Re: Some books on category and topos theory

From: Tim May <tcmay.domain.name.hidden>
Date: Tue, 9 Jul 2002 12:24:03 -0700

On Tuesday, July 9, 2002, at 11:08 AM, Bruno Marchal wrote:
>
> Me too. Now, I feel almost like you about ... knot theory.
> And this fit well with your cat-enthusiasm, for knot theory is
> a reservoir of beautiful and TOE-relevant categories
> (the monoidal one). I've just
> ordered Yetter's book: functorial(*) knot theory. It is the number 24
> of Kauffman series on Knots and Everything (sic) at World
> Scient. Publ Co. A series which could be a royal series for this
> list ...
> May I recommand the n° 1, by Louis Kauffman himself: knots and physics?
> A must for the (quantum) toes, and (I speculate now) the comp toe too!

I've looked at some of the knot series books, but have put them off for
now.

A good book to prepare for these books is Colin Adams, "The Knot Book:
An Elementary Introduction to the Mathematical Theory of Knots," 1994.

Whether knots are the key to physics, I can't say. Certainly there are
suggestive notions that particles might be some kind of knots in
spacetime (of some dimensionality)...a lot of people have played with
knots, loops, kinks, and braids for the past century.

One thing that Tegmark got right, I think, is the notion that a lot of
branches of mathematics and a lot of mathematical structures probably go
into making up the nature of reality.

This is at first glance dramatically apposite the ideas of Fredkin,
Toffoli, Wheeler1970, and Wolfram on the generation of reality from
simple, local rules. Wolfram has made a claim in interviews, and perhaps
somewhere in his new book, that he thinks the Universe may be generated
by a 6-line Mathematica program!

However, while I am deeply skeptical that a 6-line Mathematica program
underlies all of reality, enormous complexity, including conceptual
complexity, can emerge from very simple rules. A very simple example of
this is the game of Go. From extremely simple rules played with two
types of stones on a 19 x 19 grid we get "emergent concepts" which exist
in a very real sense. For example, a cluster of stones may have
"strength" or "influence." Groups of stones develop properties which
individual stones don't have. Abstraction hierarchies abound. The
Japanese have hundreds of names for these emergent, higher-order
structures and concepts. All out of what is essentially a cellular
automata.

So even if our universe is a program running as a screen saver on some
weird alien's PC, all sorts of complexity can emerge.

Getting down to earth, most of this complexity is best seen as
mathematics, I think.

I expect to take a closer look at knots after I get more math under my
belt.


> ....Now I know the z logics really should have "tensorial semantics",
> sort of many related (glued) von neumann type of logics (which are
> themselves atlases of boolean logics).
> But where (in Zs logic) those damned tensorial categories come from???
> Knots gives hints!!! This would explain the geometrical appearance
> of realities.
>
> Bruno
>
> (*) For the other: "functorial" really means categorial. Functors are
> the morphisms between categories. The first chapter of Yetter's book
> is an intro to category theory, the second one, on Knot theory, ...

Exciting stuff.


--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 Tue Jul 09 2002 - 12:26:42 PDT