Subject: categories: Alexander Grothendieck on `speculation'

From: "Ronald Brown" <[email-masked]>
Date: Sun, 5 Mar 2006 12:21:31 -0000

I am in the process of editing my correspondence 1982-1991 with AG which has
been latexed through George Malsiniotis for appearing as part of an Appendix
to a published edition of `Pursuing Stacks'.  I came across the following
extract  which seemed to me to contain points of general interest about
mathematical methodology and sociology, so I give this below, to invite
comments.

Ronnie Brown
www.bangor.ac.uk/r.brown

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 from a letter dated 14 June, 1983,

Your idea of writing a ``frantically speculative" article on
groupoids seems to me a very good one. It is the kind of thing
which has traditionally been lacking in mathematics since the very
beginnings, I feel, which is one big drawback in comparison to all
other sciences, as far as I know. Of course, no creative
mathematician can afford not to ``speculate", namely to do more or
less daring guesswork as an indispensable source of inspiration.
The trouble is that, in obedience to a stern tradition, almost
nothing of this appears in writing, and preciously little even in
oral communication. The point is that the disrepute of
``speculation" or ``dream" is such, that even as a strictly
private (not to say secret!) activity, it has a tendency to
vegetate - much like the desire and drive of love and sex, in too
repressive an environment. Despite the ``repression", in the one
or two years before I unexpectedly was led to withdraw from the
mathematical milieu and to stop publishing, it was more or less
clear to me that, besides going on pushing ahead with foundational
work in SGA and EGA, I was going to write a wholly science-fiction
kind [of] book on ``motives'', which was then the most fascinating
and mysterious mathematical being I had come to meet so far. As my
interests and my emphasis have somewhat shifted since, I doubt I
am ever going to write this book - still less anyone else is going
to, presumably. But whatever I am going to write in mathematics, I
believe a major part of it will be ``speculation" or ``fiction",
going hand in hand with painstaking, down-to-earth work to get
hold of the right kind of notions and structures, to work out
comprehensive pictures of still misty landscapes. The notes I am
writing up lately are in this spirit, but in this case the
landscape isn't so remote really, and the feeling is rather that,
as for the specific program I have been out for is concerned,
getting everything straight and clear shouldn't mean more than a
few years work at most for someone who really feels like doing it,
maybe less. But of course surprises are bound to turn up on one's
way, and while starting with a few threads in hand, after a while
they may have multiplied and become such a bunch that you cannot
possibly grasp them all, let alone follow.

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From: Krzysztof Worytkiewicz <[email-masked]>
Date: Sun, 5 Mar 2006 10:38:42 -0500

Today's motto sadly appears to be: "the presentable individuals are
precisely the discrete ones, what an exemplary society!!" (Gabriel-
Ulmer 1971)

Cheers

Krzysztof

-- my government will categorically deny the incident ever occurred

---

From: jim stasheff <[email-masked]>
Date: Mon, 06 Mar 2006 13:27:37 -0500

This suggests two possibiloities:

for the brave, start your own blog for speculations
for the timid, same input but into a file only you can access
until late in life and famous you can show how you had the ideas all
along

jim

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From: "Ronald Brown" <[email-masked]>
Date: Tue, 14 Mar 2006 22:43:17 -0000

Jim Stasheff writes:

> This suggests two possibilities:
>

The situation is more complicated in that what could be classed as
speculation may get published as theorem and proof. For example,  in
algebraic topology, sometimes proofs of continuity are omitted as if this
was an exercise for the reader, yet the formulation of why the maps are
continuous (if they are necessarily so) may contain a key aspect of what
should be a complete proof. This difficulty was pointed out to me years ago
by Eldon Dyer in relation to results on local fibration implies global
fibration (for paracompact spaces) where he and Eilenberg felt Dold's paper
on this
contained the first complete proof. I have been unable to complete the proof
in Spanier's book, even the second edition. (I sent a correction to Spanier
as the key function in the first edition was not well defined, after
Spanier had replied `Isn't it continuous?')   Eldon speculated (!) that
perhaps 50% of published algebraic topology was seriously wrong!

van Kampen's original 1935 `proof' of what is called his theorem is
incomprehensible today, and maybe was then also.

Efforts to give full details of a major result, i.e. to give a proof, are
sometimes derided. Of course credit should be given to the originator of the
major steps towards a proof.

Grothendieck's efforts to develop structures and language which would reduce
proofs to a sequence of tautologies are notable here. Colin McLarty's
excellent article on `The rising sea: Grothendieck on simplicity and
generality ' is relevant.

Some scientists snear at the mathematical notion of rigour and of proof. On
the other hand many are attracted to math because it can give explanations
of why something is true. But `explanations' need a higher level of
structural language than for what might be called proofs.

I can't resist mentioning that one student questionaire on my first year
analysis wrote `Professor Brown puts in too many proofs.' So I determined to
rectify the situation, and next year there were no theorems,  and no proofs.
However there were lots of statements labelled `FACT' followed by several
paragraphs labelled `EXPLANATION'.  This did modify the course because
something labelled `explanation' ought really to explain something! I leave
you all to puzzle this out!

In homotopy theory, many matters, such as the homotopy addition lemma, had
clear proofs only years after they were well used.

Surely much early algebraic topology is speculative, in that the language
has not yet been developed to express concepts with rigour so that a clear
proof can be written down. It would be  a curious ahistorical assumption
that there is not at this date another future level of concepts which
require a similar speculative approach to reach towards them.

Ronnie Brown

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From: Gaucher Philippe <[email-masked]>
Date: Wed, 15 Mar 2006 10:31:40 +0100

Le Mardi 14 Mars 2006 23:43, vous avez écrit :
> contained the first complete proof. I have been unable to complete the
> proof in Spanier's book, even the second edition. (I sent a correction to
> Spanier as the key function in the first edition was not well defined,
> after Spanier had replied `Isn't it continuous?')   Eldon speculated (!)
> that perhaps 50% of published algebraic topology was seriously wrong!

My guess is that most of the algebraic topologists assume that the map 
they are constructing is automatically continuous since the proof will work 
for example for simplicial sets (in which there is no continuity to check). 
And this argument is wrong : because the category of general topological 
spaces is not cartesian closed while the category of simplicial sets is 
cartesian closed. And most of these proofs of continuity become possible only 
by working in a more convenient category of topological spaces.

pg.