Re: Tackling John Baez Head-On

>>From Osher Doctorow

In our fad-oriented era in the 20th and 21st centuries, 3 fads are
especially prominent: A. algebraic geometry, B. algebraic topology, C.
category theory (algebra-based in direction although functors and
functional equations are more closely related to Set Theory and
Analysis and so on). By remarkable coincidence (well, not entirely
remarkable - they tend to adhere together despite their occasional
squabbles), John Baez likes them all very much.

It isn't that these fields with the possible exception of the "objects"
of category theory (which should really be designated and thought of as
"generalized sets") were unnecessary or useless any more than
Cartesian/Analytic geometry was useless. Analytic Geometry was quite
valuable and is quite valuable, although anybody who obsesses about it
today is probably either an under-researched teacher in a pure
"teacher's college" or hasn't gone far beyond Analytic Geometry
mathematically. And Algebraic Geometry and Algebraic Topology have
some "objects" and theorems that are quite useful, although it probably
won't be long before people who obsess about them will be behind the
times.

Why did Analytic Geometry improve things, and why did obsession with it
go out of fashion, and why did it "reach its limits"? It is a good
lesson for the "Idols of the Tribe" etc. today.

John Baez probably thinks that Analytic Geometry "showed that geometry
is algebra and algebra is geometry." Hogwash. It's no more true than
that English language is Mathematics Language and Mathematics Language
is English language. Sure, it helps very much to translate between
languages, especially beween verbal and quantitative languages, and in
fact in my opinion if you can't do the latter two it may well not exist
as they say about the Yellow Pages. But if it were merely changing
the outer forms of information rather than getting insights into
different meanings, then we would all be Syntactic Information
engineers setting up satellites for both Terrorists and
Peace-At-Any-Pricers.

Is there a correspondence between Algebra and Geometry that Analytic
Geometry discovered and "unified"? Yes. It's slightly more valuable
than the correspondence between Verbal and Quantitative languages in
general, because when Geometry guides Algebra into classes of
equations, then we can explore the Algebraic properties that equations
have in common and which distinguish them. At that point, something
else happens. The Algebraists lose it. All they can do is manipulate
equations and occasionally move back and forth between different phases
that they don't recognize like real vs complex functions and ponder
inequalities. Then the Boys and Girls from Analysis enter the picture
which their continuous and geometric and limiting orientations and save
Algebra from a "fate worse than death". And the differential equation
people make the final "coup de grace" and the physicists and other
scientists interact with them or make differential equations part of
their "data-and-theorem base". Differential Equations are
Analysis-based and Analysis-derived, not Algebra-based. Little things
like continuity, limits, geometry, etc.

I know that some members of the usenet are very impressed by Analytic
Geometry to the extent of wanting to make it central to everything. I
was like that earlier - there was nothing in the world like it.
Neither is there anything in the world like using the toilet, but we
don't make a Religion out of it. Yet with the conics and their
3-dimensional analogs and so on, there's almost a Religious conviction
that Analytic Geometry has incredible power. So does a water faucet.
It doesn't produce Probability, Logic, Physics, Engineering,
Philosophy, Geometry, Topology, Arithmetic. It's nice, and then we go
on to other things while retaining the good parts of the past.

If you really are obsessed with Analytic Geometry, your calling may be
Linguistics and Languages. You'll have a chance of making it to the
U.N., where everybody translates Syntax but Semantics hasn't yet
penetrated. Good luck.

I think that John Baez about now is reduced in his arguments to
claiming that Algebra even in Algebraic Geometry and Algebraic Topology
is a building-and-unifying-block. Building yes. Unifying? Does a
building brick unify a building? Not as much as an Architect's plans
and a Contractor's insights. And two joined bricks don't either. It's
the Meta-building, the "game" of joining, that's key. Try a picture
of a building. If somebody tries to sell you a building or a house
without knowing your ideal picture and their picture, then I just
happen to have a Brooklyn Bridge that's available at great savings. :>)

Osher Doctorow

.

Relevant Pages