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SZEMERÉDI’S THEOREM: AN EXPLORATION OF IMPURITY, EXPLANATION, AND CONTENT

Part of: Sequences and sets Proof theory and constructive mathematics Additive number theory; partitions General and miscellaneous specific topics General logic Measure-theoretic ergodic theory Philosophical aspects of logic and foundations

Published online by Cambridge University Press:  03 December 2021

PATRICK J. RYAN Open the ORCID record for PATRICK J. RYAN [Opens in a new window]
PATRICK J. RYAN*
Affiliation:
DEPARTMENT OF PHILOSOPHY UNIVERSITY OF CALIFORNIA, BERKELEY BERKELEY, CA, USA
*
E-mail: patrick.j.ryan@berkeley.edu
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Abstract

In this paper I argue for an association between impurity and explanatory power in contemporary mathematics. This proposal is defended against the ancient and influential idea that purity and explanation go hand-in-hand (Aristotle, Bolzano) and recent suggestions that purity/impurity ascriptions and explanatory power are more or less distinct (Section 1). This is done by analyzing a central and deep result of additive number theory, Szemerédi’s theorem, and various of its proofs (Section 2). In particular, I focus upon the radically impure (ergodic) proof due to Furstenberg (Section 3). Furstenberg’s ergodic proof is striking because it utilizes intuitively foreign and infinitary resources to prove a finitary combinatorial result and does so in a perspicuous fashion. I claim that Furstenberg’s proof is explanatory in light of its clear expression of a crucial structural result, which provides the “reason why” Szemerédi’s theorem is true. This is, however, rather surprising: how can such intuitively different conceptual resources “get a grip on” the theorem to be proved? I account for this phenomenon by articulating a new construal of the content of a mathematical statement, which I call structural content (Section 4). I argue that the availability of structural content saves intuitive epistemic distinctions made in mathematical practice and simultaneously explicates the intervention of surprising and explanatorily rich conceptual resources. Structural content also disarms general arguments for thinking that impurity and explanatory power might come apart. Finally, I sketch a proposal that, once structural content is in hand, impure resources lead to explanatory proofs via suitably understood varieties of simplification and unification (Section 5).

Keywords

Philosophy of mathematicsmathematical explanationhidden higher-order conceptsimpuritySzemerédi’s theoremergodic theoryproof theory

MSC classification

Primary: 00A30: Philosophy of mathematics
Secondary: 03A05: Philosophical and critical 11P99: None of the above, but in this section 11B25: Arithmetic progressions 28D05: Measure-preserving transformations 03F03: Proof theory, general 03B30: Foundations of classical theories (including reverse mathematics) 03F35: Second- and higher-order arithmetic and fragments
Type
Research Article
Information
The Review of Symbolic Logic , Volume 16 , Issue 3 , September 2023 , pp. 700 - 739
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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