Research

Proof

In philosophy of mathematics, I am interested in proofs, formalisation and mathematical rigour. My 2015 paper, and my PhD thesis, argue that informal proofs can be rigorous independent of their formalisability. My 2016 paper looks at the implications of this for thinking about Gödel’s incompleteness theorems and consistency in informal mathematics. Recently, I have been working on the recipe model of proofs, which frames proofs as giving us instructions for how to carry out a piece of mathematical reasoning.

I am under contract to write a (mini-)book on mathematical rigour for the Cambridge University Press Elements series, for the Philosophy of Mathematics section.

Epistemology

I have an ongoing project applying new tools in epistemology to the philosophy of mathematics:

• My PhD thesis argues that we need to understand the epistemic function of proofs using knowledge-how, as found in the philosophy of Gilbert Ryle. This is developed further in a current draft paper called “Go Forth and Multiply: On Actions, Instructions and Imperatives in Mathematical Proofs”, which is available on request.
• I am also working on the idea that we should analyse mathematical practice in terms of epistemic virtues. This is explored in my joint paper with Ian James Kidd, where we look at individual, collective and institutional epistemic virtues. It is also the focus of the Synthese topical collection I am editing with Andrew Aberdein and Colin Rittberg on “Virtue Theory and Mathematical Practices“.
• Epistemic injustice is injustice along a specifically epistemic dimension. In joint work with Colin Rittberg and Jean Paul Van Bendegem, we apply this to mathematical practices, and more recently we have looked into how these ideas apply mathematics education.
• Social epistemology looks at the social side of knowledge and knowledge-production. Jointly with Joshua Habgood-Coote and Ursula Martin, we have organised a series of three workshops on mathematical collaboration (in Oxford, St Andrews, and Bristol). Josh and I have a paper on the epistemology of the Classification of Finite Simple Groups, forthcoming in Episteme.

Conceptual engineering

I am interested in conceptual engineering: designing, refining and improving our concepts to better fit our purposes. My 2018 paper looks at how mathematics involves extensive conceptual engineering. This relates to my favourite philosophy of mathematics book: Imre Lakatos’s Proofs and Refutations.

I have a paper under review proposing that we need to conceptually engineer the concept of extinction, as the current concept is deficient for holding companies responsible for the widespread destruction of wildlife. The draft is available on request.