ABSTRACT
We humans know a great many mathematical truths. But it is difficult to understand and explain how we can know even very basic mathematical truths. Plato presented the challenge to explain human knowledge of numbers, and Paul Benacerraf (1973) reformulated this long-standing challenge in contemporary terms. This chapter aims to present an approach to explaining mathematical knowledge that can meet the challenge. The approach rests on two pillars: (a) the view that natural numbers are numerical properties (the property view) and (b) the thesis that humans have empirical, and in some cases even perceptual, access to numerical attributes (the empirical access thesis). This is essentially the approach that Penelope Maddy (1980, 1990) and Jaegwon Kim (1981) present in response to Benacerraf. To articulate a viable version of the approach, the chapter makes an important modification of their version of the approach. While they hold that natural numbers are properties of sets, the chapter bases the approach on the view that they are plural properties, properties that relate in a sense to many things as such (the plural property view). The chapter articulates this view and discusses recent psychological studies of numerical cognition that support the empirical access thesis.
