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 (334c–349a). 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 (339d). 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 (342a–343c) 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 (to kreittō heautou) is clarified in a stretch of text in which humans are likened to puppets pulled in opposing directions by various strings or cords. On the most natural reading of the ‘puppets passage’ (644d–645b), self-mastery consists in the victory of reason in its struggle against the opposing pulls of pleasure and pain. While the passage as a whole is presented as a clarification of virtue (645c), this is surprising given that virtue is elsewhere said to consist—not in the victory of reason over pleasure and pain, but in the harmony of reason with pleasure and pain (653b). This gives rise to a dilemma: should virtue ultimately be understood as a kind of victory, or as a kind of harmony? While various interpretive strategies have been deployed to make sense of what this passage implies about virtue, I argue that the dilemma is only apparent: each horn corresponds to a distinct grade of virtue. This is most clearly revealed, I suggest, by examining the connection between self-mastery and moderation (sōphrosunē). Two kinds of moderation, popular and exalted, are explicitly distinguished from each other (710a). Popular moderation corresponds to a lower grade of virtue characterized by victory or self-mastery. Exalted moderation, by contrast, corresponds to a higher grade of virtue characterized by harmony. Tracing this distinction across key passages and situating it within the broader Platonic corpus, I offer an interpretation that not only preserves the integrity of the puppets passage, but also clarifies the developmental arc of virtue in the Laws.

[2] In the Protagoras, Socrates endorses the hedonic calculus—the measurement of pleasures and pains—as essential for virtue (356b–357b). In the Phaedo, Socrates rejects it (69a–b). While scholars have long noted this apparent tension, I propose a novel resolution grounded in a distinction from the Statesman between two kinds of measurement: comparative and normative (283c–284e). The comparative kind uses a relational measure (pros allēlas), while the normative kind uses an appropriate or due measure (to metrion). I argue that Socrates adopts the hedonic calculus in the Protagoras under a hedonistic assumption that reduces normative to comparative measure, while in the Phaedo he merely rejects comparative measure—leaving normative measure as a viable option. This distinction reframes the debate and reconciles the dialogues by showing that virtue requires normative measure.

[3] In book four of the Metaphysics, Aristotle offers an elenctic demonstration (apodeixai elenktikōs) of the firmest (bebaiotatē) of all principles—the Principle of Non-Contradiction (PNC). I examine the implications of this elenctic demonstration for Pyrrhonian skepticism as articulated by Sextus Empiricus, who famously characterizes skepticism as “an ability to set out oppositions (dunamis antithetikē) among things that appear and are thought of in any way at all” (PH I.8). This ability is utilized by the skeptic in order to induce equipollence (isostheneia): when faced with opposing but equally weighted appearances or thoughts, the recognition of their equal weight leads to the suspension of judgment (epochē), which results in tranquility (ataraxia). I argue that the ability to ‘set out oppositions’ requires the PNC. I then argue that because the Pyrrhonian cannot set out an opposition to this principle without illicitly presupposing it, they are unable to suspend judgement about it. Part of the novelty of this approach consists in challenging the Pyrrhonian to set out an opposition to the PNC directly, rather than arguing that they are in some more indirect way committed to it.

[4] 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 ‘contingent things’ (1139a5–8). 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 (horos) of right reason (1138b32–34), I argue that the role of the contemplative part of the rational faculty is to grasp stable first-principles, which supply the standard that ultimately explains what makes the reasoning of the calculative part right (orthos) in practical deliberation.

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.