FOM: Using Universes? The expert speaks again

Colin McLarty cxm7 at
Fri Apr 9 17:34:25 EDT 1999

        Now we are getting somewhere. Well, naturally I like the expert's

>1. This statement of McLarty is true:
>>    	They [Universes] are not formally indispensible to the number
>>theoretic results. That has long been clear to mathematicians
>>and I said it in my first post on this thread. They are in fact
>>present in virtually all cohomological number theory today
>>including at least one key reference in Wiles's proof of the
>>semi-stable Taniyama-Shimura.

I disagree about two nuances of the expert's claim: 

>2. This statement of Friedman is true:
>>>*The use of universes in FLT - or any serious number theory - has never,
>>>even remotely, been any kind of issue. Nobody who understands such proofs
>>>does anything but think about very small structures from the start till the
>>>end. The number theorists are perfectly well aware of this. And they didn't
>>>have to do any work to eliminate large structures.*

        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.

        Practical nuance: even today number theorists who understand the
proofs do also think, for convenience, about large structures such as the
category of Abelian sheaves on a given scheme--only they do not think hard
about it. It is quite routine and they know they could avoid it even as a
routine. They teach the large structures to students as you can see in
MODULAR FORMS AND FERMAT'S LAST THEOREM or any introduction to etale
cohomology--or in Johan de Jong's course on etale cohomology two years ago
at Harvard.

        Certainly the very small structures are "where the action is" in the
number theoretic proofs.

        I say Friedman's analogy [with using vast set theory to introduce
algebraic numbers] fails in the historical and practical senses. Algebraic
numbers were not discovered using Friedman's parody. No one today would
refer to Friedman's parody in a proof in algebraic number theory, nor teach
it to students.

Friedman gets to the crux of my differences with him when he says:

>You are simply trying to ascribe foundational significance where there is

I am precisely NOT ascribing Friedman's kind of foundational significance to
them. I have said all along that they are formally dispensible. I am saying
there is also a historical, practical aspect to foundations: What ideas did
people need in order to create this theory?--though perhaps they can be
eliminated in hindsight. And what ideas do people need to work with a
theory?--though perhaps they can be eliminated in principle.

Someone could argue that those historical/practical issues can never be made
clear enough to matter. That could be a valuable argument to get into. But
to use universes in number theory as an example, we must agree that
universes are in historical and practical fact used in number theory. I take
it that we now have that agreement.

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