FOM: translating Grothendieck; faking Grothendieck universes

Stephen G Simpson simpson at
Sun Apr 25 15:27:28 EDT 1999

Simpson 20 Apr 1999 19:02:32:

  ... so far as I can tell, Grothendieck never suggests that what the
  number theorists came up with is mathematically meaningless.

McLarty 21 Apr 1999 12:42:13:

 > Here the phrases "le sens de ces taches elle-memes comme parties
 > d'un vaste Tout, sont oublies par tous" and "arrache d'une vision
 > qui lui donne tout son sens" are crucial.

As I said earlier, my French is not very good.  And I'm away from my
library, so I don't have a French-English dictionary handy.  Maybe
somebody ought to post a precise English translation of Grothendieck's

In the meantime, I take Grothendieck to be saying that his vision of a
``vast All'' (Grothendieck universes?  Grothendieck toposes?) is what
gives the number-theoretic applications of derived functor cohomology
(crystalline cohomology, etc.) all of their meaning or sense.  And, he
is complaining that everybody has forgotten that.

However, this is very far from saying that the number-theoretic
applications are mathematically meaningless.

Therefore, I remain far from convinced of the truth of McLarty's 16
Apr 1999 11:44:32 statement ``Grothendieck regards those proofs as
essential uses of universes since the proofs which eliminate universes
have no genuine mathematical meaning.''

Perhaps McLarty will explain.

McLarty 23 Apr 1999 08:55:14:

 > Algebraic geometry is like the French language in this respect: you
 > cannot understand it by picking out key words you recognize. You
 > must read entire sentences, even paragraphs.  And if you want to
 > understand foundational issues in it you must read them critically.

As I said, my French is not very good.  However, I did learn some
category theory and algebraic geometry as a graduate student.  (One of
my teachers was Quillen.)  Later, in the late 1970's and early 1980's,
as part of my background research for reverse mathematics, I read a
fair portion of Hartshorne's textbook on algebraic geometry.  So I
think I know what I'm talking about here, even though I'm not an
algebraic geometer.  (By the way, McLarty isn't an algebraic geometer,
either.)  I really do believe that, if Hartshorne were doing or using
Grothendieck universes, I would have noticed.  

The upshot is that I remain confident of my 22 Apr 1999 17:42:45
posting, which was entitled ``faking Grothendieck universes''.

-- Steve

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