Necessity, Contingency and Counterfactuals in Mathematics

Some facts are contingent: although actually true, they could have been false. Other facts appear to be necessary: their being false is objectively impossible. The distinction has figured prominently in systematic philosophy from its beginnings.

At least some mathematical facts appear to be necessarily true. Three plus three must equal six, for instance, and couldn't have equaled seven instead. Philosophers have long held that, in fact, all mathematical truths are absolutely necessary. This project considers whether and to what extent the received wisdom is correct. In doing so, it aims to clarify the roles of necessity, contingency and counterfactual reasoning in mathematical practice.

The project has three components. The first part, "The Necessity of the Axioms", is concerned with the axioms of Zermelo-Fraenkel set theory with Choice (ZFC), which has long served as the official foundation for mathematics. I argue that the ZFC axioms are not all necessarily true. The second part of the project, "Coincidence, Contingency and Almost False Theorems", deals with the phenomenon of mathematical facts which mathematicians judge to be only barely true. I argue that such theorems are good candidates for contingent mathematical truths, and that this insight has important consequences for our thinking about contingency and explanation. The third part of the project, "Counterfactuals in Mathematical Practice", explores the nature and epistemic goals of counterfactual reasoning in mathematics. I focus on the case of "Siegel zeros" in analytic number theory: objects which are strongly believed not to exist, but which are the subject of extensive theorizing. Together, the parts of the project represent a major challenge to long-held views on the necessity of mathematics.