Aristotle on Kind‐Crossing

Abstract

This paper concerns Aristotle's kind‐crossing prohibition. My aim is twofold. I argue that the traditional accounts of the prohibition are subject to serious internal difficulties and should be questioned. According to these accounts, Aristotle's prohibition is based on the individuation of scientific disciplines and the general kind that a discipline is about, and it says that scientific demonstrations must not cross from one discipline, and corresponding kind, to another. I propose a very different account of the prohibition. The prohibition is based on Aristotle's scientific and metaphysical essentialism, according to which a scientific demonstration must take as its starting point a set of per se properties of a subject, if these make up a single, unitary definition. The subject of demonstration here is a kind, although not the general kind associated with a discipline, but rather the particular kind that the particular demonstration is about.

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

“…Here I provide the conceptual contours of Furstenberg's ergodic proof of Szemerédi's theorem. 58 The summary here should contain all the mathematical material needed to engage with the discussion below.…”

“…And it was not the end of the story: the subsequent rigorous development of mathematical analysis (independent of geometrical proofs) added even greater weight to the hypothesis that 25 Here I am following the traditional reading of 75a38-b2. For a dissenting interpretation, see the recent article by Steinkr üger [58] in which the main lines of the traditional reading are also discussed. 26 Aristotle does allow for "exceptions" to the purity constraint, e.g., one can demonstrate an optical fact using geometry.…”

“…Theorem 3.3 Let (X, B, , T ) be a measure-preserving system and let E ∈ B with (E) > 0. Then for any k ≥ 1, 58 I have primarily followed the survey [24]. See also Chapter 7 of the textbook by Einsiedler and Ward [17] as well as Tao's discussion in [61] and the expository paper [67].…”

“…However, these domains fall within a larger structural genus, genus in question that determines the purity constraint? Again, see [58] for a survey of some answers to this question. 110 Thus, the distance metric given by the topical content of a theorem does not necessarily generate a purity constraint in an obvious way.…”

“… 25 Here I am following the traditional reading of 75a38-b2. For a dissenting interpretation, see the recent article by Steinkrüger [58] in which the main lines of the traditional reading are also discussed. …”