Addition as fuzzy mutual entropy

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. Any sum of real numbers x1,.. . , x, equals the fuzzy mutual entropy of fuzzy set F in the unit hypercube [O, 11': k xi =H(F,Fc)-H(FC/F) (1) i=l where F " is the fuzzy set complement of F in the unit hypercube I ". If you add two numbers, F and F' lie in the unit square 12. If you add three numbers, they lie in 13, and so on up. The proof of (1) maps the extended real space R " = [-M, 4 " diffeomor-phically onto the embedded unit hypercube I ". The proof views the real numbers x 1,.. . , x, as the components of the real vector x in R " and maps x to a unique point or fuzzy set F in the fuzzy space I ". The mutual entropy terms H(F/F') and H(F'/F) stem from the logarithm of the fuzziness of F. As discussed below this fuzziness depends on how much F resembles its complement fuzzy set F'. In this sense we