Hilbert's Sixth Problem and Soft Logic

Abstract

Hilbert's sixth problem calls for the axiomatization of physics, particularly the derivation of macroscopic statistical laws from microscopic mechanical principles. A conceptual difficulty arises in classical probability theory: in continuous spaces every individual microstate has probability zero. In this paper, we introduce a probabilistic framework based on Soft Logic and Soft Numbers in which point events possess infinitesimal Soft probabilities rather than the classical zero. We show that Soft probability can be interpreted as an infinitesimal refinement of classical probability and discuss its implications for statistical mechanics and Hilbert's sixth problem. In addition, we show rigorously how to construct a Mobius strip, based on the soft numbers, and we discuss how this Mobius strip representation with soft numbers allows for a deeper understanding of the nature and character of Hilbert's sixth problem. Inspired by the collapsing of that classical probability to zero, we suggest adding an axiom for an Infinitesimal Probability into the list of Kolmogorov's five Probability axioms. Furthermore, we suggest a probabilistic framework based on Soft Numbers for assigning values to probabilities of impossible events of a discrete random variable with realizations outside its support (which, in the ordinary probability, collapse to zero). This assignment of Soft Number values is based on an extension of the Pascal triangle to have soft zeros outside of the regular Pascal triangle (with real values) based on factorials of negative numbers.