Le 19-sept.-06, à 11:50, marc.geddes.domain.name.hidden a écrit :
>
> Interesting what you said about modal and category theory. I don't
> know much about category theory and I'd be interested to know how you
> would define it. So: what is category theory? As far as I can make
> out it's a highly advanced branch of algebra concerned with the
> classification of mathematical structures.
You can put it in that way. A category is a collection of objects
together with collections of arrows between each object. The arrows are
supposed to follow some rules, like the existence of an identity arrow
for each objects, and arrows should be composable (i.e. if there is an
arrow f from A to B, and g from B to C, then there is an arrow gf from
A to C.
The amazing and interesting thing is that you can learn a lot about the
objects just by the arrows coming in and out of the objects. It leads
to a sort of purely functional view of math.
Typical categories are those where the objects are mathematical
structures and arrows are the morphism (or homomorphism) between those
objects. Ex: category of set (objects = sets, arrows = functions),
category of groups, category of topological spaces. Categories of
categories play the basic role, by supplying functor (morphism of
categories) which can be used for translating
Unfortunately categories occuring in recursion theory are hard to
handle, and my opinion is that we should use category when we cannot
avoid them (but I am confident that this is the case in all
mathematical fields up to some advanced point).
The third hypostases has already a free topos associated to it (a topos
is a generalisation the the category of set, it is a category of
"variable set" actuallly).
> Very interesting that you
> put Category Theory and Modal logic together!
They are many links. See Goldblat book (ref in my url).
>
> I feel the whole area of non-classical logic is underdeveloped. Things
> like modal logic, paraconsistent logic, fuzzy logic, possibility theory
> - perhaps these logics are actually the proper logic for dealing with
> 'reflection' and consciousness?
In my opinion they are overdeveloped. That is why I like to see them
unified by ... classical logic and classical (platonist) consideration.
>
> I think that conscious experience (Qualia) is ultimately knowledge
> reflecting on itself, or, to be more precise, I think Qualia are
> *mathematics modelling itself* - a sort of 'internal model of
> mathematics'.
I am more or less OK with that, although I would link them to their
"mathematics modelling itself" features that are non 3-communicable.
> To be even more precise, I think Qualia are symbols
> representing *ontological categories*.
?
> Could Category theory be
> exactly the type of math we need to understand this? How does category
> theory fit in with the other non-classical logics mentioned above?
This is almost a branch of logic by itself. Intuitionist logic fits
nicely with the toposes.
Symmetrical monoidal categories have something to say about quantal
algebra and quantum logic (but this leads to many technical
difficulties).
But if you have a taste for algebra, don't hesitate to dig ...
Categories are very useful in topology, knot theory, ...
They can be used in computation theory, but not really in computability
theory. I have used them in "Conscience et mecanisme", but abandon them
in subsequent work, if only because there are already too much "hard
math" to swallow before.
>
> We're closing in Bruno. We're closing in on the answers. I feel I
> vaguely understand the general principles behind everything now and
> it's just a matter of 'working out the math'. If course, all the devil
> is in 'working out the math' , and it could take decades. But I feel
> there's hope now. Real hope for answers in our lifetime. It might
> just be that one of us will break through and then all the powers of
> the universe will be humanities to hold...
We can hope ;)
Bruno
http://iridia.ulb.ac.be/~marchal/
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Received on Wed Sep 20 2006 - 06:25:32 PDT