Subject: categories: WHY ARE WE CONCERNED? I

From: F W Lawvere <[email-masked]>
Date: Sun, 26 Mar 2006 16:43:31 -0500 (EST)

WHY ARE WE CONCERNED? I
	When Saunders Mac Lane penned his hard-hitting 1997 Synthese
article, he was defending mathematics from an attack many of us hoped
would just go away. But Saunders was aware of the seriousness of the
threat, which indeed is still here with greater determination.
Although the title of that article was "Despite physicists, proof is
essential in mathematics", he was not opposing physics, nor even that
immediate handful who, assuming the mantle of "mathematical physicists",
gave themselves license to insult generations of scrupulously serious
physicists and to demand that mathematics adopt a culture that considers
conjecture as nearly-established truth. In essence it was an attack on
science itself, as the highest form of knowing, that Saunders was
opposing.
	The increased determination of that attack is expressed in two
ways. To equip and organize the attack, finance capital has set up several
institutions, some of which rather openly proclaim their goal of
submitting science to the service of medieval obscurantism. Others say
that they support mathematical research, but encourage a barrage of
"popular" writings to shock and awe the public into continuing in the
belief that they will never understand mathematics and hence never be able
to actively participate in science.
	The contempt for Mac Lane's fight, recently expressed in articles
supposedly memorializing him, takes the form of the claim that category
theory itself is a "cool" instrument for deepening obscurantism. Not only
Harvard's "When is one thing equal to another thing?" and the Cambridge
"morality" muddle, but also a 2003 article aimed at teachers of
undergraduates, quite explicitly support that claim. In the MAA Monthly, a
Clay Fellow states as fact that category theory "is mathematics with the
substance removed". Mastering the technique of disinformation whereby the
readers are first told that now finally they will be informed, the article
suggests that some raising of the level of understanding of the
relationship between space and intensively variable quantity is going to
be achieved. Then the author short-circuits any such understanding via the
simplifying assumption that omits the distinction between covariant and
contravariant functors as "unwieldy". As final display of the mastery of
expositional technique, the categorical object which has, for nearly
twenty pages, been heralded as simple, is revealed in the final pages in
the most complicated and unexplained form possible. (Totally passed over
is the issue that had led Grothendieck to the considerations allegedly
being treated: not only the category of affine schemes, but also the
category of all its presheaves, where the author implicitly wants us to
work, fails to have the geometrically correct colimits needed to define
projective space.)
	Another level of attack was launched when Cornell University was
given very large sums of money to develop methods of teaching geometry
without mentioning any geometrical concepts. No proof of the desirability
of such a draconian excising of content needed to be given, beyond some
phrases from the Dalai Lama.
	"Dumbing down" is an attack not only on school children and on
undergraduates, but also one taking measured aim at colleagues in adjacent
fields and at the general public. The general public is thirsty for
genuinely informational articles to replace the science fiction gruel
served constantly by journals like the Scientific American and the New
York Times "Science" section. Those journals have never published anything
resembling a mathematical proof and hence have rarely actually explained
any scientific subject in a usable way. Nor have they even undertaken any
program to raise the level of knowledge of calculus or linear algebra
among their readers in a way which would make such explanations feasible.
Instead, they provide games and amusements to divert the
mathematically-interested public.
	In January of 2005 the Notices of the AMS announced that they had
for a full ten years been strictly following a certain editorial policy.
There had been a widespread demand for expository articles. To that
demand, the response was a new definition of "expository": all precise
definitions of mathematical concepts must be eliminated. Authors of
expository articles were forced to compromise their presentation, or to
withdraw their paper. Mathematicians, who were for several years
becoming aware that these new expository articles are absolutely useless
for developing a mathematical thought, were shocked to learn that a
conscious policy had forced that situation.
	A peculiar sort of anti-authoritarianism seems to be the only
justification offered for degrading the role of definition, theorem, and
proof; certainly, serious expositors have never considered that the use of
those three pillars of geometrical enlightenment excludes explanations and
examples. Others have urged, however, that those instruments be
eliminated even from lectures at meetings and from professional papers.
	That threat is part of the background for the concern expressed in
the many messages to the categories list over the past weeks. Deeply
concerned mathematicians ask me "How can we know?". Indeed, how can we
know whether it is worthwhile to attend a certain meeting or a certain
talk, and how can a scientific committee know whether a proposed talk is
scientifically viable? If the "you don't want to know" culture of no
proofs, no definitions, is accepted, we will truly have no way of knowing,
and will be pressured to fall back on unsupported faith.

************************************************************
F. William Lawvere
Mathematics Department, State University of New York
244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA
Tel. 716-645-6284
HOMEPAGE:  http://www.acsu.buffalo.edu/~wlawvere
************************************************************

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From: jim stasheff <[email-masked]>
Date: Tue, 28 Mar 2006 15:51:45 -0500

I beg to differ - a little

F W Lawvere wrote:
> York Times "Science" section.

so far so good

  Those journals have never published anything
> resembling a mathematical proof

why should they?

and hence have rarely actually explained
> any scientific subject in a usable way.

a math proof is hardly necessary to explain a scientific subject in a
usable way.

now for a mathematical subject a math proof is sometimes but not always
necessary

> withdraw their paper.

Not all of us

and notice you are talking about the NOTICES
not the Bulletin
Mathematicians, who were for several years
> for developing a mathematical thought,

developing a mathematical thought,

depends what you mean by that
developing in the sense of enough to be active in the field - of course not

developing a sense of what the thought of the experts are so that one
might want to learn more or NOT
or
might see relevance to ones own disparate research - they work fine
  were shocked to learn that a
> eliminated even from lectures at meetings and from professional papers.

Examples ? I certinaly have not seen such
In fact as an editor and referee and all the referees I've used
have never tolerated such elimination.  in fact, due to cross
fertilization, even some physics papers now have defintions

> and will be pressured to fall back on unsupported faith.
Me thinks thou doth protest too much

or you've run into some alternate universe I'm unfamiliar with

;-D  jim

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From: Vaughan Pratt <[email-masked]>
Date: Wed, 29 Mar 2006 12:10:40 -0800

> now for a mathematical subject a math proof is sometimes but not always

Absolutely.  I would add publication date as a factor here.  As an
example, a few decades ago an elementary exposition of the Fundamental
Theorem of Algebra would not be expected to include an elementary proof
since the extant proofs were either lengthy arguments or nonelementary
appeals to the minimum modulus principle, properties of holomorphic
functions such as Liouville's theorem, or other results the reader would
be unlikely to be on top of.  The dominant belief was that the only
short proofs were nonelementary ones.

But for an audience aware only that z^i for any nonnegative integer i
maps circles at the origin to i-fold circles of radius r^i at the
origin, an entirely elementary notion, an expositor today would be
morally obligated to include a full proof since there is hardly anything
left to explain.  The polynomial a_d z^d + ... + a_0 maps little circles
to the neighborhood of a_0 and big circles to a loop tending to a very
big d-fold circle of radius a_d r^d, whence the smoothly growing image,
under the polynomial, of a smoothly growing circle is obliged to cross
the origin at some stage.  Still a topological argument, but now an
entirely elementary one.

Except, that is, for the theorem that a loop wound d times around the
hole in the punctured plane cannot be continuously retracted to a point,
which was tacitly smuggled in there.   But that statement is less
intimidating than anything based on holomorphic functions.

This slick proof seems only to have emerged in the past couple of
decades.  It is an interesting commentary on mathematics that it took
this long for people to come up with an argument "for the rest of us."
Maybe some people "knew" it all along, but in that case they were
keeping pretty quiet about it.

Vaughan Pratt