Hilbert's Sixth Problem: Descriptive Statistics as New Foundations for Probability

TL;DR

This paper proposes a new foundation for physical probability based on descriptive statistics and auto-correlation functions of time series, addressing Hilbert's Sixth Problem and challenging traditional probability theory.

Contribution

It introduces a novel approach to axiomatize physics using auto-correlation functions, emphasizing the role of descriptive statistics over classical probability foundations.

Findings

Auto-correlation functions of linear dynamical systems are approximately equal regardless of initial conditions.

The approach addresses the physical interpretation of probability, linking it to statistical properties of data.

It provides a new perspective on the axiomatization of physics inspired by Hilbert's Sixth Problem.

Abstract

Hay esbozos seg\'un los cuales las probabilidades se cuentan como la fundaci\'on de la teor\'i a matem\'atica de las estad\'isticas. Mas la significaci\'on f\'isica de las probabilidades matem\'aticas son oscuros, muy poco entendidos. Parec\'i era mejor que las probabilidades f\'isicas se fundaran en las estad\'isticas descriptivas de datos fisicales. Se trata una teor\'i a que as\'i responde a una cuestiona de Hilbert propuesta en su Problema N\'umero Seis, la axiomatizaci\'on de la F\'isica. Esta est\'a basada en las auto-correlaci\'ones de los series temporales. Casi todas las funciones de auto-correlaci\'on de las trayector\'i as de un sistema din\'amico lineal (con un n\'umero de grados de libertad bastante grande) son todas aproximadamente iguales, no importan las condiciones iniciales, a\'un si el sistema no sea erg\'odico, como conjetur\'o Khintchine en 1943. Usually, the theory of probability has been made the foundation for the theory of statistics. But the physical significance of the concept of probability is problematic, with no consensus. It would seem better to make the descriptive statistics of physical data the foundations of physical probability. This will answer a question posed by Hilbert in his Sixth Problem, the axiomatization of Physics. It is based on the auto-correlation function of time series. Almost all trajectories of a linear dynamical system (with sufficiently many degrees of freedom) are approximately equal, no matter their initial conditions, even when the system is not ergodic, as conjectured by Khintchine in 1943.