Rips et al. have rightly pointed out a lack of fit between the properties of the natural numbers as defined by axiomatic systems in number theory and the unlearned representations of magnitude in infants. However, to conclude from this that there is “no dependency whatsoever” (sect. 6, para. 2) between them seems premature. Typically, the domains of intuitive knowledge that developmental psychologists have uncovered (such as intuitive psychology or number) are underdetermined. For example, although children are born with cognitive biases that lead them to attend to the actions and goals of others, it remains as yet unclear how they reliably develop a fully-fledged belief-desire psychology that differentiates an agent's mental states from the actual state of the world.
As an alternative to Rips et al.'s and to traditional bootstrapping approaches, it may be fruitful to examine number from the perspective of the epidemiology of representations. Its basic idea is that there is a strong causal link between the mental representations of the individual members of a culture and the public representations they share – in this case natural number concepts and their symbolic denotations (number words, numerical notation systems). To acquire a novel concept, learners partly draw on pre-existing knowledge. Thus, each time a cultural representation is transmitted, it has to pass the bottleneck of the pool of mental representations within the minds of individual learners. Representations with a poor fit to the pool of knowledge are less likely to be understood, and hence transmitted, than those with a good fit. Cognitive biases that are universal in humans likely play an important role in this process. As Nichols (Reference Nichols2002) has demonstrated for etiquette norms, our universal feel of disgust for bodily excretions makes rules that limit contact with them (e.g., prohibitions to spit in public) more attractive than norms that do not stir our evolved emotional responses (e.g., placement of the napkin to the left or right of the plate). Importantly, Nichols (Reference Nichols2002) does not claim that etiquette norms are directly based on or constructed from universal human predispositions. Rather, their good fit with our evolved drive to avoid disgusting situations has promoted their cultural success.
In the case of number, unlearned quantificational skills might similarly constrain and guide the cultural transmission of numerical concepts. If number concepts were based on axiom-like schemas, as Rips et al. suggest, we would expect some cultures to develop nonstandard numbers – which satisfy Peano's axioms in all respects but which we would yet not call numbers; however, apart from Western mathematics, there is no evidence that nonstandard models of arithmetic were ever developed. Unlearned intuitions of number may promote the cultural fitness of some numerical representations in favor of others. Evidence from educational psychology (Vlassis Reference Vlassis2004) suggests that adolescents have difficulties grasping the concept of negative integers: they make more mistakes when solving equations that involve negative terms, and especially those that yield negative solutions. Although adults can compare the magnitudes of pairs of natural numbers quasi-automatically, their performance drops markedly when one or both digits are negative (Fischer Reference Fischer2003). These difficulties are hard to explain from a purely formal point of view, as the negative integers' properties are in many respects similar to the natural numbers', such as closure (i.e., a+b is a natural number/integer for any natural number/integer a and b), commutativity, and associativity. From an ecological point of view, however, conceptualizing negative integers is less relevant for organisms than conceptualizing positive quantities. If our evolved intuitions of number continue to play a role in learning processes, it becomes easier to understand why negative integers were historically less widespread than positive integers. Indeed, negative numbers were actively resisted despite their usefulness in calculations in cultures as disparate as 16th-century Europe, Han-dynasty China, and the medieval Islamic world.
What, then, is the relationship between our innate magnitude representations and natural numbers? One possibility which seems consistent with anthropological data is that although natural numbers are supported by unlearned inductive inferences (De Cruz Reference De Cruz2006), there is considerable cultural variation in the degree to which public representations of number actually support them. For example, humans are equipped with the ability to discriminate between continuous and discrete quantities (Castelli et al. Reference Castelli, Glaser and Butterworth2006). In some cultures (e.g., Western culture), children are confronted with a variety of symbolical representations for number, such as number words, Arabic digits, or even finger counting. These public representations provide external instantiations of the discreteness of natural numbers, leading children to understand that large numbers that are close together are yet distinct. Indeed, Western 5-year-olds who typically only count to 20 infer that numbers above their counting range apply to specific, unique cardinal values: if a set has 61 members, it cannot contain 65 elements (Lipton & Spelke Reference Lipton and Spelke2006). In contrast, in some Amazonian or Australian aboriginal cultures this distinction is not made, leading people growing up in these communities to rely on approximate numerical skills only. Consequently, they cannot discriminate between quantities if the ratio between them is small. While our intuitive quantification skills are not sufficient for natural number concept formation, they do support inductive inferences that promote an understanding of natural numbers in cultures that use symbolic representations that denote exact cardinalities.
