Rips et al. have written a valuable critique of current theory and research, but in concluding that mental magnitudes form no part of the foundation of natural number concepts, the authors consider only briefly theories positing a partial role for mental magnitudes. These theories deal with children's early counting: Because mastery of verbal counting implies knowledge of critical aspects of natural number, studies of counting acquisition should be useful to researchers interested in hunting down the origins of natural number.

Gelman, Gallistel, and colleagues have argued that “very early in the prolonged process of learning verbal counting children recognize that the structure and function of verbal counting are the same as the structure and function of their pre-verbal counting system. Both processes honor the one-one, stable order, and cardinal principles, and both processes deliver symbols that are subject to arithmetic processing” (Gallistel Reference Gallistel2007, p. 439). They hold that this recognition of common structure and function allows children to link count words to their corresponding mental magnitudes, but that the count words are numerically meaningful prior to the mapping (Gallistel & Gelman Reference Gallistel and Gelman1992).

Other researchers have criticized this account on various grounds. First, a serial enumeration process is often assumed to be critical to the idea that young children (like nonhuman animals) have access to a nonverbal counting process (e.g., Le Corre & Carey Reference Le Corre and Carey2008). There is evidence that representations of the numerosity of spatial arrays are not constructed serially, in adults and in infants (Barth et al. Reference Barth, Kanwisher and Spelke2003; Wood & Spelke Reference Wood and Spelke2005), but as Cordes and Gelman (Reference Cordes, Gelman and Campbell2005) point out, a serial counting process is not required at all. A parallel enumeration mechanism (e.g., Dehaene & Changeux Reference Dehaene and Changeux1993) that adheres to the formal structure of a counting process would do the trick. Of course, a parallel process also makes the relation between verbal and nonverbal counting less apparent – but for Gelman and her colleagues, a parallel process does count.

A potentially more serious problem may arise if analog magnitude representations of numerosity are generated by a process that could not reasonably be called counting. Barth et al. (Reference Barth, Kanwisher and Spelke2003) and Wood and Spelke (Reference Wood and Spelke2005) suggest that the best explanation of experimental results to date may rest in a mechanism that computes numerosity through global non-numerical properties, rather than by enumerating individuals, but their data do not speak directly to this point. This is an important empirical question which remains open; despite the arguments of Rips et al., there is no a priori reason to reject density-based models. The visual system's mechanism for extracting density need bear no resemblance to the common definition of density as number per unit area, or to the authors' description of “visual crowding.”

Would direct evidence against the existence of any nonverbal counting mechanism (serial or parallel) defeat the analog magnitude structure-mapping hypothesis (e.g., Gelman & Lucariello Reference Gelman, Lucariello, Pashler and Gallistel2004)? Clearly the version of the hypothesis that depends specifically upon nonverbal counting would not survive such evidence. Important components of the hypothesis could nevertheless remain: we need not choose between a nonverbal counting system and an unstructured conglomeration of disconnected magnitudes. An analog magnitude system with representations generated by computations over global display properties would possess an implicit ordinal structure, with larger magnitudes systematically corresponding to larger sets in the world. Structure mapping would be a more difficult problem with such a system, but it would still be possible – and children's prolonged learning of these mappings does imply that they are solving a difficult problem (Wynn Reference Wynn1992b).

The nature of the process that generates mental magnitudes may not necessarily decide the fate of the analog mapping hypothesis, broadly conceived. What evidence would argue forcefully against it? What is certainly critical is that children must recognize the structures of the mental magnitude and verbal counting systems and the relation between these structures. Researchers have criticized the hypothesis in part because children do not appear to exhibit an early understanding of the rule that positions later in the verbal count list should correspond to larger magnitudes (Condry & Spelke Reference Condry and Spelke2008; LeCorre & Carey Reference Le Corre and Carey2007; Reference Le Corre and Carey2008; an understanding of this “later/greater” rule is also important for related proposals, e.g., Spelke Reference Spelke, Gentner and Goldin-Meadow2003; Wynn Reference Wynn1992b). These findings would seem to spell trouble for structure-mapping accounts (LeCorre & Carey Reference Le Corre and Carey2008).

However, it is possible that children's failure to demonstrate any later/greater knowledge in the tasks used to date reflects something other than incompetence. These studies have shown that beginning counters do not have an early understanding of the fact that numbers around 10 should map to larger sets than numbers around 5. Does this finding necessarily mean that such children have no understanding of the parallel structures of the counting words and the magnitude system? Possibly not, for a number of reasons – one of which is the relatively compressed numerical range tested. Consider that children might initially map “the next number in the verbal count list” to “the noisy mental magnitude associated with the next easily discriminable numerosity.” If so, the corresponding change in magnitude that initially maps to the next number in the count list may be too great for the 5–10 range to reveal children's later/greater knowledge. Children might possess knowledge of the later/greater rule even while failing utterly to demonstrate that knowledge in the 5–10 range: their extreme underestimation, in combination with noise in their estimates, could conceal all evidence of a later/greater understanding. Somewhat paradoxically, larger sets might be needed to reveal this knowledge. Therefore, additional converging evidence from similar tasks using larger sets would be desirable in order to rule out the analog mapping hypothesis (but see Gallistel Reference Gallistel2007, for discussion as to why such evidence would not be sufficient). Of course, the account lacks the critical notion of a unique next number, but it may be premature to reject the possibility of any role for the analog magnitude system, especially if the idea of a mental unit magnitude is included in such a model (e.g., Leslie et al. Reference Leslie, Gallistel, Gelman, Carruthers, Laurence and Stich2007).