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.

Figures (6)

• Figure 1: The Soft coordinate axis (example of a line connected between $x=\frac{1}{2}$ and $\frac{1}{x}=2$)
• Figure 2: The lines between $x=1/n$ (horizontal axis) and $1/x=n$ (vertical axis). We illustrate for some natural numbers $n$, but the connections lines are valid for all real numbers $n\ge1$.
• Figure 3: The intersection point
• Figure 4: The distinction between $-0$ and $+0$
• Figure 5: The complete Soft coordinate system. The soft number point $C$ is presented by the height $A$ (=5) and width $B$ ($\approx$0.5).

Theorems & Definitions (10)

• Definition 1
• Definition 2: Order
• Definition 3: Soft Number
• Proposition 1
• Proposition 2
• Lemma 1
• proof
• Lemma 2
• proof
• Definition 4