Mathematics and the Structure of Reality

From: Tim May <tcmay.domain.name.hidden>
Date: Tue, 3 Dec 2002 15:30:11 -0800

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 evidence for black holes has mounted dramatically,
faster than many of us 30 years ago thought it would.

The reasons for this fall into two main categories (no pun intended).
One, the low-hanging fruit point. A lot of bright minds have churned
over things, with perhaps 1000 times as much effort as we had when
Einstein was theorizing in his patent office, or even when the
community of quantum mechanics experts was still small enough that they
could meet a few times a year (a la Solvay Conferences) or meet with
David Hilbert, Emmy Noether, Emil Artin, and John von Neumann in their
offices at Gottingen and Heidelberg.

The second reason is the energy one. It took a few thousand dollars's
worth of equipment in 1900-1920 to produce interesting new particles,
to prove the existence of electrons, protons, etc. Then it took maybe
10-20 that to build Cockroft-Walton and Van de Graf accelerators to
find more. Then another similar increment in cost and energy for the
early cyclotrons to find the interesting particles of the 1930s. Then
another such factor to build nuclear piles and larger cyclotrons. And
thus the 1950s saw a huge expenditure to build the Bevatron, where the
antiproton and other particles were discovered/confirmed/created. (By
the way, Dirac's prediction of antiparticles was a very "category
theory"-like process of looking at symmetries in a commutative diagram
and essentially saying "to make this diagram commute, I need this leg
of the diagram.") And then we had multibillion dollar Brookhaven AGS
machines, SLAC, and on to machines like Fermilab and CERN which cost
many tens of billions. The Superconducting Supercollider was ultimately
dropped due to incredible cost and low bang for the buck (perhaps only
one new particle, according to many predictions.)

Yeah, it's conceivable that the SSC would have produced some new realm
of physics, as the enhancements going into CERN may still do. But the
odds are against it. (John Cramer, he of the transactional
interpretation of quantum mechanics, has some nice science fiction
about this sort of thing: "Twistor." But it's SF, not anything that is
likely to come out of CERN.)

_These_ are the reasons we are in a sense "filling in the details,"
fleshing out the tree whose branches grew to nearly their present form
by the 1970s.

It's one reason, I think, a lot of us who got started in physics in the
60s and 70s moved into other fields. (There was nothing of interest to
me in S-matrix theory and Regge calculus, the "hot" areas of
theoretical physics in 1972.)

On your second point, about "how are you at depth?", I hope this wasn't
a cheap shot. Assuming it wasn't, I dig in to the areas that interest
me. As I have said more than a few times here, I am just in the past 8
months or so digging in deeply to areas of logic (especially modal),
brushing up on my math (algebra, topology, algebraic topology,
analysis, etc.), and using topos and category theory as my touchstones.
When I have gone as far as I wish to, perhaps I will move on to other
areas.



>
> I will give an example. Socrates would rank in my estimation as a
> Creative
> Geniuses of Maximum Depth. The world of Athens was very superficial,
> facially and bodily and publicly oriented but with relatively little
> depth,
> and when push came to shove, rather than ask what words meant, it
> preferred
> to kill the person making the inquiries.

The death of Socrates notwithstanding, this does not sound like the
Athens I studied. Socrates was a strange bird, and there is much
evidence that he did everything he could to ensure his own death
sentence. Equally profound thinkers like Plato and Aristotle faced very
little pressure.

> What it was afraid of was going
> deep, asking what the gods really were,

This doesn't match my reading of history. Aristotle asked profound
questions about the nature of reality, metaphysics, belief, etc.

> You mentioned, Tim, that the Holographic Model is still very
> hypothetical.
> Are we to understand that G. 't Hooft obtained the Nobel Prize for a
> very
> hypothetical idea [question-mark] among others.

Ah, but 't Hooft did not get the Nobel for his work with Susskind and
others on the holographic model. He got it for his work on the
electroweak force.
>
> I will conclude this rather long posting with an explanation of why I
> think
> Lawvere and MacLane and incidentally Smolin and Rovelli went in the
> wrong
> direction regarding depth. It was because they were ALGEBRAISTS -
> their
> specialty and life's work in mathematics was ALGEBRA - very, very
> advanced
> ALGEBRA. Now, algebra has a problem with depth because IT HAS TOO MANY
> ABSTRACT POSSIBILITIES WITH NO [MORE CONCRETE OR NOT] SELECTION
> CRITERIA
> AMONG THEM. It is somewhat like the Ocean - if an explorer worships
> the
> Ocean, then he will go off in any direction that Ocean seems to be
> leading
...

Sorry, but this is a silly argument. Smolin and Rovelli may in fact be
wrong in their theory of loop quantum gravity (and the closely related
theories of spin foams, etc., along with Penrose, Susskind, Baez,
Ashketar, and the whole gang), but it is almost certainly not for some
simplistic reason that they were "ALGEBRAISTS."

In fact, Penrose is a geometer's geometer. See, for example, the essays
in his Festschrift. Now the geometry focus of Penrose does not prove
_anything_ about either the internal consistency or the ultimate truth
of some of his spin network and spinor models, nor about the truth or
falsity of spin foams and so on.

As for Lawvere and Mac Lane being "ALGEBRAISTS," I neither see your
point nor its relevance. What Mac Lane may or may not be is open to
debate...his work on homology theory tends to mark him as an algebraic
topologist. And Grothendieck and Lawvere were looking into
generalizations of the concept of a space--and they succeeded.
(Personally, and speculatively, when the concept of a space is
generalized so nicely, I think in terms of "this probably shows up in
the physical world or its description someplace." If this ain't
geometry affecting a physics outlook, what is?)

Anyway, it's silly to argue along these lines. You ought to take a look
at one of his recent books (co-authored when he was around 80):
"Sheaves in Geometry and Logic: A First Introduction to Topos Theory."
I'd call sheaves, presheaves, and locales some pretty deep
geometrical/topological ideas, albeit at a level of abstraction that
takes a lot of effort to master.

What the structure of reality really is depends on a couple of
important things:

1. What aspect we are looking at, whether the local causal structure of
spacetime or the "explanation" of the particles and their masses, or
even at some grossly different scale, such as fluid turbulence (still
not understand, in many ways, and yet almost certainly not depending on
theories of branes or strings or the Planck-scale structure of
spacetime).

2. Scales and energies, whether the cosmological or the ultrasmall.

3. Our conceptual biases (if we only know geometry, we see things
geometrically, and so on).

One of the reasons I like studying math is to expand my conceptual
toolbox, to increase the number of conceptual basis vectors I can use
to build models with.


--Tim May
Received on Tue Dec 03 2002 - 18:35:12 PST