On Tuesday, August 13, 2002, at 06:16  PM, Wei Dai wrote:

> On Tue, Aug 13, 2002 at 10:08:50AM -0700, Tim May wrote:
>> * Because toposes are essentially mathematical universes in which
>> various bits and pieces of mathematics can be assumed. A topos in which
>> Euclid's Fifth Postulate is true, and many in which it is not. A topos
>> where all functions are differentiable. A topos in which the Axiom of
>> Choice is assumed--and ones where it is not assumed. In other words, as
>> all of the major thinkers have realized over the past 30 years, topos
>> theory is the natural theory of possible worlds.
> How does this compare to the situation in classical logic, where you can
> have theories (and corresponding models) that assume Euclid's Fifth
> Postulate as an axiom and theories that don't?

Because such a dichotomy ("and theories that don't") means the logic is 
ipso facto modal. The very form tells us that a modal (and hence 
intuitionist) assumption is at work: "If it were the case that the 
parallel postulate were valid, then..." and "Suppose the parallel 
postulate is not true, then..."

If the Fifth Postulate is independent of the others, then within the 
framework of the other postulates one may have one "branch" where the 
Fifth holds (Euclidean Geometry) and another branch where it doesn't 
hold (all of the various non-Euclidean geometries).

Now this turns out to be a not very important example, as various 
geometries with various geodesics on curved surfaces are sort of 
mundane. And the details were mostly worked out a hundred years ago, 
starting with Gauss, Bolyai, Lobachevsky, Riemann, and continuing to 
Levi-Cevita, Ricci, and Cartan. The fact that by the mid-19th century we 
could _see_ clear examples of geometries which did not "obey" the 
parallel postulate, e.g., triangles drawn largely enough on a sphere, 
great circles, figures drawn on saddle surfaces and trumpet surfaces, 
etc., meant that most people didn't think much about the modal aspects. 
But they are certainly there.

(I believe it's possible to cast differential geometry, including the 
parallel postulate or its negation, in topos terms. Anders Kock has done 
this with what he calls "synthetic differential geometry," but I haven't 
read his papers (circa 1970-80), so i don't know if he discusses the 
parallel postulate explicitly.)

Both category theory and topos theory have been used to prove some 
important theorems (e.g., the Weyl Conjecture about a certain form of 
the Riemann zeta function, and the Cohen "forcing" proof of the 
independence of the Continuum Hypothesis from the Zermelo-Frenkel 
logical system), but it is misleading to think that either will give 
"different results" from conventional mathematics. It is not as if 
Fermat's Last Theorem is true in conventional logic or in conventional 
set theory but false in intuitionist logic or category theory.

I'm going to have to slow down in my writing. You ask a lot of short 
questions, but these short questions need long answers. Or, perhaps, I 
feel the need to make a lot of explanations of terminology and 
motivations. I'll have to tune the length of my responses to the length 
of your questions, I think!

--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, 
Background: physics, Intel, crypto, Cypherpunks

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