On Wednesday, July 3, 2002, at 07:30  AM, Bruno Marchal wrote:

> Welcome to the list Tim. I'm not sure the everything people
> will be so glad to add Category and Topos Theories among the
> branch of math which are needed in the search of a TOE :)
> (They "know" that computability (recursion) theory, provability
> logics, modal logics, quantum mechanics, etc. can be needed.

Thanks for the welcome. I've been occasionally reading the list via the 
archives for a while, and subscribed a couple of weeks ago to get a more 
direct feel.


>
> But category could be a good move. After all it is a sort
> of algebraic approach to a mathematical form of Everything ...

It'll be interesting and useful or it won't be (the future is already 
set? Boolean logic really _does_ rule?).

What I mean is that it's the particular area that's interesting to me 
now. In fact, it's the most interesting thing I've seen in many years.

I mentioned it in some long articles on another list I've been active in 
(Cypherpunks, a crypto list), but their issues and concerns are pretty 
far-removed from logic, category theory, and math (other than number 
theory). This list seems to have more points of intersection. So, here I 
am.



>
> In my older version of my thesis (a very long french technical
> report) I have make many sections showing that topos is *the*
> mathematical structure needed to provide rich models for what
> I call the first person discourse or the first person universe).
> This is quasi obvious if you know the relationship between
> intuitionist logic or Heyting algebra and toposes.

Yes, in a hand-waving way it sounds obvious.

There's a great book by Paul Taylor, "The Practical Foundations of 
Mathematics." He covers a lot of ground, giving a lot of historical and 
anecdotal references (something I wish more math book authors would do, 
even if it inflated their dry texts by 20% in length). One thing he 
says, is, I'll quote:

"Category theory provides the technology for creating new worlds. This 
is quite simple so long as we do not require them to be classical. 
...One application is to provide the generic objects ...in such a way 
that we may reason with them in the ordinary way in their worlds, and 
then instantiate them." (p. 59)

[Aside: I'm very intrigued by the possibility of building category-based 
class systems which go way beyond object-oriented class systems such as 
seen in Smalltalk, Java, ML, CLOS. Since categories are generalized 
objects, with more of a focus on _relations_ than objects, and since 
toposes are in some sense still more general (logical worlds instead of 
just worlds), I look for the computer languages of 20 years from now to 
look even more categorical, or even more toposophical, than today's 
crude cuts.]

> Now concerning the third person discourse (the "scientifically
> communicable" propositions) I have been lead toward symmetric
> monoidal categories not unlike those used as models of linear
> logic. Unfortunately I have very strong and non trivial constraints
> (due to the fact that I use the comp hypothesis in the "philosophy
> of mind") and I cannot really choose the mathematical structures.
> Actually I have derived from comp a "theory of physical reality",
> (Z1*) and I am in search of a semantics for it. It *should* be
> a braided category in which I could use the Hopf-algebraic tools
> for solving a (re)normalisation problem in Z1*.

Sounds interesting, but a paragraph obviously can't convey much. I'll 
look forward to hearing more "shadows on the wall," more "forgetful 
functors" giving glimpses.

>
> In the "new" much shorter version of my thesis I have suppressed
> all references to categories tough.

I'm curious why. I know a handful of people still think of category 
theory as "generalized abstract nonsense," but this is clearly not the 
case. At least no more so than algebra and topology are "generalized 
abstract nonsense."

Baez makes a point that the process of "decateforification," the process 
of ignoring categories and concentrating on simple measurables, is now 
being reversed by a revolution in physics of categorizing. Geroch's 
excellent "Mathematical Physics" text, 1985, starts with categories in 
Chapter 1 and treats relativity, QM, and later physics exclusively in 
categorical terms. Wonderful for seeing the connections and 
commonalities.


>
>
>>  Pearl, by the way, the UCLA professor who is the father of the 
>> murdered journalist Danny Pearl (in Pakistan).
>
>
> Gosh! I didn't knew that! I tought PEARL was a common name.

As soon as I heard mention on CNN that Pearl's father was a professor at 
UCLA, I suspected. A quick search confirmed the details.
Pearl's book is one that would profit immensely from a "collision" with 
other areas, especially the causal set work of Isham, Markopoulou, etc. 
(As in a lot of areas, the "invisible colleges" (communities of 
researchers) are often operating in micro-universeses of discourse, 
referencing from the same set of papers, using the same basic ideas. For 
example, the only real point of commonality between the universes of 
[Pearl-Dempster-Shafer-cognitive-AI]  and  
[Smolin-Markopoulou-Isham-physics] seems to be the term "causality" and 
varius references to Saul Kripke (who is better known in linguistic 
circles!).
>
>> [SNIP] What's fascinating is that a topos is a kind of "micro 
>> universe.' Not in a physical sense, a la Egan or Tegmark, but in the 
>> sense of generating a consistent reality. More on this later [SNIP].
>
>
> In the sense of (brouwerian-like) mental construction?
>

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.

A good example is throwing out Euclidean geometry. Or, rather, truly 
understanding that Euclidean geometry is just one of _many_ geometries.

Euclidiean geometry, Riemannian geometry, Newtonian physics, Lorentzian 
transforms in a Minkowski space (aka Einsteinian physics), etc. are just 
special kinds of universes. Are they "real"?

(This gets into Tegmark territory, about "actual" (whatever that means!) 
physical universes with different mathematics and then, obviously, 
different physical laws. Egan, in "Distress," calls this the "all 
topology  model.")

Personally, I'm a partial Platonist (these things in mathematics have 
some kind of existence) and a partial formalist (we are pushing symbols 
around on paper and in our minds). I increasingly view the 
constructivist (Intuitionist/Brouwerian) point of view as being 
consistent with what we see in the world and what we can model or 
simulate on computers.

There are aspects to the world which are Newtonian, which are Boolean, 
which use the discrete topology, which use only continuous functions, 
which are beyond Boolean, which are compact (in the compact space 
sense), which are non-compact, which are Banach, Hilbert, Fock, etc. 
spaces, and so on.

While I think the Universe is remarkably understandable, I don't think 
it makes much sense to talk about what "the" topology or laws of 
mathematics of the Universe is/are. (I apologize if this is not 
clear...this is just a glimpse.)

There is much, much more to say on all of these topics.

--Tim



--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

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