Publications
"A Quick and Easy Recipe for Hypergunk," Australasian Journal of Philosophy, forthcoming.
"Labyrinth of Continua," Philosophia Mathematica, forthcoming.
"Infinitesimal Comparisons: Homomorphisms between Giordano’s Ring and the Hyperreal Field," Notre Dame Journal of Formal Logic, forthcoming
"Zeno's Arrow and the Infinitesimal Calculus," Synthese, Vol. 192, No. 5, May 2015, doi: 10.1007/s11229-014-0620-1
"Parts of Singletons," with Ben Caplan and Chris Tillman, Journal of Philosophy, Vol. CVII, No. 10, Oct 2010, (click to download)
"A Scientific Enterprise?: Penelope Maddy’s Second Philosophy," with Stewart Shapiro, Philosophia Mathematica, Vol. 17, No. 2, 2009, (click to download)
Presentations
"Locating Success and Failure in Applying Nonstandard Analysis to Euler's Introductio," Joint International Meeting of the American Mathematical Society and the Israeli Mathematical Union, special session on history and philosophy of mathematics, Bar-Ilan University, 2014
"The Labyrinth of the Continuum," Kenyon College, 2013
"A 'Non-standard Analysis' of Euler's Introductio in Analysin Infinitorum,'' Midwestern Philosophy of Mathematics Workshop, Notre Dame, 2012
"Parts of Singletons," with Ben Caplan and Chris Tillman, APA Pacific Division Meeting, 2010
Honors
Early Career Survey Article Competition, Philosophia Mathematica (One of Six Winners), 2013
Dissertation Abstract
Infinitesimals for Metaphysics: Consequences for the Ontologies of Space and Time
In my dissertation, I defend unorthodox conceptions of the continuum as both conceptually viable and philosophically fruitful. In the first chapter, I argue that the orthodox conception of the continuum---which comes to us from Georg Cantor and Richard Dedekind, and which uses the real numbers as a model---doesn't satisfy all of our intuitions about the continuum; indeed, none of the conceptions of the continuum does. This opens up conceptual room for my defense of unorthodox conceptions of the continuum. In the second chapter, I argue that an unorthodox conception of continuum, one that is based on the infinitesimals---numbers that are as small as infinity is large---provides a novel solution to what I call the "problem of contact," which is that pairs of material bodies with certain topological profiles are unable to touch without counterintuitive consequences. My solution accounts for the possibility that material bodies can come into contact regardless of their topological profiles. In the third chapter, I argue that two other unorthodox conceptions of the continuum can provide the basis for a solution to Zeno's paradox of the arrow; specifically, I argue that the present moment is infinitesimally extended in time. Given that the orthodox conception of the continuum conflicts with our intuitions and that the unorthodox conceptions can be usefully deployed in philosophical debates, I conclude that the unorthodox conceptions should be granted equal consideration in solving relevant philosophical puzzles.
Thesis Abstract (for MSc)
Internal Set Theory and Euler's Introductio in Analysin Infinitorum
In Leonhard Euler's seminal work Introductio in Analysin Infinitorum (1748), he readily used infinite numbers and infinitesimals in many of his proofs. We aim to reformulate a group of proofs from the Introductio using concepts and techniques from Abraham Robinson's celebrated Nonstandard Analysis (NSA); in particular, we will use Internal Set Theory, Edward Nelson's distinctive version of NSA. We will specifically examine Euler's proofs of the Euler formula, the Euler product, the Wallis product and the divergence of the harmonic series. All of these results have been proved in subsequent centuries using epsilontic (ε-δ) arguments. In some cases, the epsilontic arguments differ significantly from Euler's original proofs. We will compare and contrast the epsilontic proofs with those we have developed by following Euler more closely through NSA. We claim that NSA possesses the tools to provide appropriate proxies of some--but not all--of the inferential moves found in the Introductio.