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 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 ‘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 Charmides, Socrates examines the proposal that moderation is self-knowledge, a proposal then developed into the claim that moderation is knowledge-of-knowledge: the only knowledge both of other kinds of knowledge and of itself (166c). So understood, knowledge-of-knowledge is both second-order and reflexive. Socrates raises two challenges: whether such knowledge is possible and whether it can be beneficial while remaining second-order. I argue that these challenges identify constraints on any adequate account of knowledge-of-knowledge and that the account of dialectic developed in the Republic and the Sophist can satisfy them: the Republic supplies resources for meeting the benefit challenge, while the Sophist supplies resources for meeting the possibility challenge. My reading also clarifies the sense in which moderation is self-knowledge: dialectic is the reflexive and beneficial knowledge, while moderation is the condition of the soul insofar as it possesses that knowledge and thereby knows itself.

[2] In Plato’s Laws, an unnamed Athenian presents self-mastery as the victory of reason over the opposing pulls of pleasure and pain in what is commonly called the ‘puppets passage’ (644d–645b). Although he calls this passage both a tale and an articulation of virtue (645b–c), he elsewhere characterizes virtue as the harmony of reason, pleasure, and pain (653b). Does virtue consist in victory or harmony? Many interpretations privilege conflict, explain it away, or treat it as merely developmental. I argue that the dilemma dissolves once victory and harmony are understood as characterizing different grades of virtue—a hierarchy illuminated by the contrast between popular and exalted moderation (710a). When popular moderation is reliably guided by reason or law, it is dispositional self-mastery: the lowest grade of virtue, in which non-rational pulls remain opposed to reason. When habituated pleasures and pains harmonize with stable true belief, popular moderation develops into an intermediate grade plausibly possessed by guardians with true belief. Exalted moderation is wisdom-guided harmony, the highest grade, possessed by guardians with wisdom. This graded account preserves the puppets passage as a genuine articulation of virtue while reconciling it with the account of virtue as harmony.

[3] Plato twice connects virtue with what is commonly called the hedonic calculus: the art of measurement as applied to pleasures and pains. In the Protagoras, Socrates presents such measurement as the knowledge on which virtue depends, describing it as a comparative exchange aimed at obtaining the greatest net balance of pleasure (356b–357b). In the Phaedo, however, he maintains that such exchanges, even if successful, are insufficient for virtue (69a–b). I resolve this apparent conflict using the distinction between two kinds of measurement found in the Statesman: comparative and normative (283c–284e). The comparative kind measures things in relation to one another (pros allēla), while the normative kind measures them in relation to what is due or appropriate (pros to metrion). I argue that the assumption in the Protagoras that pleasure is the good makes the greatest net balance of pleasure the due measure, thereby reducing normative measure to comparative measure. No such assumption is present in the Phaedo, where comparative exchanges are assessed as correct or incorrect in relation to wisdom. The Phaedo therefore rejects the sufficiency of comparative measure for virtue—not the hedonic calculus as such.

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.