Math Schemata and the Origins of Number Representations.

The contrast Rips et al. draw between "bottom-up" and "top-down" approaches to understanding the origin of the capacity for representing natural number is a false dichotomy. Its plausibility depends upon the sketchiness of the authors' own proposal. At least some of the proposals they characterize as bottom-up are worked-out versions of the very top-down position they advocate. Finally, they deny that the structures that these putative bottom-up proposals consider to be sources of natural number are even precursors of concepts of natural number. This denial depends upon an idiosyncratic, and mistaken, idea of what a precursor is.