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/


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