Seeing Numbers

In his influential book The principles of Psychology (1890) the well-known psychologist and philosopher William James listed seven “elementary mental categories” that he postulated as having a natural origin [Jam07, p. 629]. In an alleged order of genesis he listed, on the third place, together with ideas of time and space, the idea of number. His mentioning the idea of number along with the ideas of time and space as something natural is truly interesting both philosophically and from the viewpoint of cognitive science. In this respect it is worth noting that one of the symptomatic features of the mainstream cognitive science is the tendency not to talk so much about ideas as about their representations, either in the mind or, even better, in the brain. Just notice the maxim by Randy Gallistel in the above motto (quoted in [Cal09]). The hypothesis that the human sense of number and the capacity for arithmetic finds its ultimate roots in a basic cerebral system has been frequently proposed and elaborated, for instance by the neuroscientist Stanislas Dehaene [Deh97, Deh01, Deh02]. In this essay I am going to take up somewhat different perspective. Instead of following up with various extraneous concerns offered by brain research and behavioral and/or developmental psychology I will base my consid-