page 273 using the maximal ideal principle.
Lang's ALGEBRA is not a textbook on algebraic numbers. People also study
algebraic extensions of finite fields, for perfectly concrete reasons such
as encryption, and want to know these fields have algebraic closures. People
also look at function fields over these fields, and thus need to know about
inseparable extensions. Lang covers all this and more because it is used.
Sorry, no tutorial here on finite field methods in encryption. References
are well known and well catalogued in university libraries.
Generality has real uses. Indeed it also has fake uses and you may find some
textbook somewhere on algebraic number theory that only works on numbers
algebraic over the rationals and yet proves every field has an algebraic
closure. But you may not either. Certainly most texts on this neither prove nor
refer to that theorem.
Apparently all texts, and probably all research papers, on cohomological
number theory rely on proofs using universes. Your experts have not
mentioned any exceptions. Well, your first expert said Wiles's proof was an
exception. But in fact the "turning point in the whole proof", to use
Wiles's own words, came when he noticed a consequence of the Grothendieck
duality theorem, and for that theorem he directly cites a proof using universes.
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