Seminário Permanente de Filosofia das Ciências | Representations and Conceptual Understanding in Mathematics: A Case Study | José Ferreirós (Universidade de Sevilha)

A 26 de Janeiro de 2021, às 16h em videoconferência (Zoom), o Professor José Ferreirós, da Universidade de Sevilha, apresenta a comunicação “Representations and Conceptual Understanding in Mathematics: A case study”.

Sumário_ It is well known that visual representations are not allowed in the modern mathematical practice of proof (from 1900 on). This is a strong contrast with ancient mathematics, which was often built around diagrammatic representations. The modern idea of proof is purely logico-linguistic, but several studies have shown that visuo-spatial representations keep playing important roles in the practice of maths. At the same time, the proof-theoretic approach, despite its successes, has been unable to fully justify mathematical knowledge. The question becomes particularly pressing and difficult with respect to analysis and set theory, which make strong existence claims (rejected by constructivist mathematicians). It’s precisely at this point that deep issues concerning ‘conceptual understanding’ emerge: (most) mathematicians are convinced of the acceptability and consistency of analysis and basic set theory, yet their conviction must have a source different from mere deductive justification. We shall analyze in this connection a case study from basic set theory, in an attempt to understand where conceptual understanding comes from.