Un dialogo tra matematica e filosofia

Abstract

A Dialogue between Mathematics and Philosophy

The Editorial Board of Nóema invited three contemporary mathematicians, well-known – not only in the specialist field – who have always been particularly attentive to the dialogue with philosophy, to a reflection. In these pages, Giuseppe Longo (ENS, Paris), Alessandro Sarti (EHESS, Paris), and Fernando Zalamea (University of Bogotá) answer some questions posed by the members of the journal, conversing with each other and clarifying how, in their view, one should consider research that sheds light on the genesis of mathematical meaning, taking into account the suggestions coming from some philosophers and the operations conducted with a synthetic, rather than an analytical and objectivist, outlook.

In the first contribution, Giuseppe Longo answers questions about mathematics and its meaning, and touches upon how it reshapes the world in its own way, in fact, in different ways, following different histories. The invention of concepts and structures is at the heart of this: the audacity, for example, of proposing “contours” and “edges” of the world that do not exist, but which cut it out and qualify it with great effectiveness and conceptual stability. In physics, in our cultures, mathematics has proposed a very solid theological framework that is still relevant today, but certainly inadequate for talking about living beings. Longo mentions attempts to overcome these metaphysical schemes and change perspective.

In the second contribution, Alessandro Sarti clarifies that mathematics not only establishes rules and invariants, but also opens up new fields of inquiry and new possibilities. Is it possible to conceive of a heterogeneous differential becoming in space and time, where causality and composition are not in contradiction? That is, the dynamics that Deleuze and Guattari called heterogenesis, in which spaces of possibilities are not invariant but part of the process.  Here, Sarti proposes a journey through differential dynamics ranging from physics to dynamic structuralism to heterogenesis as an imaginative materialism. These “mathématiques de la pensée” are closely intertwined with the “pensée des mathématiques” by Giuseppe Longo and Fernando Zalamea.

In the third contribution, Fernando Zalamea illustrates how mathematics is a complex science that integrates 1) imagination (abduction in Peirce’s terms), 2) reason (deduction), and 3) adequacy (induction). Mathematicians redesign the world in a constant oscillation between Peirce’s 1-2-3, applied to mathematical structures. A rigid mathematics, which seeks only its own foundations, has a very narrow meaning, valuable for studying fragments of classical logic, set theory, and elementary numbers with extreme care, but useless for observing true mathematics in action (advanced number theory, abstract algebra, topology, complex variables, algebraic geometry, differential geometry, functional analysis). This is why other perspectives are fundamental, such as non-classical logics, category theory, and sheaf theory, which serve as the basis for non-restrictive and non-analytical philosophical views.