Le 05-juin-05, à 17:30, Stephen Paul King a écrit :
> FAR AWAY IN THE HEAVENLY ABODE OF THE GREAT GOD INDRA, THERE IS A
> WONDERFUL NET WHICH HAS BEEN HUNG BY SOME CUNNING ARTIFICER IN SUCH A
> MANNER THAT IT STRETCHES OUT INDEFINITELY IN ALL DIRECTIONS. IN
> ACCORDANCE WITH THE EXTRAVAGANT TASTES OF DEITIES, THE ARTIFICER HAS
> HUNG A SINGLE GLITTERING JEWEL AT THE NET'S EVERY NODE, AND SINCE THE
> NET ITSELF IS INFINITE IN DIMENSION, THE JEWELS ARE INFINITE IN
> NUMBER. THERE HANG THE JEWELS, GLITTERING LIKE STARS OF THE FIRST
> MAGNITUDE, A WONDERFUL SIGHT TO BEHOLD. IF WE NOW ARBITRARILY SELECT
> ONE OF THESE JEWELS FOR INSPECTION AND LOOK CLOSELY AT IT, WE WILL
> DISCOVER THAT IN ITS POLISHED SURFACE THERE ARE REFLECTED ALL THE
> OTHER JEWELS IN THE NET, INFINITE IN NUMBER. NOT ONLY THAT, BUT EACH
> OF THE JEWELS REFLECTED IN THIS ONE JEWEL IS ALSO REFLECTING ALL THE
> OTHER JEWELS, SO THAT THE PROCESS OF REFLECTION IS INFINITE
> THE AVATAMSAKA SUTRA
> FRANCIS H. COOK: HUA-YEN BUDDHISM : THE JEWEL NET OF INDRA 1977
> ***
> I am suggesting that these "jewels" give us an excellent way to
> think of OMs. If we are to allow for a value K {ranging from 0 to 1}
> to represent the degree to which one "jewel" "reflects" or "is similar
> to" or "implies", it seems that we get a very neat way to span a whole
> lot of logics and math with a simple picture. And, to top it off, we
> have a way to deal with infinite regress and circularity without
> paradox. (BTW, this is what Non-Well founded set theory is trying to
> explain!)
And Lee wrote in the same vain:
> As for circular, too bad your theories aren't circular! They'd
> explain more.
"My theories" are full of circular constructions! But as it is well
known circular construction can lead to paradoxes or even to frank
contradictions. Recursion theory, and then theoretical computer science
have provided founded semantics for most unfounded mathematical
structure appearing in computer science.
Don't forget I postulate comp which does give some importance to the
founded notion of bits and numbers. The magic is that bits and numbers
leads automatically and naturally to non-founded (circular) structure
with respect to universal machine/environment.
This is illustrated by the last post on combinators, which I have
introduced in part as an introduction to computer-theoretical circular
structure. I don't want to use Non-Well-founded set theory (nor any set
theory), nor category theory because the minimum of logic I use is
considered as already too abstruse to many. But those are very
interesting of course.
Note that John Case, one of the master of computer self-reference,
refers to the INDRA NET to introduce its generalization of Kleene fixed
point theorem. My whole approach is based on similar circular
self-reference, but, being programs or sets, mathematicians can use
them only when they have founded model of it. Look at the combinators:
it is only when Dana Scott provide founded models that the work on the
circular combinatory structures explodes in the literature.
Bruno
PS Lee, I will take some time to comment your posts. Thanks for your
patience.
http://iridia.ulb.ac.be/~marchal/
Received on Mon Jun 06 2005 - 03:24:28 PDT