Le 08-nov.-05, à 18:48, uv a écrit :
Bruno wrote
I don't know about the work of Heather and Rossiter, except some
thought on quantum computation I just found by Googling. Perhaps you
could elaborate a little bit.
I can answer you briefly on that one immediately by giving URL
http://computing.unn.ac.uk/staff/CGNR1/advstudiesmathsmonism.pdf
Please let me know if it disappears before you get there, nowadays
they sometimes do unfortunately.
I got it and have printed it. Interesting (especially for the Category
Theory minded people, which I am a little bit) but I do think it is a
little bit out of topic, at least for the moment. In my "Brussels'
thesis" I have use a bit of category theory, but I have decided to
suppress it when I realized that asking referees simultaneous
knowledge in
basic cognitive science/philosophy of mind,
+ mathematical logic
+ quantum theory
is almost impossible (even if only a familiarity with only the most
basic introduction is enough). Adding "category theory" to that panel
makes things worst as you can imagine.
That is very close to implying a
TOE. My own group is http://groups.yahoo.com/group/ttj It also
gives my blog and URL.
Some work has also been done by Heather and Rossiter on quantum
computing, with some comments on Deutsch's work.
By the way Johnson
Johnson ? Do you mean Johnstone?
seems to be the really important man in category
theory, "Sketching the Elephant" being the big book but afraid I am
still reading Lawvere and Schanuel
That is a good one. A very rare elementary introduction to category
theory.
Actually I have a much more rare and implausible book: an introductory
course in category theory from Kinshasa University Press (Congo), quite
nice but no more on the market. I have also the notes by Lawvere before
Shanuel makes the book.
I really love category theory (especially for logic and computer
science), and eventually, when I will come back to the combinators (if
I do) category will appears naturally by themselves, but I do think it
could be premature now.
A good book on Category Theory is the book by Robert Goldblatt "Topoi".
Some categorist (like Johnstone) criticize it, because it does not
stick on pure diagrammatic chasing, but then Goldblatt is a (modal)
logician, and actually it is that which makes the book understandable
(at least for logician).
The *must* remains the MacLane's book "categories for the working
mathematician" (takes me year to grasp just the preface, though!, but
then I learn a lot).
In relation with my work, and oversimplifying a little bit, categories
appears mainly as generalisation of the modal S4, or S4Grz logics, and
as such correspond to "first person notion"(*) and their intuitionistic
logic.
Contrariwise, the 3-person notions, which with comp are based on
recursion theory, are the notion which fits the less with the category
approach (but with the Combinators some light appears in the dark ...).
(*) Kripke models of S4 are multiverse with a reflexive and transitive
relation of accessibility (between universes/states/observer-moments).
A category is just the same except that more than one arrows are
allowed among the "points/states...", and arrows must be composed.
Bruno
http://iridia.ulb.ac.be/~marchal/
