Reformulating the Map Color Theorem

A post to the [lawsofform] mailing list:

Reformulating the Map Color Theorem

Louis H. Kauffman

Department of Mathematics, Statistics and Computer Science University of Illinois at Chicago


This paper discusses reformulations of the problem of coloring plane maps with four colors. We include discussion of the Eliahou-Kryuchkov conjecture, the Penrose formula, the vector cross product formulation and the reformulations in terms of formations and factorizations due to G. Spencer-Brown.

… to move to a different computing paradigm

Michael A. Nielsen and Isaac L. Chuang (2004) wrote in their book "Quantum Computation and Quantum Information" [ISBN 0-5216-3503-9] on page 4:

"One possible solution to the problem posed by the eventual failure of Moore's law is to move to a different computing paradigm. One such paradigm is provided by the theory of quantum computation, which is based on the idea of using quantum mechanics to perform computations, instead of classical physics."

Another such paradigm is provided by the theory behind Pile Systems (see Peter Krieg's paper "The Red Brick Wall" deals with technology trends in the computer industry and their implications in relation to Pile.

Logic of Forms and lambda calculus

Loet Leydesdorff wrote (in the Luhmann mailing list):

"Beyond Varela, it has been Louis Kauffman who has taken lambda calculus to its consequences. The lambda calculus was developed in a series of papers with Goguen. Kauffman coauthored with Varela a paper entitled "Form dynamics" in the Journal of Social and Biological Structures (1984). There is a follow-up in 1987 with a paper entitled "Self-reference and recursive forms" in the same journal. Recently, Kauffman returned to the issues in "The Mathematics of Charles Sander Pierce," Cybernetics & Human Knowledge 8 (2002), 79-110."

That is not the first time, that Loet contributed on the subject ([1], [2]). In August 2005 he wrote (see [1]):

"Spencer Brown's specification of an observer is part of a mathematical discourse. It is a logic (static) and not a calculus (dynamic). Varela has tried to extend it to his so-called lambda calculus from a biological perspective, but that project has failed. In my opinion, we have a calculus available in Shannon's information theory, but the latter needs then to be extended to communication systems (instead of communication channels) and a theory of meaning. Elements are to be found (and have been found) in Maturana's theory of autopoiesis. Furthermore, there is beautiful work in biology about systems evolution and probabilistic entropy (e.g., Brooks & Wiley) and the theory of anticipatory systems is relevant."

If you google for <"Form dynamics" in the Journal of Social and Biological Structures (1984)> you will come across Louis' paper "Time, Imaginary Value, Paradox, Sign and Space" (see [3]).


[1] LUHMANN Archives — August 2005 (#83)

[2] LUHMANN Archives — October 2005 (#296)


Quantum Computation as Geometry :

"Quantum computers hold great promise for solving interesting computational problems, but it remains a challenge to find efficient quantum circuits that can perform these complicated tasks. Here we show that finding optimal quantum circuits is essentially equivalent to finding the shortest path between two points in a certain curved geometry. By recasting the problem of finding quantum circuits as a geometric problem, we open up the possibility of using the mathematical techniques of Riemannian geometry to suggest new quantum algorithms or to prove limitations on the power of quantum computers."