Hilbert's Sixth Problem and Soft Logic

2026-03-31

AI Summary

This study addresses the conceptual difficulties arising in Hilbert’s sixth problem from the classical probability theory’s assignment of zero probability to single points in continuous spaces. To resolve this, the paper proposes a novel probabilistic framework grounded in soft logic, soft numbers, and nonstandard analysis. This approach endows microstates with nonzero infinitesimal “soft probabilities,” enabling a refined mathematical reconstruction of the foundations of statistical mechanics. Innovatively employing soft numbers, the framework rigorously constructs topological structures such as the Möbius strip, thereby establishing a probability system in which point events possess nonzero infinitesimal probabilities. Furthermore, it offers a new geometric and logical perspective for the axiomatization of physical laws.

Technology Category

Reasoning under Uncertainty: Other Foundations of Reasoning under UncertaintyMachine Learning: Probabilistic Circuits and Graphical ModelsConstraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsWeb Mining and Content Analysis: Models for Web evolutionSecurity and Privacy: Large-scale security measurements

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.

Problem

• Hilbert's sixth problem
• statistical mechanics
• classical probability theory
• microscopic states
• macroscopic laws

Innovation

• Soft Logic
• Soft Numbers
• Soft Probability
• Hilbert's Sixth Problem
• Statistical Mechanics