Szemerédi’s Theorem: An Exploration of Impurity, Explanation, and Content

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).

Ontological Purity for Formal Proofs

“…That is, we suggest each theorem has structural content in the form of a mathematical structure, which has as its instances the ontological content of a theorem, as well as surrogate versions of the ontological content. Structural content has been proposed before as "the instantiation of a particular fundamental mathematical structure by the entities intuitively involved in the statement" [29], but for us the structural content will refer exactly to the structure itself. There are several variants of structuralism, with a distinction between eliminative structuralism (there are possible structures, but not actual ones) and non-eliminative structuralism (there are actual structures) [30].…”