On the contribution of cognitive science to the philosophy of mathematics

A number of recent developments of cognitive sciences are concerned with some aspects of our cognitive activity that seem to be connected with our mathematical competences. Are these developments able to contribute to the solution of some classical problems in the philosophy of mathematics? The paper critically examines certain positive answers, tries to compare them with older attempts to connect psychological researches and philosophical analysis about mathematics, and endeavors to establish the framework of a possible collaboration between these disciplines. Such a collaboration concerns the study of the cognitive conditions of constitution of mathematical theories understood as being different from ordinary mathematical skills. Dehaene’s thesis asserting the existence of an innate capacity to perceive and represent numerical quantities is firstly considered, by showing that it is different from the capacity of enumeration. G. Lakoff’s et R. E. Núñez’s attempt to reduce the constitution of mathematical theories to the establishment of a “metaphori­cal mapping” is then studied, by showing that it consists of a conceptual analysis of already constituted mathematical theories to which some pretended empirical models are simply associated. These attempts are compared with Piaget’s. Finally H. Poincaré’s and J. Nicod’s reconstructions of the cognitive origins of geometry and Maddy’s thesis about the cognitive origins of set theory are considered. These analy­ses suggest that the contribution of cognitive sciences to the philosophy of mathematics would be more valuable if it was able to account for the specific features of particular mathematical theories and to cooperate with other researches concerned with the logical and historical aspects of their constitution.