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)

This article offers a comprehensive and critical analysis of Eric Heinze’s book Hate Speech and Democratic Citizenship (Oxford University Press, 2016). Heinze’s project is to formulate and defend a more theoretically complex version of the idea (also defended by people like Ronald Dworkin and James Weinstein) that general legal prohibitions on hate speech in public discourse compromises the state’s democratic legitimacy. We offer a detailed synopsis of Heinze’s view, highlighting some of its distinctive qualities and strengths. We then develop (...) a critical response to this view with three main focal points: (1) the characterisation of democratic legitimacy as something distinct from (and whose demands aren’t identical with those of) legitimacy per se; (2) the claim that the requirements of democracy are hypothetical, rather than categorical, imperatives; and relatedly (3) the question of how we should reconcile the requirements of democratic legitimacy with the costs that may follow from prioritising democratic legitimacy. We argue that there are significant difficulties for Heinze’s account on all three fronts. (shrink)