The first part of the project investigated the notions of formal and informal proof, and how they are connected to our acceptance of mathematical statements, with a particular focus on arithmetical reasoning. Our everyday use of arithmetic seems to imply that we implicitly believe that arithmetical reasoning will not lead to any contradictions and that every conclusion we obtain by rigorous arithmetical reasoning is true. These properties can be expressed formally in the language of arithmetic through ‘reflection principles’. Such principles, however, do not follow from the axioms of arithmetic, highlighting an asymmetry with our informal reasoning. This project developed an epistemic account of the justification of reflection principles, which dispenses with any appeal to a formal truth theory.
The second part of the project was devoted to the development of a rigorous account of the notion of ‘human-effective computability’: computability by an idealised mathematician that incorporates the non-deterministic character of mathematical reasoning. It was argued that the epistemic version of Church’s Thesis holds for this notion, and the hypotheses advanced were tested in models that embed natural idealisations. This laid the grounds for fruitful applications to Gödel’s Disjunction. This work package led to a promising new line of research on the use of computability-theoretic hierarchies to measure the complexity of mathematical objects and mathematical reasoning. Preliminary results from this research formed the basis of a plenary lecture at the Logic Colloquium 2018.
The third part of the project developed an original version of mathematical naturalism which accounts for parts of mathematics that are far removed from applications in scientific practice and is compatible with an empiricist approach to mathematics. The result of this novel view is a naturalist framework that fits with mathematical practice and is ontologically parsimonious. The project then applied formal tools from reverse mathematics to the analysis of the informational content of mathematical proofs, with the aim of explaining cases in which, after a purely existential result is obtained, mathematicians expend considerable efforts to obtain more ‘informative’ proofs. The major innovation of this approach consisted in applying the distinction between ‘de re’ and ‘de dicto’ predication of properties to the analysis of existential statements in mathematics, to obtain notion of the informational content of a proof that applies to both classical and constructive mathematics.
Invited talks disseminating the research conducted within the project were given at numerous universities and world-leading centres for philosophical and mathematical logic, including the universities of Glasgow, Konstanz, Manchester, Oxford, Lorraine, Salzburg, St. Andrews, Tübingen, and Udine within the EU, as well as Boise State, UC Berkeley, UC Irvine, and the University of Utah. The results of the project are also being disseminated in the form of articles in peer reviewed journals, chapters in collected volumes, and edited volumes.
