Hilbert's Sixth Problem: the endless road to rigour

TL;DR

This paper reviews the ongoing efforts to formalize the axiomatic foundations of physics and related sciences, highlighting recent advances in quantum probability, fluid dynamics, and machine learning inspired by Hilbert's Sixth Problem.

Contribution

It provides a comprehensive overview of modern developments addressing Hilbert's Sixth Problem across various scientific disciplines.

Findings

Continuum limits in atomistic kinetics may differ from classical fluid dynamics.

The curse of dimensionality in machine learning can be mitigated by statistical physics insights.

Quantum probability aids in modeling geological uncertainties.

Abstract

Introduction to the special issue of Phil. Trans. R. Soc. A 376, 2018, `Hilbert's Sixth Problem'. The essence of the Sixth Problem is discussed and the content of this issue is introduced. In 1900, David Hilbert presented 23 problems for the advancement of mathematical science. Hilbert's Sixth Problem proposed the expansion of the axiomatic method outside of mathematics, in physics and beyond. Its title was shocking: "Mathematical Treatment of the Axioms of Physics." Axioms of physics did not exist and were not expected. During further explanation, Hilbert specified this problem with special focus on probability and "the limiting processes, ... which lead from the atomistic view to the laws of motion of continua". The programmatic call was formulated "to treat, by means of axioms, those physical sciences in which already today mathematics plays an important part." This issue presents a modern slice of the work on the Sixth Problem, from quantum probability to fluid dynamics and machine learning, and from review of solid mathematical and physical results to opinion pieces with new ambitious ideas. Some expectations were broken: The continuum limit of atomistic kinetics may differ from the classical fluid dynamics. The "curse of dimensionality" in machine learning turns into the "blessing of dimensionality" that is closely related to statistical physics. Quantum probability facilitates the modelling of geological uncertainty and hydrocarbon reservoirs. And many other findings are presented.