On Sunday, December 1, 2002, at 11:21 AM, Ben Goertzel wrote:
FWIW, I studied category theory carefully years ago, and studied toposYou should study what interests you. If this type of math does not interest you, you should of course study something else.
theory a little... and my view is that they are both very unlikely to do
more than serve as a general conceptual guide for any useful undertaking.
(Where by "useful undertaking" I include any practical software project, or
any physics theory hoping to make empirical predictions).
As for measures of usefulness, it's certainly the case that most engineers, programmers, and even working physicists need little more than some linear algebra and bit of differential equations to handle 99% of the situations they encounter. From what I see of the current AI literature, maybe a bit of logic, a smattering of graph theory, and whatever specialty math in the chosen subfield of AI. For physics, the mix is different but the same applies.
And then I look to those doing the kind of physics of interest to me: John Baez, Fotini Markopoulou, Carlo Rovelli, Chris Isham, Lee Smolin. Then it's a lot of algebraic topology, n-category theory, homology, noncommutative geometry, topos theory (sheaves, locales, etc.), lattice theory, and so on. (Those interested in strings and field theories would deal with a slightly different mix.)
Certainly I agree that most advanced math doesn't lead to specific theories having specific predictions. And before anyone jumps on me for saying this, this is a fact of our times. The "low-hanging fruit" of 50-80 years ago, where a few weeks or months spent brushing up on the then-abstract area of Hilbert spaces could lead to some testable predictions are over.
My hunch is that the insights and understandings will come from those doing math and mathematical physics for the love of it, with occasional new testable predictions.
I'd count the Adjoint Functor Theorem as very deep, and useful. Likewise, Grothendieck's seminal work in the late 50s and early 60s on generalized spaces (Grothendieck topologies), along with the work by Serre, Lawvere, and others, eventually led to a category-theoretic proof by Deligne in 1974 of the Weil Conjecture.
My complaint is that these branches of math are very, very shallow, in spite
of their extreme abstractness. There are no deep theorems there.
(All of these terms can be explored on the Web, of course, with some of them actually explained pretty well. A basic text like Mac Lane's "Categories for the Working Mathematician," will help. Here's one of many Wikipedia-type articles on adjoint functors and the Adjoint Functor Theorem. http://www.wikipedia.org/wiki/Adjoint_functors )
The early 60s saw three parallel developments, all deeply interrelated: the aforementioned Grothendieck topologies work, the efforts by Lawvere to provide an alternative foundation to mathematics besides set theory, and Cohen's forcing methods to prove the independence of the Axiom of Choice from much of the rest of mathematics. This was the birth of topos theory. (And the connections with intuitionistic logic, that is, the logic of Brouwer and Heyting, notably, is closely tied to these areas. And, I believe and so do others, closely tied to issues of partial knowledge, quantum cosmology, the holographic model (still very hypothetical), and issues with black holes, information content, and quantum mechanics.)
What in mathematics is "deep," anyway? To some, the proof of the Taneyama Conjecture was deep, while to others it was just number theory. To others, proving certain properties about pullbacks in categories of objects and morphisms is deeper than proving something about the zeroes of a polynomial or calculating a Mersenne prime. To each their own.
There areWhen I look at the expanse of mathematics, I find it useful to see that category theory (and topos theory) tells us that a proof for semigroups, for example, automatically applies to an ostensibly different domain. This language allows us to "move up a level" and see the underlying similarities, even the isomorphisms. To see that while we may have several different names for things, they are actually the same thing.
no surprises. There are abstract structures that may help to guide thought,
but the theory doesn't tell you much besides the fact that these structures
exist and have some agreeable properties. The universe is a lot deeper than
Robert Geroch bases his text "Mathematical Physics" around category as a unifying approach...I hope this becomes the norm as this century unfolds, similar to the way the differential forms version of general relativity took over from the index gymnastics of ordinary tensors. (And apropos of your point about the new math not leading to many new predictions, did the Cartan-influenced differential forms approach to GR lead to new predictions that the classical, tensor-oriented approach did not? Probably few, if any, as most of the accessible predictions of GR were made a long time ago. But should students learning GR learn the methods for raising and lower indices in tensors or the more modern differential forms approach? The time saved, and the unity gained, may lead to new syntheses, such as in quantum gravity.
Likewise, is the Hilbert space formulation of QM dramatically different in making predictions that the Schrodinger wave equation formulation? Working chemists still calculate Hamiltonians and wave equations--they don't need to think in terms of Hilbert space abstractions. (And in the area of observables, the great Von Neumann actually got it _wrong_ in his formulation, as Bell proved several decades later...)
Geroch says this in his introduction:
"In each area of mathematics (e.g., groups, topological spaces) there are available many definitions and constructions. It turns out, however, that there are a number of notions (e.g., that of a product) that occur naturally in various areas of mathematics, with only slight changes from one area to another. It is convenient to take advantage of this observation. Category theory can be described as that branch of mathematics in which one studies certain definitions in a broader context--without reference to the particular area to which the definition might be applied. It is the "mathematics of mathematics."
"Although this subject takes some getting used to, it is, in my opinion, worth the effort. It provides a systematic framework that can help one to remember definitions in various areas of mathematics, to understand what many constructions mean and how they can be used, and even to invent useful definitions when needed."
And apropos of one of the direct themes of this list, the chart on page 248 is a better chart of the categories which are of direct (known) relevance to modern physics than Max Tegmark's chart of what he thinks of as the branches of mathematics. (I don't mean this to sound snide...it's just a statement of my opinion. Further, Tegmark and others working on All Math Models need to get up to speed on this "mathematics of mathematics.")
Sure, there's juicy stuff in the details of octonions. John Baez would agree with you. Getting down to making exact calculations is almost always necessary, and sometimes illuminating. But he also connects quaternions, octonions, etc. to n-categories and more generalized truths. Read his stuff for details--he writes more about both of these areas, various algebras and various categories, and their connections to physics, than anyone I know.
Division algebras like quaternions and octonions are not shallow in this
sense; nor are the complex numbers, or linear operators on Hilbert space....
Anyway, I'm just giving one mathematician's intuitive reaction to these
branches of math and their possible applicability in the TOE domain. They
*may* be applicable but if so, only for setting the stage... and what the
main actors will be, we don't have any idea...
Look, I'm happy that you looked at category theory and didn't find it to your taste. I had the opposite experience. Diversity is good.