Publications

[1] How can a line segment with extension be composed of extensionless points?
Synthese 200, no. 85 (2022): 1-28 (with Michael Vazquez and Scott Weinstein)

We provide a new interpretation of Zeno’s Paradox of Measure that begins by giving a substantive account, drawn from Aristotle’s text, of the fact that points lack magnitude. The main elements of this account are 1) the Axiom of Archimedes which states that there are no infinitesimal magnitudes, and 2) the principle that all assignments of magnitude, or lack thereof, must be grounded in the magnitude of line segments, the primary objects to which the notion of linear magnitude applies. Armed with this account, we are ineluctably driven to introduce a highly constructive notion of (outer) measure based exclusively on the total magnitude of potentially infinite collections of line segments. The Paradox of Measure then consists in the proof that every finite or potentially infinite collection of points lacks magnitude with respect to this notion of measure. We observe that the Paradox of Measure, thus understood, troubled analysts into the 1880’s, despite their knowledge that the linear continuum is uncountable. The Paradox was ultimately resolved by Borel in his thesis of 1893, as a corollary to his celebrated result that every countable open cover of a closed line segment has a finite sub-cover, a result he later called the “First Fundamental Theorem of Measure Theory.” This achievement of Borel has not been sufficiently appreciated. We conclude with a metamathematical analysis of the resolution of the paradox made possible by recent results in reverse mathematics.

[2] Being and Becoming Good in Plato’s Protagoras
Illinois Classical Studies 47, no. 2 (2022): 244-68

At the heart of Plato’s Protagoras is a lengthy interlude in which Socrates is compelled to offer an interpretation of a poem by Simonides (334c5–349a4). Protagoras, a notable sophist, challenges Socrates to explain away an alleged inconsistency: Simonides claims that it is hard for someone to become good or virtuous, but then criticizes another poet for saying what seems to be the same thing (339d1). In taking up this challenge, Socrates offers a protracted exegesis of the poem. While this so-called ‘poetic interlude’ has long attracted the attention of commentators, I uncover new and important ways in which it is connected to broader Platonic arguments and themes. I begin by marking a threefold distinction between ignorance, competence, and expertise, which both Protagoras and Socrates seem to utilize, but which has not been noted in the literature. I argue that this distinction helps to explain why Socrates is concerned to distinguish ‘being’ good from ‘becoming’ good in his own exegesis of the poem. I also argue that it helps to explain the significance of the preamble Socrates offers to his own interpretation (342a6–343c7) since it reveals the kind of knowledge that he believes is required for ‘being’ good or virtuous. The dialogue concludes with an explicit dilemma about virtue, but the resources I uncover allow for a satisfying resolution to it.

Active Projects

[1] In Plato’s Laws, the concept of self-mastery is clarified in a stretch of text known as the ‘puppets passage’ (644d7–645b1). On the most natural reading of this passage, self-mastery consists in the victory of reason in its struggle against the opposing pulls of pleasure and pain. The victory of reason is said to issue in virtue (645b1). This claim is surprising given that virtue had earlier been said to consist, not in victory, but in the harmonious relations among potentially conflicting parties (628c9–e1). This leads to a dilemma: should virtue be understood as a kind of victory, or as a kind of harmony? While commentators have produced and defended a number of interpretations aiming to resolve this dilemma about virtue, the concept of self-mastery has often been sidelined in the process. This, I argue, is a mistake. An adequate understanding of self-mastery dissolves the dilemma by revealing how virtue can legitimately be understood both as a kind of victory and as a kind of harmony.

[2] In book six of the Nicomachean Ethics, Aristotle distinguishes between two parts of the rational faculty: one part contemplates ‘things whose principles cannot be otherwise,’ while the other calculates ‘variable things’ (1139a6). Although both of these parts have received considerable treatment in the literature, they tend to be analyzed either in isolation or in contradistinction. Both approaches obscure the fact that these two parts comprise a single, unified faculty. Noting that the stated aim of book six is to determine the standard that ‘fixes’ right reason (1138b33), I argue that the role of the contemplative part of the rational faculty is to grasp stable first-principles, which not only ‘fix’ but ultimately explain what makes the reasoning of the calculative part ‘right’ in practical deliberation.

[3] Sextus Empiricus famously defines skepticism as the ability to “set out oppositions among things that appear and are thought of in any way at all” (PH I.8). This ability is then utilized by the skeptic in order to induce equipollence. When faced with opposing but equally forceful arguments or appearances, the recognition of their equal strength leads to the suspension of judgment, resulting in tranquility. I argue that the ability to ‘set out oppositions’ requires that the skeptic make use of the principle of non-contradiction; since the Pyrrhonian cannot set out an opposition to this principle without in some sense begging the question, they cannot remain skeptical about it.

Completed Projects

[1] My dissertation (supervised by Susan Sauvé Meyer) addresses an interpretive puzzle involving Plato’s seemingly incompatible accounts of psychic conflict. I argue that a better understanding of the soul helps to resolve this puzzle, and that we can properly understand Plato’s conception of the soul by focusing on his account of self-mastery (to kreittō heautou). Self-mastery is a concept well-suited to this project because it is treated consistently across dialogues and features centrally in discussions of psychic conflict.

Committee: Charles Kahn, Errol Lord, Rachel Singpurwalla

[2] My master’s thesis (supervised by Thomas Johansen) examines parts and wholes in Plato’s Parmenides. In this dialogue, an escalating series of one-many puzzles yields the surprising result that forms are wholes composed of parts. In order to explain how each form can be ‘one’ whole composed of ‘many’ parts, Plato introduces a new sort of unity: a one that is not also a many. I explore what this simple unity entails for Plato’s ethics, focusing on his enigmatic claim that the one is the good.