These are interesting times for psychologists who research number development. Ingenious studies are revealing capabilities in infants and nonhuman species that are challenging to describe and explain. Whether and how these studies relate to later development is an important issue.
Rips et al. are not the first to think that the philosophy of mathematics will prove helpful in guiding the psychology of number development. It is true that explaining the nature of mathematical knowledge and how it can be acquired has concerned some philosophers since the beginnings of Western philosophy. Mathematical concepts and relations have been variously portrayed in perspectives such as Platonism, nominalism, intuitionism, constructivism, logicism, set theory, and structuralism. My understanding is that no position has successfully dealt with the issues philosophers feel a philosophy of mathematics should manage. Some problems with using philosophy as a guide are that there are several possibilities, and they point in different directions; and it is unclear whether any have reached the destination of interest. The existence of such continuing disputes suggests that there can be no simple connection between mathematical proficiency and philosophical conceptions of natural number.
Another problem is that philosophical discourse is particularly ill-suited for operationalisation into empirical tests. Any conversation with children about the natural numbers may be fraught with ambiguity: they may be talking about numbers as abstract entities defined only in relation to each other, or they may be tacitly referring to quantities of objects. One can be more confident when discussing the multiplication of negative numbers that reference to the real world has been abandoned, but there are ingenious real-world analogies for this, which would need ruling out.
Whereas philosophers are concerned with explicitly conscious and communicable constructions of number systems, the authors are proposing to use their ideas to characterise the unconscious intuitions of children. This risks treating a conception of natural number as equivalent to a set of components. A set of intuitions is no more an implicit concept of number than a set of components is an implicit computer.
The most serious disadvantages of trying to base the psychology of number development on the philosophy of mathematics derive from their fundamentally different purposes. The philosophy of mathematics is not interested with the development of the individual or even in explaining the history of mathematics.
In contrast, any psychology of number development is bound to be deficient unless it recognizes that children do not develop their ideas about number in a vacuum: They grow up in environments replete with uses of number and numerals. The natural number sequence, also known as the counting numbers, is a cultural product that has developed over a long period and embodies considerable knowledge of number. Some time after acquiring it, children learn the knowledge implicit in it, for example, the relative magnitude of numbers (Siegler & Robinson Reference Siegler, Robinson, Reese and Lipsitt1982) and the number-after rule (Baroody Reference Baroody1995).
It is not just exposure to cultural tools such as the natural number sequence that is responsible for children's development. If it were, then they would be in the same predicament as archaeologists who are still unsure about the mathematical significance of the Ishango bone. Instead of reconstructing number on their own, children benefit from conversations with number system users and activities which may be deliberately or incidentally instructive (Harris & Koenig Reference Harris and Koenig2006). Not enough is known about how others support children in appreciating the properties of cultural tools, but the success of interventions such as Rightstart (Griffin et al. Reference Griffin, Case, Siegler and McGilly1994) demonstrates how effective such support can be. When children progress from conceiving of number as an adjective to thinking of number as a noun (Resnick 1992), it is likely to be through their participation in classroom communities that create mathematics such as those described by Ball and Bass (Reference Ball, Bass, Kilpatrick, Martin and Schifter2003).
