Re: Tackling John Baez Head-On

>>From Osher Doctorow

Elie Cartan is a study in probability-statistics. I kid you not. He
is a living (well, actually dead) example of what's wrong with physics
and to a lesser degree mathematics.

Despite John Baez' love of Clifford "algebra" and Division Algebras and
related objects, he's gotten his love object slightly wrong. Professor
Hestenes of Arizona State U. (ASU) is in my opinion the world's
greatest expert on Clifford "algebra", and regards the whole mismash as
Spacetime Calculus or Spacetime Geometry or both. Read his many papers
on the internet to see how some ______ (expletive deleted) got the
story backward.

But Elie Cartan - surely there we have an interdisciplinary person
worthy of John Baez' praise of combining algebra with topology,
geometry, and God knows what else as interdisciplinary? Not exactly.
The usual University of St. Andrews, Scotland site biography, School of
Mathematics and Statistics, University of St Andrews, Scotland, JOC/EFR
June 2005,
http://www-history.mcs.st-andrews.ac.uk/~history/Mathematicians/Cartan.html,
tells us that after 3 years at the Sorbonne he was appointed to the
Chair of Differential and Integral Calculus in Paris (what, not
algebra?), then appointed Professor of Rational Mechanics in 1920
(what, not algebra?), and then Professor of Higher Geometry from 1924
to 1940, and he retired in 1940 (what, not algebra?).

Yes, he started in algebra and his Ph.D. thesis contributed greatly to
Lie algebras where he classified them - oops! Shades of John Baez!

But Saint Andrews (to use the long form :>) says: "By 1904 Cartan was
writing papers on differential equations and in many ways this work is
his most impressive." Also: "From 1916 on he published mainly on
differential geometry." These are both from page 2. What the heck, no
algebra?

Admittedly, Saint Andrews tells us that his work achieved a synthesis
between/among continuous groups, Lie algebras, differential equations,
geometry.

Now, if you see a man/woman starting in algebra and ending up in
geometry, what do you have? Another Garrett Birkhoff. (See my various
discussions of him on sci.physics.) You don't have a Saunders MacLane
ending up in the Category Theory Happy Hunting Ground.

But wait - aren't Lie algebras and groups in algebra? Ach zo! But
we're talking group representation theory. They're representing things
outside algebra largely. In fact, they're mostly used to represent
physics, differential operators, etc. You don't have Clifford
Algebras being really Spacetime Geometry and Lie algebras/groups being
non-geometric objects.

Readers should have gotten the hint in the directions of his
publications as time progessed and his jobs and also the word
"continuous groups" - continuous as in differential equations and
analysis! These guys imitated geometry! They also imitated
physics. Like Nobel Laureate Steven Weinberg, they got tired of
imitating (he moved from Harvard to U. Texas Austin, from his own
Effective Gauge QFT and QFT in general to String Theory). Weinberg was
an interesting character, like Cartan, who kept changing even his own
theories, quite out of step with the sci.physics.relativity quick-march
I guess.

Of course, you might say: there is residual doubt, like the chicken or
the egg, as to whether Elie Cartan used algebra for geometry-DEs or
geometry-DEs for algebra despite the coincidence of his moving into
geometry and doing his best stuff in DEs and working most of his life
as Professor of Higher Geometry. That's where Probability-Statistics
comes in. When in doubt, make a Probability-Statistics model. Weigh
the various factors as best you can (preferably while not drinking
alcohol or taking dope) as one does in Probability-Statistics, and get
an estimate subject to some error.

How do I know anything? The Lie bracket - the most distinctive
feature of Lie algebras and Lie groups! It's not only physics but
probability-statistics (see my postings to sci.stat.math over the
years).

Am I saying that any use of two or more tools can be claimed as
"unification"? Or that Cartan may not have believed that he was
unifying things? Yes and maybe. It was a temporary "meeting of
tools", on the level of Analytic Geometry, but the tools were "impure"
by then and weren't even basically Algebraic even in Lie terms! They
were like the Jews - sufficiently raped and murdered over the millenia
so that they were mostly of whatever blood their "host" nations
contained.

Osher Doctorow

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