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. (shrink)

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. 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. 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, 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. (shrink)