Proof and understanding in mathematics (PUMa). Purity of methods, simplicity, and explanation in mathematical reasoning.

As generally acknowledged, the fundamental notion in the methodology of mathematics is that of proof which constitutes a special and evolving cognitive artifact of the human brain. Conventional wisdom has it that any rigorous proof displays a logical dependency among the mathematical concepts involved, in the service of both its own repeatability and the establishment of a mathematical truth. Yet, the cognitive importance of a proof exceeds such a service. An infallible oracle answering "true" or "false" to any mathematical conjecture would be completely useless. What is accomplished by a proof beyond the warrant of a mathematical truth? The basic idea of PUMa project is that an encompassing investigation of the notion of mathematical proof deserves a substantial enlargement in the direction of concepts other than truth, justification, and logical form. The question: “what is a proof?” should be replaced by the more salient but also more neglected question: “what does it mean to understand a proof?”. Proofs, indeed, are central in mathematics as bearers of understanding, which is delivered by factors that are not exhausted by logic, as the logical acceptance of inferences within a proof does not coincide with their strategic acceptance relative to the proof as a whole. In brief, the essence of a proof remains elusive. Understanding proofs requires (at least) the grasping of explanatory relationships in a comprehensive body of mathematical information. The challenge, then, is to unravel the interplay between the explicitness of deduction and the implicitness of cognition, which spans a range of sophisticated automatisms (including visual thinking and the management of implicatures and presuppositions). However, technical contributions from proof-theory – the discipline professionally dealing with proofs as mathematical objects – will also be expected. We will focus specifically on Hilbert’s 24th problem asking for criteria for ‘simplicity’ of a proof which actually mean identity criteria for it. In particular, we shall be concerned with the conceptual interaction between simplicity of proofs and purity of methods (an issue inaugurated by Aristotle’s Posterior Analytics and thematized among others by Newton, Gauss, Bolzano, Frege). We will address these aspects, comparatively, from both the point of view of ordinary mathematical practice and formalized mathematics. Other topics about the epistemic and linguistic dimension of proofs – constrained by matters of epistemology, history of mathematics, philosophy of language, linguistics – will be also considered. These topics include reasoning with generic objects, paradoxes, as well as the semantic nature of proofs playing the role of truthmakers for mathematical propositions. The Research Team will provide innovative theoretical approaches on the PUMa project themes, consolidating and fostering national and international networks of collaborations in the philosophy of logic and mathematics.