On Tuesday, December 3, 2002, at 02:15  PM, Osher Doctorow wrote:
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
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

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