As adults, we treat the concept of cardinal number as an endogenous support for children's engagement with mathematical reasoning. This is fine, as long we can then exogenously disengage children from treating number words solely as the answer to the question “How many?” on the grounds that reasoning about integers cannot be reduced to operations on set sizes (Haylock & Cockburn Reference Haylock and Cockburn2003). We read Rips et al. as endorsing a notion of disengagement of some part of the child's representational activity from physical numerousness as a prerequisite to representing natural number within an abstract generative system that reaches up to infinity. However, Rips et al.'s speculative proposal amounts to a non-engagement mechanism being endogenously provided. The appeal for an innate specification of natural number seems surprisingly premature given Rips et al.'s insistence that an account of number concepts must be developmental, but where the developmental inputs are yet to be specified.

We cannot see that Rips et al.'s radically innate proposal generates testable hypotheses, though we hope that it will, in the interests of adversarial testing. Still, Rips et al. assume that the two ways of looking through the lens – through a philosophical telescope towards infinity or through a developmental microscope at the earliest cognitive building blocks – have been resolved from the perspective of mathematics. We, however, would prefer to hold off, pending an evaluation of an orderly set of modelings that vary in the richness of initial-state representations (as propounded by Margolis & Lawrence Reference Margolis and Laurence2008) and of subsequently reiterated representational operations on them (as propounded by Karmiloff-Smith Reference Karmiloff-Smith1992). If magnitude projection entails the extension of a number concept onto external items (e.g., the fingers on my hand) to yield the cardinal “five,” then the proposed disengagement looks like yielding an analysis of the abstract structure that got me to “five.”

On the other hand, there is every reason to welcome Rips et al.'s focus on reasoning with – and about – numerical concepts. Disengaging mental representations of number from the real-world objects that provide numerical input is a start, but one must also be in a position to operate over such representations. Consider the following test concerning the successor function. If a child watches a puppet miscounting five items, “1, 2, 3, 4, 6,” and if he or she knows that (a) the puppet thinks there are 6, and that (b) there are really 5, the child is starting to disengage one of the numerical representations of this event from the veridical how-many representation. A child who answers correctly multiple variations of these questions (i.e., where the miscounts and the discrepancy between factual and counterfactual representations of cardinality vary), conceivably appreciates the precise way in which the errors violate the unique successor function (Freeman et al. Reference Freeman, Antonucci and Lewis2000).

One naturally wants to look for symptoms of children spontaneously disengaging themselves from the numerical representation of plurality of objects, towards a concept of natural number as specified by Rips et al., and there is certainly a paucity of evidence within psychology. As Rips et al. astutely point out, counting ability is not a good predictor of this. We can add that neither is it likely to be (Muldoon et al. Reference Muldoon, Lewis and Freeman2003). But insight does appear to be associated with an emerging ability to reflect on how the accuracy of the counting procedure both identifies ordinal relations and determines the validity of the cardinal representation of the set (Muldoon et al. Reference Muldoon, Lewis and Freeman2003). It is from this perspective that we start to get a picture of how discourse provides domain general endogenous input for the mental activity necessary to “disengage.”

Age-appropriate tests can also be amended to link in with set theory's insights into the relational nature of sets. A weakness in Rips et al.'s position is that they appear to be concerned primarily with the structure of a single infinite sequence. But number has a relational aspect to sequence that is equally important, and an appreciation of this rests on the criterion that is rightly identified as second-order mathematical induction. Preschoolers' engagement with the logical necessity of number is evident when they share objects out using a “one-for-you, one-for-me” procedure to produce sets in 1-1 correspondence and show some understanding that by counting one set, the number of objects in the other set can be inferred. Crucially, children as young as 4 years are sensitive to (1) the fact that numerical comparisons between two or more counted sets rest on adherence to the rules governing within-set enumeration (i.e., a unique first term and a unique successor); and (2) how the logic underpinning the ordinal relations between sequential count terms is tied to the concept of object-to-object correspondence (Muldoon et al. Reference Muldoon, Lewis and Towse2005; Reference Muldoon, Lewis and Francis2007). As with their appreciation of the successor function, this is not associated with procedural mastery of counting or sharing – although these are probably necessary skills – but with the ability to reason appropriately about numerically relevant versus irrelevant inputs.

In the final analysis, the presence of any predictors of number development begs the question of whether prompting children to reason about numerical data feeds into, or draws from, a schema for recursion and its extension to problems of numerical relationships. This is the key question that needs answering. There is evidence that such reasoning does not emerge spontaneously (Muldoon et al. Reference Muldoon, Lewis and Francis2007), and the processes of learning – particularly in social settings – appear to us equally valid candidates for schema input as the innate representations that Rips et al. propose. Just because psychology has not revealed the origins of the natural number concept in its work to date, does not mean yet that we have to abandon a bottom-up approach in favor of a top-down one. We do not have to turn the telescope round and look through the smaller end towards infinity; we need to train the developmental microscope on the right target and carefully adjust the focus.