Ben Goertzel <[EMAIL PROTECTED]> wrote:
>Tim May wrote:
>> As I hope I had made clear in some of my earlier posts on this, mostly
>> this past summer, I'm not making any grandiose claims for category
>> theory and topos theory as being the sine qua non for understanding the
>> nature of reality. Rather, they are things I heard about a decade or so
>> ago and didn't look into at the time; now that I have, I am finding
>> them fascinating. Some engineering/programming efforts already make
>> good use of the notions [see next paragraph] and some quantum
>> cosmologists believe topos theory is the best framework for "partial
>> The lambda calculus is identical in form to cartesian closed
>> categories, program refinement forms a Heyting lattice and algebra,
>> much work on the fundamentals of computation by Dana Scott, Solovay,
>> Martin Hyland, and others is centered around this area, etc.
>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).
>My complaint is that these branches of math are very, very shallow, in spite
>of their extreme abstractness. There are no deep theorems there. 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
>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...
Although I would agree that there is an atom of truth in the idea that
categories are shallow structures, I do think they will play a more and more
important role in the math, physics and (machine) psychology of the future.
1: Shallowness is not incompatible with importance. Sets are shallow
structures but are indispensable in math for example.
2: Categories are just sets, in first approximation, where morphism
are taken into account, and this has lead to the capital notion
of natural transformation and adjunction which are keys in universal
3: Categories are non trivial generalisation of group and lattice,
so that they provide a quasi-continuum between geometry and logic. This
made them very flexible tools in a lot of genuine domains.
4: Special categories are very useful for providing models in logic,
like *-autonomous categories for linear logic, topoi for intuitionist
logic, etc. Some special categories appear in Knot Theory, and gives
light on the role of Quantum field in the study of classical geometry.
Despite all this, some domain are category resistant like Recursion
Theory (I read the 1987 paper by Di Paola and Heller "Dominical Categories:
Recursion Theory Without Element" The journal of symbolic logic, 52,3,
594-635), but I still cannot digest it, and I don't know if there has
been a follow-up.
So my feeling is that category theory and some of its probable "quantum
generalisation" will play a significant role in tomorow's sciences.
In fact, categories by themselves are TOEs for math. Topoi are
mathematical universes per se.
At the same time, being problem driven, I think
category theory can distract the too mermaid-sensible researcher.
'Course there is nothing wrong with hunting mermaids for mermaids sake,
but then there is a risk of becoming a mathematician. Careful!