My PhD examined the programme of reverse mathematics from a philosophical point of view, with a particular eye towards its relevance for our understanding and evaluation of different foundations for mathematics. As well as continuing to develop this project, I’m now starting to study applications of reverse mathematics within philosophy, as a new tool for philosophers interested in the unexamined mathematical commitments of different philosophical views.
If you’re interested in any of the manuscripts listed below, do .
“Set existence and closure conditions: unravelling the standard view of reverse mathematics”, submitted.
Abstract. It is a striking fact from reverse mathematics that almost all theorems of ordinary mathematics are provably equivalent to just five subsystems of second order arithmetic. The standard view is that the significance of these equivalences lies in demonstrating the set existence principles which are necessary and sufficient to prove a given theorem. In the first part of this article I analyse the concept of a set existence principle and suggest three constraints which any account of this concept should satisfy: non- triviality, comprehensiveness, and unity. I then propose an account of set existence principles according to which they are closure conditions on the powerset of the natural numbers, and argue that it satisfies these three constraints. The final part of the article presents some counterexamples to the standard view, and sketches two ways to resolve them: one which rules them out, and another which accommodates them. I conclude by arguing that the second, more deflationary path is the more promising.
“Computational reverse mathematics and foundational analysis”, submitted.
Abstract. Reverse mathematics studies which natural subsystems of second order arithmetic are equivalent to key theorems of ordinary or non-set-theoretic mathematics. The main philosophical application of reverse mathematics proposed thus far is foundational analysis, which explores the limits of various weak foundations for mathematics in a formally precise manner. Shore (2010, 2013) proposes an alternative framework in which to conduct reverse mathematics, called computational reverse mathematics. Despite some attractive features, computational reverse mathematics is inappropriate for foundational analysis, for two major reasons. Firstly, the computable entailment relation employed in computational reverse mathematics does not preserve justification for all of the relevant foundational theories, particularly a partial realisation of Hilbert’s programme due to Simpson (1988). Secondly, computable entailment is a Π 11-complete relation, and hence employing it commits one to theoretical resources which outstrip those acceptable to the stronger foundational programmes such as predicativism and predicative reductionism.
“Tarski”, forthcoming in A. Malpass and M. Antonutti Marfori (eds.), The History of Philosophical and Formal Logic: From Aristotle to Tarski, Bloomsbury, 2016 (preprint).
Review of Denis R. Hirschfeldt, Slicing the Truth: On the Computability Theoretic and Reverse Mathematical Analysis of Combinatorial Principles, in preparation.