The constructed objectivity of mathematics and the cognitive subject

Giuseppe Longo

pp. 433-463

in: Mioara Mugur Schchter (ed), Quantum mechanics, mathematics, cognition and action, Berlin, Springer, 2002

Abstrakt

Mathematics is engendered in conjunction with other forms of knowledge, physics in particular. It is a "genealogy of concepts" (Riemann), that stems from our active reconstruction of the world. Mathematics organizes space and time. It stabilizes notions and concepts as no other language, while isolating by them a few intelligible fragments of "reality" at the phenomenal level. Thus an epistemological analysis of mathematics is proposed, as a foundation that departs from and complements the logico-formal approaches: Mathematics is grounded in a formation of sense, of a congnitive and historical nature, which preceeds the explicit formulation of axioms and rules. The genesis of some conceptual invariants will be sketched (numbers, continua, infinity, proofs, etc.). From these, categories as structural invariants (objects) and "invariant preserving maps" (morphisms, functors) are derived, in a reflective equilibrium of theories that parallels our endeavour to gain knowledge of the physical world.