FOM: the ``mysterious dimension'' of Grothendieck universes

Stephen G Simpson simpson at
Sun Apr 25 16:09:51 EDT 1999

Summary of the discussion so far:

Colin McLarty tried to puff up the importance of Grothendieck
universes by claiming they are integral to modern algebraic geometry.
Harvey Friedman and I deflated this claim.  In the process of
deflating McLarty's claim, Harvey and I cited (1) Hartshorne's
textbook of algebraic geometry, (2) informal conversations with
prominent algebraic geometers.  Walter Whiteley 23 Apr 1999 14:52:31
and Robert Tragesser 23 Apr 1999 17:54:43 took McLarty's side of the
argument, attacking f.o.m. on the grounds that f.o.m. allegedly
neglects the ``mysterious dimension'' of mathematical thought, in
particular geometrical intuition, which allegedly doesn't often find
its way into textbooks and lectures.

Continuation of the discussion:

The first thing to notice about the Whiteley/Tragesser complaint is its
vagueness.  It has absolutely no mathematical content.  It is silent
vis a vis Grothendieck universes, algebraic geometry, and all other
specific mathematical issues.  Obviously the f.o.m. standpoint
irritates Whiteley and Tragesser, but apparently they are not ready to
articulate a coherent objection to it, so they settle for grumbling
(Whiteley) and f.o.m.-bashing (Tragesser).

I call on Whiteley and Tragesser to put aside grumbling and
f.o.m.-bashing and focus on the mathematical issue at hand: the
alleged role of Grothendieck universes in algebraic geometry.

In that mathematical context, the second thing to notice about the
Whiteley/Tragesser complaint is its presumably unintentional irony.
Didn't it occur to Whiteley and Tragesser that many geometers might
make precisely the opposite complaint?  If geometers were called on to
name the one thing that tends to deprive modern algebraic geometry of
visual/pictorial intuition, I conjecture they would name the
cohomological and category-theoretic formalism.  And Grothendieck
universes would represent an extreme in this direction.

It's interesting that Whiteley and Tragesser seem unaware of the
existence of this countervailing viewpoint.  I even have to wonder
whether the countervailing viewpoint might have been shared by
Gian-Carlo Rota himself.  I wish I could ask Rota whether Whiteley and
Tragesser speak for him, as Whiteley and Tragesser claim.  It's a
shame that Rota can no longer speak for himself.

I also wonder how McLarty views the Whiteley/Tragesser complaint.  I'm
sure McLarty finds the anti-f.o.m. aspect congenial, but what about
the expert-insight aspect?  Rota 23 Apr 1999 14:52:31 stresses the
value of informal conversations with experts, but McLarty 23 Apr 1999
08:55:14 expresses doubts:

 > I doubt the value of studying foundations by playing "whisper down
 > the lane" with experts no matter how skilled.

-- Steve

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