Philosophy of Mathematics

My research in the philosophy of mathematics focuses on understanding mathematics in a way that agrees with mathematical and scientific practice. I think that the best way to do this is through a combination of structuralism and nominalism. But this approach raises questions that cut across many areas of philosophy, such as the philosophy of science, the philosophy of language, epistemology, and the philosophy of mind. The research questions I focus on include:

 

  • What role does mathematics play in our explanations of empirical phenomena?

  • How should we account for the applicability of mathematics to the empirical world?

  • What, if anything, do mathematical terms refer to, and what makes mathematical truths true? 

  • What is mathematical knowledge knowledge of?

  • What is mathematical abstraction?

Here is a video of a talk I gave on some of these topics at the Munich Center for Mathematical Philosophy

  • 'Non-Eliminative Structuralism, Fregean Abstraction, and Non-Rigid Structures', Erkenntnis, DOI: 10.1007/s10670-018-0096-3, 2018. (Journal - open access)

  • 'The Structuralist Thesis Reconsidered', with Georg Schiemer, The British Journal for the Philosophy of Science, DOI: 10.1093/bjps/axy004, 2018. (Journal - open access)

  • 'Grounding in Mathematical Structuralism', in Bliss, R. and Priest, G. (eds.) Reality and its Structure:  Essays in Fundamentality, Oxford University Press, 217 - 236, 2018. (PDF)

  • Guest column ‘What’s Hot in Mathematical Philosophy’, The Reasoner 12: 22 - 23, 2018. (PDF)

  • 'Scientific Realism without Mathematical Platonism: A Structuralist Approach', under review. 

  • 'Mathematical Structuralism and Metaphysical Dependence', video presentation, LMU Munich, 2015.

Philosophy of Logic

My research in the philosophy of logic focuses on the idea that different formal logics are theories of consequence. Applying tools from the philosophy of science, I specifically focus on the view that formal logics are scientific theories of consequence, and consider questions such as: When are two theories of consequence equivalent to one another? Why choose one logic over another as the (or a) correct theory of consequence? How do the standard theoretical virtues (simplicity, strength, etc.) apply when treating formal logics as theories of consequence? What features should a formal logic have in order to deal with inconsistent information or data? I also have interests in formal theories of truth, an area that serves as a nice case study to examine many of these questions in the philosophy of logic. 

  • 'Individuating Logics: A Category-Theoretic Approach', Thought: A Journal of Philosophy 8: 200 - 208, 2019. (Journal - open access)

  • 'Logical Anti-Exceptionalism and Theoretical Equivalence', Analysis, 77: 759 - 767, 2017. (PDF) (Journal)

  • Review of Ian Rumfitt The Boundary Stones of Thought. Philosophical Quarterly, 67: 219 - 221, 2017. (PDF) (Journal)

  • Review of Penelope Rush (ed.) The Metaphysics of Logic. Philosophy, 90: 710 - 715, 2015. (PDF) (Journal)

  • 'Bayesian Confirmation of Logical Theories', in preparation.

Modal Set Theory

I am interested in whether a modal conception of sets, according to which sets are merely possible with respect to their members, can serve as a satisfactory foundation for mathematics. This research examines the fruitfulness of a modal version of the naive comprehension axiom. Combining this axiom with the axiom of extensionality, one arrives at a modal theory of sets from which one can derive modal versions of the Zermelo-Fraenkel axioms. One can also derive the (possible) existence of enough sets to do mathematics. But most importantly, this can be done without resulting in contradiction. 

  • 'Bi-Modal Naive Set Theory', The Australasian Journal of Logic 15: 139 - 150, 2018. (Journal - open access)

  • 'Set-Theoretic Dependence', The Australasian Journal of Logic 12: 159 - 176, 2015. (Journal - open access)

Dissertation

My dissertation examines claims of dependence in the philosophy of mathematics.  I focus on two areas where such claims are made: set theory and mathematical structuralism.  In set theory, a relation of dependence between a set and its members is often used to justify the iterative conception of set, a conception that gives the universe of sets a hierarchical structure.  Mathematical structuralists often claim that mathematical objects depend on the structures that they belong to, and on other objects in those structures.  I propose and justify necessary and sufficient truth conditions for the dependence claims made in these areas.