There is a widespread belief that by studying the basis of the “counting” numbers we learn about mathematics itself (Butterworth 1999; Dehaene Reference Dehaene2002). Many experimentalists in child psychology and number neuroscience think that the concept of natural number is bootstrapped from early quantitative skills such as estimating magnitudes and enumerating. These basic skills, however, cannot be directly extended to provide the richness, precision, and sophistication of the concept of natural number, let alone that of more complex mathematical concepts. In their target article, Rips et al. lucidly explain this point and conclude that children construct the concept of natural number and arithmetic relying on top-down processes and by constructing “mathematical schemas.” I agree. However, between facts and conclusion–as in good panini–the interesting stuff is in the middle: Where do these schemas come from?

A major problem in explaining the “acquisition” of the concept of number is that scholars often introduce crucial elements of the explanans in the explanandum (e.g., taking number systems as pregiven). Gallistel et al. (Reference Gallistel, Gelman, Cordes, Levinson and Jaisson2006), for instance, speak of “mental magnitudes” referring to a “real number system in the brain” (p. 247); the very real numbers are taken for granted. The system of (infinitely precise) real numbers is an extremely sophisticated concept, shaped over centuries with technical notions such as completed order field and the least upper bound axiom. How could such a system be simply “in the brain”? For the purposes of a biological brain dealing with magnitudes in the real world, the dense ordered field of rational numbers–with infinitely many rationals between any two rationals–would suffice. But, again, rational numbers cannot be taken for granted, either. The point is that in explaining the acquisition of the concept of number, not even the natural numbers and their properties can be taken for granted. Rips et al. are aware of this, and go on to propose a top-down approach based on mathematical schemas. Their characterization of schema, however, is quite abstract and generic, leaving unanswered questions such as: (1) What known cognitive mechanisms make such schematic abstractions possible? (2) What is the biological plausibility of these schemas? (3) Can top-down constraints established beyond the child's mind shape the consolidation of such schemas? If yes, how?

Embodied cognition and cognitive semantics have been concerned with questions of precisely this kind. Regarding mathematical concepts, George Lakoff and I have suggested that mathematical abstractions and idealizations such as Rips et al.'s mathematical schemas can be investigated through everyday cognitive mechanisms for human imagination such as conceptual metaphor and blending (Fauconnier & Turner Reference Fauconnier and Turner2002), among others (Lakoff & Núñez Reference Lakoff and Núñez2000). Not only do these mechanisms sustain human imagination (Q. 1) (Lakoff Reference Lakoff and Ortony1993; Núñez Reference Núñez and Hersh2006), but they are also biologically plausible (Q. 2) (Gallese & Lakoff Reference Gallese and Lakoff2005; Núñez et al. Reference Núñez, Huang and Sereno2007). Moreover, they can be culturally shaped (Núñez & Sweetser Reference Núñez and Sweetser2006) with their inferential organization established beyond the individual proper (Q. 3). Rips et al. are right that the concept of natural number cannot be, via empirical induction alone, a mere extension of everyday object manipulation. However, their dismissal of conceptual mapping theory is based on an incomplete and superficial understanding of it. We suggest that the laws of arithmetic for natural numbers (e.g., additive commutativity) come not just from one empirical mapping involving object manipulation (i.e., Arithmetic is Object Collection, as in gathering two and three balls to get five), but rather, from the isomorphic structure of four different source domains of primary experiences, one of which does not deal with objects but with motion along a path (as in taking two steps and then three to land five steps away; Lakoff & Núñez Reference Lakoff and Núñez2000, pp. 71–80). Such isomorphism provides structural correspondences across the source domains of four different grounding metaphors, yielding equivalent numerical results. This understanding is not about a mere phrasing of operations on physical objects, it is about the abstracted numbers. Moreover, from a neural perspective, the conflation of these isomorphic primary experiences presumably involves coactivations of brain areas that sustain those experiences, resulting in relevant neural links. Finally, this approach explains the precise and ubiquitous nature (even in technical domains) of numerical metaphorical expressions such as “greater” than, “smaller” than, “between” π and -π, and so on. Rips et al.'s mathematical schema does not.

Regarding conceptual mappings, Rips et al. rightly ask why selected inferences are more convincing than potential competing ones (i.e., the “No-competing-inference test for psychological explanations”; see sect. 3). Arguing that our theory fails this test, they miss that this concern is indeed central in conceptual mapping theory; and in mathematics, it is this very issue that shows how the field is largely driven by top-down processes. Consider an essential property of natural, rational, and real numbers: order. Then, like Tartaglia in sixteenth-century Italy, you bump into √-1, which, being a nonzero entity, is neither greater nor smaller than 0. √-1 simply fails the usual order test. Two competing inferences stand out: √-1 is not a number, because it violates order, or it is a number, but “number” must be redefined by giving up order. Both inferences are perfectly viable, but it is the latter one that over centuries the mathematical community has sanctioned as the more desirable one. Clashing inferences leading to competing bodies of knowledge permeates the history of mathematics, from post-Pythagorean mathematics with irrational numbers (Lakoff & Núñez Reference Lakoff and Núñez2000, p. 71) to Georg Cantor (but not Galileo) creating transfinite numbers (Núñez, in press). Often the mathematical community ends up adopting one inferential avenue and burying the other one, but this is not always the case. Competing-inference cases can also coexist. There are internally consistent but mutually inconsistent set theories, including one, for instance, that rules out self-membership (axiom of foundation) and another–hypersets, which allows self-membership (anti-foundation axiom; Lakoff & Núñez Reference Lakoff and Núñez2000; Núñez Reference Núñez, Calvo and Gomila2008).

Looking at how children “acquire” the concept of integers (natural numbers including zero and their negatives) with addition and multiplication may be informative here. The usual “negative times negative yields positive” rule cannot be derived either from early number skills or from direct mappings of physical experience. It is a matter of top-down mechanisms that impose precise, previously sanctioned inferential structure (which usually is taught dogmatically). Rarely explained in textbooks, this rule is a convention sanctioned by (adult) mathematicians–not children–who saw the fruitfulness of extending the distributive property of multiplication over addition to negative numbers. For the developing child who had no say in the sanctioning, the rule seems arbitrary. Similar, although more subtle, top-down dynamics must be accounted for in explaining the child's “acquisition” of the concept of natural number.