Re: Bourbaki and Categories

["Zinovy Diskin" <zdiskin@swen.uwaterloo.ca>]

Let me add my two cents :)

On Tue, Sep 16, 2008 at 4:57 AM, Vaughan Pratt <pratt@cs.stanford.edu> wrote:

> If the subject Bourwiki is proposing to serve is mathematics, then
> perhaps it is time that the American Mathematical Monthly, along with
> the Putnam Mathematical Competition, the International Mathematics
> Olympiad, and the Journal of the AMS, abandon their pretense of being
> about mathematics and come up with a suitable name for their subject.

We can think of two definitions of math. The first is based on the
subject matter ("what"): math is the study of Bourbaki's structures.
The other is based on "how:"  math is the study of structures in a
well-structured way. With this definition, a good computer programmer
or, say, Henry Ford, who applied conveyor to assembling, are more
mathematicians than some of the guys responsible for what Vaughan
wrote about.

If we imagine a two dimensional plane with the "what" axis being
vertical and the "how" horizontal, then we get two mathematics and
resp. two sorts of mathematicians: vertical and horizontal. (CT and
CT-rists go, of course, along the harmonized diagonal :).
Historically, Bourbaki put a bold point on the line of
Euclid-Peano-Hilbert and indeed formed the vertical dimension of the
modern mathematical space.  However, while the very texts written by
Bourbaki are mostly enjoyable, his epigones have created a special
literary style, which is good for writing/producing but hardly for
reading mathematical papers.  Bourbaki should not be blamed for wide
dissemination of this indigestible style but...Vladimir Arnold once
said that  "bourbakization" of modern mathematics should perhaps be
called "oBourbachivanie" in Russian (which rhymes with the Russian
"oDourachivanie", which refers to fooling with someone :).

> Not only do categories, functors, natural transformations, adjunctions,
> and monads go unused in these 20th century icons of mathematics, they go
> unacknowledged.  Clearly they have not gotten with the modern
> mathematical program and fall somewhere between a throwback to a golden
> age and a backwater of mathematics.  When they die off like the
> dinosaurs they are, real mathematics will be able to advance unfettered
> into the 21st century and beyond.
>
Dinosaurs would not normally die off themselves. Some causes are
needed, and here's one  (somewhat speculative though). CT can change
the very notion of what a formal definition is. In the modern style,
the notion of ordered pair/tuple and its derivatives like formula and
term are central.  This quite simple syntax (as is often happens with
simple syntax, think, for example, of a Java program)  can hide
complex structures so that a tuple-based formal definition is not
actually formal and implicitly involves intuitive concepts. If CT will
sometime indeed reshape the criteria of being a formal specification
(of a Bourbaki's structure), then dinosaurs would be forced to acquire
CT (or die off).

Zinovy


> Judging from the talks at BLAST in Denver last month (B = Boolean
> algebras, L = lattices, A = (universal) algebra, S = set theory, T =
> topology), at least the algebraic community is moving very slightly in
> this direction.  Things will hopefully improve yet further when
> algebraic geometry gets over its snit with equational model theory.
>
> Meanwhile if you need a witness for seven degrees of separation, look no
> further than AMM and CT.
>
> (I confess to being an unreconstructed graph theorist and algebraist
> myself.  I may have to preemptively volunteer myself for re-education
> before it becomes involuntary.)
>
> Vaughan Pratt
>
>
>