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If you can distinguish, you can express: Galois theory, Stone--Weierstrass, machine learning, and linguistics

Ben Blum-Smith, Claudia Brugman, Thomas Conners, Soledad Villar

TL;DR

The paper investigates a unifying principle connecting algebra, analysis, and linguistics by interpreting Stone–Weierstrass and the Fundamental Theorem of Galois Theory as two instantiations of distinguishing vs expressing. It presents an elementary theorem showing how SW-distinguishing data can be extended to subgroups via invariants that Galois-distinguish the subgroup, yielding $L^H = k(f_1,...,f_r,f^*_1,...,f^*_s)$. The authors apply this lens to machine learning contexts—graph learning, point clouds, and orbit recovery—showing that degree-3 invariants can generically separate orbits and generate invariant fields via Rosenlicht’s framework. They illustrate the linguistic side with Saussurean signs, demonstrating how expressive capacity emerges from structured distinctions across kin terms, pronouns, numerals, and color terms.

Abstract

This essay develops a parallel between the Fundamental Theorem of Galois Theory and the Stone--Weierstrass theorem: both can be viewed as assertions that tie the distinguishing power of a class of objects to their expressive power. We provide an elementary theorem connecting the relevant notions of "distinguishing power". We also discuss machine learning and data science contexts in which these theorems, and more generally the theme of links between distinguishing power and expressive power, appear. Finally, we discuss the same theme in the context of linguistics, where it appears as a foundational principle, and illustrate it with several examples.

If you can distinguish, you can express: Galois theory, Stone--Weierstrass, machine learning, and linguistics

TL;DR

The paper investigates a unifying principle connecting algebra, analysis, and linguistics by interpreting Stone–Weierstrass and the Fundamental Theorem of Galois Theory as two instantiations of distinguishing vs expressing. It presents an elementary theorem showing how SW-distinguishing data can be extended to subgroups via invariants that Galois-distinguish the subgroup, yielding . The authors apply this lens to machine learning contexts—graph learning, point clouds, and orbit recovery—showing that degree-3 invariants can generically separate orbits and generate invariant fields via Rosenlicht’s framework. They illustrate the linguistic side with Saussurean signs, demonstrating how expressive capacity emerges from structured distinctions across kin terms, pronouns, numerals, and color terms.

Abstract

This essay develops a parallel between the Fundamental Theorem of Galois Theory and the Stone--Weierstrass theorem: both can be viewed as assertions that tie the distinguishing power of a class of objects to their expressive power. We provide an elementary theorem connecting the relevant notions of "distinguishing power". We also discuss machine learning and data science contexts in which these theorems, and more generally the theme of links between distinguishing power and expressive power, appear. Finally, we discuss the same theme in the context of linguistics, where it appears as a foundational principle, and illustrate it with several examples.

Paper Structure

This paper contains 10 sections, 4 theorems, 6 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. What does the Fundamental Theorem of Galois Theory have to do with the Stone--Weierstrass theorem?
  3. A precise connection between the two notions of distinguish in Stone--Weierstrass and Galois Theory
  4. Distinguish $\Leftrightarrow$ express in data science and machine learning
  5. Stone--Weierstrass in graph learning
  6. Almost universal invariant machine learning on point clouds via Galois theory
  7. Orbit recovery and field generation
  8. The number of (almost) distinguishing invariants
  9. Distinguish $\Rightarrow$ express in linguistics
  10. Summary

Key Result

Theorem 2.1

Let $X$ be a compact Hausdorff topological space. Let $C(X,\mathbb{R})$ be the Banach space of continuous real-valued functions on $X$, with the sup norm; it is a Banach algebra under pointwise multiplication of functions. Let $S\subset C(X,\mathbb{R})$ be a subset. Then the algebra $\mathbb{R}[S]$

Figures (2)

  • Figure 1: 3D objects represented as point clouds from shapenet2015.
  • Figure 2: Figures from zaslavsky2022evolution depicting the Nafaanra color naming system in 1978 (a) and in 2018 (b) for the colors in the chip chart from the World Color Survey (WCS) used as a stimulus grid. (a) The 1978 system: fiNge 'light', wOO 'dark', and nyiE 'warm or red-like.' (b) The 2018 system: the three terms from 1978 have smaller expressive ranges, and new terms have emerged--- wrEnyiNge 'green', lomru 'orange', Ngonyina 'yellow-orange', mbruku 'blue', poto 'purple', wrEwaa 'brown', and tOOnrO 'gray'.

Theorems & Definitions (12)

  • Theorem 2.1: Stone--Weierstrass
  • Theorem 2.2: Fundamental Theorem of Galois Theory
  • Theorem 2.3: Rosenlicht's theorem rosenlicht1956some
  • Remark
  • Definition 3.1: generically Stone--Weierstrass-distinguishing set of functions for a group $G$
  • Definition 3.2: Galois-distinguishing set of functions for a subgroup $H\subset G$
  • Theorem 3.3: Theorem 3.1 in blum2024galois
  • proof
  • Example 5.1: Kin terms
  • Example 5.2: Personal pronouns
  • ...and 2 more