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History of Archimedean and non-Archimedean approaches to uniform processes: Uniformity, symmetry, regularity

Emanuele Bottazzi, Mikhail G. Katz

TL;DR

The paper compares Archimedean and non-Archimedean treatments of uniform chances using a fair spinner as a focal example. It analyzes how symmetry, uniformity, regularity, and the requirement that every outcome be possible constrain models in Archimedean (Lebesgue-based) versus non-Archimedean (hyperfinite/infinitesimal) frameworks, showing that in ZFC no single model is forced by these hypotheses. It highlights a route to uniqueness via a minimal Archimedean model and its regular non-Archimedean counterpart, while also detailing Parker’s haecceitistic critiques and how axiomatic NSA approaches (e.g., BSPT', SCOT) can mitigate these concerns. The work underscores that the choice of foundational assumptions, including the use of Solovay-type models or nonstandard analysis, profoundly shapes which models of chance are considered valid, with implications for both mathematical philosophy and probabilistic modeling of vast sample spaces.

Abstract

We apply Nancy Cartwright's distinction between theories and basic models to explore the history of rival approaches to modeling a notion of chance for an ideal uniform physical process known as a fair spinner. This process admits both Archimedean and non-Archimedean models. Advocates of Archimedean models maintain that the fair spinner should satisfy hypotheses such as invariance with respect to rotations by an arbitrary real angle, and assume that the optimal mathematical tool in this context is the Lebesgue measure. Others argue that invariance with respect to all real rotations does not constitute an essential feature of the underlying physical process, and could be relaxed in favor of regularity. We show that, working in ZFC, no subset of the commonly assumed hypotheses determines a unique model, suggesting that physically based intuitions alone are insufficient to pin down a unique mathematical model. We provide a rebuttal of recent criticisms of non-Archimedean models by Parker and Pruss.

History of Archimedean and non-Archimedean approaches to uniform processes: Uniformity, symmetry, regularity

TL;DR

The paper compares Archimedean and non-Archimedean treatments of uniform chances using a fair spinner as a focal example. It analyzes how symmetry, uniformity, regularity, and the requirement that every outcome be possible constrain models in Archimedean (Lebesgue-based) versus non-Archimedean (hyperfinite/infinitesimal) frameworks, showing that in ZFC no single model is forced by these hypotheses. It highlights a route to uniqueness via a minimal Archimedean model and its regular non-Archimedean counterpart, while also detailing Parker’s haecceitistic critiques and how axiomatic NSA approaches (e.g., BSPT', SCOT) can mitigate these concerns. The work underscores that the choice of foundational assumptions, including the use of Solovay-type models or nonstandard analysis, profoundly shapes which models of chance are considered valid, with implications for both mathematical philosophy and probabilistic modeling of vast sample spaces.

Abstract

We apply Nancy Cartwright's distinction between theories and basic models to explore the history of rival approaches to modeling a notion of chance for an ideal uniform physical process known as a fair spinner. This process admits both Archimedean and non-Archimedean models. Advocates of Archimedean models maintain that the fair spinner should satisfy hypotheses such as invariance with respect to rotations by an arbitrary real angle, and assume that the optimal mathematical tool in this context is the Lebesgue measure. Others argue that invariance with respect to all real rotations does not constitute an essential feature of the underlying physical process, and could be relaxed in favor of regularity. We show that, working in ZFC, no subset of the commonly assumed hypotheses determines a unique model, suggesting that physically based intuitions alone are insufficient to pin down a unique mathematical model. We provide a rebuttal of recent criticisms of non-Archimedean models by Parker and Pruss.

Paper Structure

This paper contains 30 sections, 12 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. History of regular chances
  3. History of critiques of regular chances
  4. Critiques of regular probabilities
  5. History of the three realms
  6. Basic mathematical models in probability
  7. The issue of countable additivity
  8. The fair spinner
  9. Choosing hypotheses on the model
  10. Strengthening the hypothesis Po
  11. The A-track approach
  12. When is the Lebesgue measure total?
  13. Extensions of the Lebesgue measure in ZFC
  14. Assessment of the A-track models
  15. Possible path to unicity: minimal Archimedean model
  16. ...and 15 more sections

Key Result

Proposition 4.1

An Archimedean probability measure $\mu$ on an infinite sample space $X$ cannot satisfy simultaneously the hypotheses Un$_p$, Po and Reg.

Theorems & Definitions (22)

  • Definition 3.1
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • Proposition 4.3
  • proof
  • Proposition 4.4
  • Proposition 4.5
  • proof
  • Corollary 4.6
  • ...and 12 more