Visual Thinking in Mathematics: An Epistemological Study

concepts. . . Inattention is a most efficacious logical faculty; presumably this accounts for the absentmindedness of professors. [5, pp. 84–85] The extent to which Husserl’s early philosophical work is susceptible to Frege’s criticisms, as well as the extent to which these criticisms has substantial influence on Husserl’s later work, is still subject to debate (see, for example, [18]). But Husserl clearly felt the need to respond to Frege’s invective and distinguish his project from brute psychology, and the Prolegomena to the first volume of his Logische Untersuchungen [8] contains long passages renouncing psychologism. Husserl’s later transcendental idealism reinforced the distinction between his philosophical program and psychology by characterizing the former as a determination of the essential capacities of an idealized mind, rather than the determination of our own mental capacities. With Frege, analytic philosophy took a different tack. When one reads his characterization of the foundational project in the Grundlagen, there is a striking absence of explanation as to whence the concept of number that he presents derives its normative force. Rather, the work can be read as an analysis of the concept of number as it is used in our scientific and informal practice, and a clarification of the norms that govern that use. However one interprets the project of the Grundlagen, Frege made it clear that psychology has nothing to do with it: his first fundamental principle is that there “must be a sharp separation of the psychological from the logical, the subjective from the objective” [4, p. 90]. That attitude has held firm, at least in analytic philosophy, to the present day. But a lot has changed since the turn of the twentieth century. Traditional research in the philosophy of mathematics has begun to stagnate, and the various metaphysical and foundational “isms” on offer have by now sprouted so many prefixes, modifiers, riders, and caveats that it is often hard to tell what the discussion is about. Many now feel that if the philosophy of mathematics is to have any bearing on mathematics itself, the subject has to attend to the kinds of value judgments that govern everyday mathematical practice, and provide a more realistic description of the goals and purpose of mathematical activity. In this last respect, the goal of navigating a complex world with limited cognitive resources seems to be a reasonable candidate, in which case the nature of those cognitive resources becomes relevant. Moreover, psychology now seems much better equipped to deliver. Cognitive science offers a robust vocabulary and methodology for analyzing our