Rips et al.'s extensive analyses adopting a “top-down approach” not only question the bootstrapping hypotheses, but also strengthen the possibilities that some defining properties of natural numbers are innate to human cognitive structure. In the authors' proposal, innate resources – including the ability of recursion – enable the construction of a math schema that embodies the one-to-one successor function, a critical property of natural numbers (sect. 5.2). In an alternative proposal (Leslie et al. Reference Leslie, Gallistel, Gelman, Carruthers, Laurence and Stich2007), the innate basis of natural-number concepts is a mental magnitude system (“the speedometer”), which incorporates a discrete digital representation that is aligned and calibrated to, but distinct in nature from, the continuous analog representation. Current data, however, are too limited to evaluate these proposals. Here I suggest some directions for investigation.

Infant studies still play a critical role. Although mathematics is typically learned late, Rips et al. have argued that to children, math principles are not necessarily consciously available, expressible in language, or acquired all at once (sect. 5.3, para. 1), leaving it open whether some knowledge of such principles is present in infancy. Accordingly, predictions can be developed and tested. However, there are great practical difficulties in testing infants, because the tests must be directly about number concepts, as Rips et al. have emphasized.

Neuropsychological data are also relevant. According to Rips et al., mental magnitudes are too imprecise to be associated with specific natural numbers (sect. 3.1), suggesting that construction of the natural-number schema is relatively independent from the magnitude-representation system. In contrast, the model proposed by Leslie et al. (Reference Leslie, Gallistel, Gelman, Carruthers, Laurence and Stich2007), which incorporates a digital representation as part of the magnitude-representation system, suggests a strong tie between operation of the system and acquisition of natural-number concepts. As such, neural imaging data and specific clinical cases may differentiate the two proposals.

Evolutionary perspectives may offer a plausibility check. Several domains of knowledge are considered to be both essential to survival and ones for which innate mental modules have evolved. For knowledge of natural numbers, Rips et al. have proposed that recursion is a critical underlying capacity, which may have undergone a domain-specific-to-domain-general shift in human evolutionary history (Hauser et al. Reference Hauser, Chomsky and Fitch2002), whereas Leslie et al. (Reference Leslie, Gallistel, Gelman, Carruthers, Laurence and Stich2007) have proposed that a digital-integer representation is a built-in aspect of the evolved magnitude-representation system. These proposals for innate structures that embody properties of natural numbers can be examined with respect to which one could plausibly argue that such structures would have adaptive values.

Lastly, it should be pointed out that some findings seem to be inconsistent with the processes depicted in Rips et al.'s model (Fig. 1 of the target article). These findings deserve consideration in building a more complete model and for examining its implications. The model in Figure 1 is primarily based on infants' processing of visual arrays, such that the attentional mechanisms for parallel individuation function is the starting point, and the outcome, depending on the number of objects (>4 or not), follows separate routes of processing. However, numerous studies have shown that infants can quantify sequences of items such as sounds and actions. For example, 6-month-olds discriminate between sequences at a 1:2 ratio when the numbers of items were≥4 (e.g., 4 vs. 8, 8 vs. 16), and the discrimination precision increases to a 2:3 ratio (e.g., 4 vs. 6, 8 vs. 12) by age 9 months. In sharp contrast, when the number of items was≤4, infants did not discriminate at the ratio that they did when the number was≥4: 6-month-olds failed discrimination at the 1:2 ratio (e.g., 2 vs. 4), and 9-month-olds succeeded at that but failed at the 2:3 ratio (2 vs. 3) (e.g., Lipton & Spelke Reference Lipton and Spelke2003; Reference Lipton and Spelke2004; Wood & Spelke Reference Wood and Spelke2005). Note that earlier studies have shown that 6-month-olds can discriminate between 2- and 3-action sequences (Sharon & Wynn Reference Sharon and Wynn1998; Wynn Reference Wynn1996), although some have argued that non-numerical factors may account for these results (e.g., Wood & Spelke Reference Wood and Spelke2005). In sum, none of these results can be accounted for by parallel individuation processes of attention; such processes cannot be the starting point for the quantitative processing of sequences of items.

In addition, the failure to discriminate between small-number sequences does not provide direct evidence that magnitude representation is not formed of small numerosities. This is because how many items a sequence consists of is not determined until the sequence ends and becomes defined – magnitude representation of a sequence, if formed, must be a result of constant updating from the input of the very first item on. Thus, other factors may be responsible for this finding.

The same possibility applies to infants' processing of visual arrays. The model in Figure 1 implies that for sets containing fewer than 4 objects, magnitude representations of set size are not formed, and thus object representations (object-files) must be responsible for infants' discrimination, as suggested by the findings that 10- and 12-month-old infants succeeded in discriminating among 1 to 3 objects, but failed with comparisons involving 4 objects (e.g., 1 vs. 4) (Feigenson & Carey Reference Feigenson and Carey2005; Feigenson et al. Reference Feigenson, Carey and Hauser2002a). However, a recent study has shown that 5-month-olds can discriminate between arrays of 2 versus 8 and 3 versus 12 objects (Cordes & Brannon Reference Cordes and Brannon2007). This suggests that infants do form magnitude representations of small-number sets, but require a greater ratio of 1:4 (as compared with that of 1:2 for large-number sets) for successful discrimination. It also casts some doubt on the claim that numerical discrimination between small-number arrays is based only on representations of individuals that are subject to a 3-item limit.

The model also implies that for arrays containing 4 or fewer objects, discrete object representations are maintained only if they have distinctive properties, allowing number-based responses; otherwise, infants' responses are driven by magnitude representation of continuous quantities of the array (Feigenson Reference Feigenson2005). However, such a deficit in numerical discrimination of homogenous arrays has not been found in most studies with infants or preschool children, and many studies actually reported a relative advantage for homogeneous sets (for discussion, see Gelman & Gallistel Reference Gelman and Gallistel1978 and Cantlon et al. Reference Cantlon, Fink, Safford and Brannon2007). For instance, Starkey (Reference Starkey1992) found that pre-counting 24- and 30-month-olds performed better on a nonverbal addition-and-subtraction task with homogeneous sets than with heterogeneous sets. Thus, Rips et al.'s model is in need of revision to address these findings.