On Tuesday, December 3, 2002, at 02:15 PM, Osher Doctorow wrote:
The
theorems that Tim has cited are one counterexample class to this, but
where
are the great predictions, where is there anything like the Einstein
Field
Equation, the Schrodinger Equation, Newton's Laws, Fermat's numerous
results, Maxwell's Equations, the Gauss-Bonnet Theorem and its
associated
equation that ties together geometry and topology,
Non-Euclidean/Riemannian
Geometry, Euclidean Geometry, the Jacobsen radical, Gauss-Null and
related
Well, sure, these examples are typical of the massive breakthroughs in
our understanding of the universe, and of the core fields of
mathematics, that came in the 19th and early 20th centuries, roughly
from 1850 to 1950. (There was a similar phase a bit earlier, with
Newton, Leibniz, Laplace, Lagrange, etc.)
This is an example of how the rate of discovery is _slowing down_.
(I'll wait a moment for the jeers to subside...)
By slowing down I mean of course that we have basically mapped out the
larger structure of physics, and also chemistry, geology, math, and so
on. Biology is perhaps much less mapped out.
It's not likely that any theory, whether algebraic topology or model
theory or whatever is going to give us anything like QM or relativity,
for the reason that we have them and we have no evidence that some
large body of experimental evidence awaits explanation in the way that
the early QM reearchers knew they were explaining aspects of reality
all around them (hydrogen atom, electron diffraction, spectral lines in
emissions, radioactive decay, new and obvious particles like alphas,
neutrons, positrons, muons, etc., and so on. Likewise, Einstein and
others knew full well the import of the Michelson-Morley experiment.
And the bending of light around the sun was predicted and then observed
less than 2 years later. Finally, both theories came together with the
atom bomb.
Theories today are much, much further removed from experiment and from
everyday implications. I don't need to say more about this, I presume.
sets in geometric nonlinear functional analysis, Godel's theorems, or
even
Hoyle's Law or the Central Limit Theorems or the almost incredible
theorems
of Nonsmooth Analysis and Kalman filters/predictors and Dynamic
Programming
and the Calculus of Variations and Cantor's cardinals and ordinals and
Robinson's infinitesimals and Dirac's equations and Dirac's delta
functions
and Feynmann's path history integrals and diagrams and the whole new
generation of continuum force laws and on and on.
Here you're getting more modern areas, areas which in fact are deeply
connected with topos theory, for example. Besides logic, which remains
an active field with active researchers, you ought to look into
"synthetic differential geometry," explored by Anders Kock and others.
SDE reifies the infinitesimals. I would strongly argue that the
nonsmooth analysis and infinitesimal analysis you crave is _more_
closely related to Grothendieck toposes than you seem to appreciate.
Likewise, check out papers by Crane, Baez, and others on the
connections between Feynman diagrams and category theory. (One of them
is "Categorical Feynmanology. The arXive site has them all.
Kalman filters are just an applied tool. Check out "support vector
machines" to see that work continues on new and improved tools of this
sort.
Expecting category theory to be a theory of dynamic programming or
linear algebra practical programming is not reasonable.
Sure, category theory can go in to many fields and find a category and
then
take credit for the field being essentially a category, and I can go
into
many fields and find plus and minus and division and multiplication
analogs
and declare the field as an example of Rare Event Theory [RET] or
Fairly
Frequent Event Theory [FFT or FET] or Very Frequent Event Theory [VFT
or
I confess that I have never understand your "Rare Event Theory."
Science is well-equipped to deal with events measured with large
negative exponents, even probabilities with 10^-50 or whatever.
I don't think the world's nonacceptance of "RET" means it is on par
with category theory, just because some here don't think much of it.
But string and brane theory are suffering from precisely what category
theory is suffering from - a paucity of predictions of the Einstein and
Schrodinger kind mentioned in the second paragraph back, and a paucity
of
depth. Now, Tim, you certainly know very very much, but how are you at
depth [question-mark - my question mark and several other keys like
parentheses are out].
On your first point, yes, many current theories are very far from
having testable predictions. Including, of course, the Tegmark theory,
the Schmidhuber theory, and all sorts of universe-as-CA theories.
String and loop quantum gravity theories may be hundreds of years away
from being tested...or a test could surprise us within the next 10
years, much as the eviden
