On Sunday, December 1, 2002, at 11:21 AM, Ben Goertzel wrote:
> FWIW, I studied category theory carefully years ago, and studied topos
> 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).
You should study what interests you. If this type of math does not
interest you, you should of course study something else.
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.
>
> My complaint is that these branches of math are very, very shallow, in
> spite
> of their extreme abstractness. There are no deep theorems there.
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.
(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 are
> 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
> that....
When 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.
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."
(p. 3)
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.")
>
> 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...
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.
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.
--Tim May
Received on Tue Dec 03 2002 - 14:36:20 PST