
629Parts of singletonsJournal of Philosophy 107 (10): 501533. 2010.In Parts of Classes and "Mathematics is Megethology" David Lewis shows how the ideology of set membership can be dispensed with in favor of parthood and plural quantification. Lewis's theory has it that singletons are mereologically simple and leaves the relationship between a thing and its singleton unexplained. We show how, by exploiting Kit Fine's mereology, we can resolve Lewis's mysteries about the singleton relation and vindicate the claim that a thing is a part of its singleton.

133A scientific enterprise?: A critical study of P. Maddy, Second Philosophy: A Naturalistic Method (review)Philosophia Mathematica 17 (2): 247271. 2009.For almost twenty years, Penelope Maddy has been one of the most consistent expositors and advocates of naturalism in philosophy, with a special focus on the philosophy of mathematics, set theory in particular. Over that period, however, the term ‘naturalism’ has come to mean many things. Although some take it to be a rejection of the possibility of a priori knowledge, there are philosophers calling themselves ‘naturalists’ who willingly embrace and practice an a priori methodology, not a whole …Read more

62Zeno’s arrow and the infinitesimal calculusSynthese 192 (5): 13151335. 2015.I offer a novel solution to Zeno’s paradox of The Arrow by introducing nilpotent infinitesimal lengths of time. Nilpotents are nonzero numbers that yield zero when multiplied by themselves a certain number of times. Zeno’s Arrow goes like this: during the present, a flying arrow is moving in virtue of its being in flight. However, if the present is a single point in time, then the arrow is frozen in place during that time. Therefore, the arrow is both moving and at rest. In “Zeno’s Arrow, Divisi…Read more

60Labyrinth of ContinuaPhilosophia Mathematica 26 (1): 139. 2018.This is a survey of the concept of continuity. Efforts to explicate continuity have produced a plurality of philosophical conceptions of continuity that have provably distinct expressions within contemporary mathematics. I claim that there is a divide between the conceptions that treat the whole continuum as prior to its parts, and those conceptions that treat the parts of the continuum as prior to the whole. Along this divide, a tension emerges between those conceptions that favor philosophical…Read more

51Interpreting the Infinitesimal Mathematics of Leibniz and EulerJournal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 48 (2): 195238. 2017.We apply Benacerraf’s distinction between mathematical ontology and mathematical practice to examine contrasting interpretations of infinitesimal mathematics of the seventeenth and eighteenth century, in the work of Bos, Ferraro, Laugwitz, and others. We detect Weierstrass’s ghost behind some of the received historiography on Euler’s infinitesimal mathematics, as when Ferraro proposes to understand Euler in terms of a Weierstrassian notion of limit and Fraser declares classical analysis to be a …Read more

39Quick and Easy Recipes for HypergunkAustralasian Journal of Philosophy 98 (1): 178191. 2020.I argue for the possibility of hypergunk: that is, it is possible that there exists an x such that every part of x has a proper part and, for any set S of parts of x, there is a set S′ of parts of...

23Infinitesimal Comparisons: Homomorphisms between Giordano’s Ring and the Hyperreal FieldNotre Dame Journal of Formal Logic 58 (2): 205214. 2017.The primary purpose of this paper is to analyze the relationship between the familiar nonArchimedean field of hyperreals from Abraham Robinson’s nonstandard analysis and Paolo Giordano’s ring extension of the real numbers containing nilpotents. There is an interesting nontrivial homomorphism from the limited hyperreals into the Giordano ring, whereas the only nontrivial homomorphism from the Giordano ring to the hyperreals is the standard part function, namely, the function that maps a value to…Read more
Ohio State University
PhD, 2012
Areas of Specialization
Science, Logic, and Mathematics 
Metaphysics and Epistemology 
Areas of Interest
Metaphysics and Epistemology 
Science, Logic, and Mathematics 