On Thursday, July 4, 2002, at 07:21 AM, Bruno Marchal wrote:
>
> This list is based on the idea that -more is simpler-. We are open to
> all many-things idea (many worlds, many computations, etc.).
> Category theory is interesting in that respect, but beware
> mathematical mermaids! :)
Or sirens beckoning us onto (1-to-1) the rocks?
> Mmmh ... Years ago I found a reference on "Categoros" (?) a programming
> language based on category theory. I loose the reference (I remember
> only
> that it was a japanese work).
> I guess you know that typed lambda calculus have nice natural semantics
> in term of cartesian closed category. I prefer UNtyped lambda calculus,
> like
> pure lisp or like (axiomatic) recursion theory. I have study the Hyland
> topos
> which manage to have models both linked to recursion theory and sort of
> lambda
> calculus, but eventually I leave it because those models are not well
> fitting
> the problems I am working on.
I think you are well ahead of me on this stuff (which is _good_, as it's
nice to have people around to ask questions of). I know in handwaving
ways that the lambda calculus is closely related to cartesian closed
categories...I have a book by Lambek and Scott I've looked through, but
I need more basics first.
By the way, I started my real programming with Lisp, on a nice Symbolics
Lisp Machine. A lot of folks talk about the (true) point that any
recursively-complete language is sufficient to perform anything that can
be computed, with just constant factors between implementations, but I
strongly believe that the "conceptual gap" (or semantic gap) between
ideas and implementations is best bridged with rich languages, possibly
domain-specific languages. And that a lot of "knowledge" (about a
domain) ought to be accessible through languages. For example, C++,
Lisp, and Mathematica may all be in some sense equivalent, but the
usefulness varies tremendously. I have Mathematica, for example, and I
like the way it includes vast libraries of functions which reify or
encapsulate knowledge and give us "mathematical objects" to manipulate.
The tensor packs available for it are a good example, giving structures
like "Ricci" and "Riemann" to manipulate directly and efficiently.
A good class library is the same thing, of course, in an object-oriented
language like Smalltalk or Java. (Apparently a big part of the successes
of both Perl and Visual Basic--gag!--is in the extensive function or
subroutine libraries.)
It's intriguing to think of more abstract structures, such as the
categories of algebra, topology, algebraic topology, etc., being
implemented in the same way.
> My current "hobby" is Knot Theory. Curiously some "quantum categories"
> seem to
> appear in knot theory ... Louis Kauffman wrote quite readable papers on
> that.
> My interest in knots stems from my reading of Kitaev Papers on anyonic
> quantum
> computing (see also Freedman about his "modular functor").
I saw a streaming video talk given by the topologist Michael Freedman at
MSRI. URL for his talk "Anyons in Mathematics, Computer Science, and
Physics" is:
http://www.msri.org/publications/ln/msri/2000/subfactors/freedman/1/
Some interesting stuff on quantum computation and the braid category,
but inasmuch as I know even less about knots than about category theory,
I can't say much about his work.
>> I mean in the sense that the history of modern science seems to me to
>> be a succession of "throwing out the "centered" object," throwing out
>> a world centered around the Sun, or centered around God, or centered
>> around even Newtonian physics.
>
>
> "throwing out the "centered" object" (or "subject" perhaps?) is quite in
> the spirit of this list.
> Have you read Everett?
Hugh Everett, I assume you mean. Yes, indeed. I have the book edited by
Bryce DeWitt and Neill Graham, "The Many-Worlds Interpretation of
Quantum Mechanics," 1973. I think this is how many in the physics
community encountered MWI, through DeWitt's late 60s, early 70s
re-analysis.
Ironically, my general relativity instructor at UC Santa Barbara was Jim
Hartle, known then (1973) for his work on photon black holes and such,
and later famous for collaborations with Gell-Mann on "consistent
histories" and 'wave function of the universe" and with Hawking.
> Quite important. He just embeds the physicist in the
> physical world. My own work is a (radical) generalisation of that idea
> in the sense that I embed the "arithmetician" in the arithmetical world,
> making it a first order citizen.
Sounds intriguing. I'm currently less-focused on the role of human (or
machine) observers.
Isham makes an excellent point about time-varying sets, echoed by
Smolin. In a nutshell, while the logic of a quantum universe (or
cosmological universe, perhaps) may follow a Heyting logic where "the
cat is neither alive nor dead," once _any_ observation or measurement,
whether a machine or a written note or a memory or whatever, then the
logic is Boolean, as we "are used to."
Now obviously we're all familiar with this as the basic "measurement
collapses the wave function" model, so there is at first glance nothing
new here (you skeptics out there are right to be skeptical). However,
the topos-theoretical point of view, in which topos logic (Heyting) is
used instead of Boolean logic, seems to me to make the "interpretation"
problem (Copenhagen vs. MWI vs. Cramer vs. ...) largely go away.
The "naive realism" view is that whether we can see the cat or not, it
"must" be "really" either alive or dead. The Heyting/Isham/Smolin/etc.
point of view is that speculating about whether the cat is alive or dead
is as meaningless as speculating about what the "actual number of cats
living at this moment in Andromeda" is, given that that place is outside
our light cone (our causal past) and that the earliest we could even
conceivably answer that question is two million years from now.
Smolin covers this territory convincingly, for me, in his "Three Roads
to Quantum Gravity."
In fact, the elimination of the absolute view is refreshing.
Take, for example, the very model of past and future light cones. We are
familiar with the conventional world line of, say, me or you. Our world
line moves from out past to our future in this Minkowski (or some
variant) space-time. This is the point of view of the "outside,
omniscient, sees all events and objects in all parts of space-time"
point of view. The God viewpoint.
This very point of view encourages (some) people to think in
deterministic terms. "The future" and all that (emphasis on "the"). One
thing reading a lot of science fiction has done for me is to disabuse me
of any notion of "the" future. Instead, sheafs of possible futures.
Locally determistic, and past-deterministic (pace the point about
Heyting-->Boolean), but various possible worlds of various futures are
unknowable to observers in the real universe.
(Smolin makes the case that the universe is everything there is, that it
is pointless to speak of external observers who can see the entire
structure of space-time. The links between this viewpoint and other
areas are fascinating.)
We are finite beings in an effectively finite, though very large and of
effectively unlimited potential complexity, universe. The logic of
time-varying sets (essentially topos logic) is the natural way to
describe such systems. Locally, and in most everyday situations, Boolean
logic works very well in physical situations (all honest observers will
agree on any observation)...just as Euclidean geometry works very well
in most situations, just as other theories work very well in most
situations.
I'm amazed at how well humans can understand reality.
As I said, lots of people are way ahead of me in understanding the math.
Seeing how once obscure parts of mathematics turn out to be very useful
for Theories of Everything, I'm more convinced than ever that
essentially all branches of mathematics are somehow "built in" to the
structure of reality.
(And this is one reason I'm skeptical of models that reality is just a
cellular automaton running local rulesets on some computer. I have a
hard time conceiving of how so much interesting mathematics would exist
with simple local CA rules. But I could be wrong. :-) )
--Tim May
(.sig for Everything list background)
Corralitos, CA. Born in 1951. Retired from Intel in 1986.
Current main interest: category and topos theory, math, quantum reality,
cosmology.
Background: physics, Intel, crypto, Cypherpunks
Received on Fri Jul 05 2002 - 12:38:03 PDT