Addition as fuzzy mutual entropy

A sum of real numbers equals the mutual entropy of a fuzzy set and its complement set and the proof uses the logistic map, a diffeomorphism that maps extended real space onto the fuzzy cube embedded in it.

Abstract

A sum of real numbers equals the mutual entropy of a fuzzy set and its complement set. A “fuzzy” or multivalued set is a point in a unit hypercube. Fuzzy mutual (Kullback) entropy arises from the logarithm of a unique measure of fuzziness. The proof uses the logistic map, a diffeomorphism that maps extended real space onto the fuzzy cube embedded in it. The logistic map equates the sum of a vector's components with the mutual entropy of two dual fuzzy sets. Diffeomap projection offers a new way to study the fuzzy structure of real algorithms.