As Rips et al. have rightly underlined, humans are not only able to approximate numerosities as many other animal species do, but they also have the specific possibility to develop the concept of natural numbers. For Rips et al., natural numbers correspond to sequences that have a unique first element, a unique immediate successor for each element in the sequence, and a unique immediate predecessor for all but the first element. For them, this type of information would be included in an innate math schema. Yet, this theoretical proposition cannot account for different facts.

First, it cannot account for the fact that the construction of word number systems in all cultures is a very slow process (Hurford Reference Hurford1987) that shows evidences of a progressive elaboration with common stages across cultures. For instance, in most languages, small numerosities are expressed by numerals that do not express, in their morphological structure, a successor function (e.g., two is not expressed as “one unit after one”) and do not correspond to the name of fingers. This suggests that the first numerals have probably not been immediately used as items of an ordered series characterized by a successor principle, but as independent lexical units corresponding to the immediate perception of small numerosities (the subitizing range). The schema proposed by Rips et al. does not predict such a distinction between the first three number words and the other ones. Moreover, when a language possesses a complete system of numerals organized in series, a regular structure is observed: a lexical segment of variable length and then an additive structure that ends at some point at which a multiplicative structure appears. Interestingly, when a multiplicative structure appears, it modifies the way of representing the additive structure just at that point in the series. For example, in many languages, the first way to indicate numerosities beyond the base 10 is the creation of particular words reflecting, in their structure, an additive relation (for instance, “third-teen” in thirteen). But this way of progressing in the structure will be modified just where the multiplication structure appears: after that point the additive structure is created by adding the unit to the decade word (e.g., twenty-two). These irregularities indicate that, in the past, individuals have introduced an additive structure without being aware that many years later, it would be necessary to introduce a multiplicative structure. If we had an innate schema of natural numbers, why would it take so long and be so chaotic to build a well-structured numeral system? Hence, this description of the different numeral systems indicates that the conquest of natural numbers with their complete formal characteristics has been a long and effortful story and not simply the expression of an innate schema.

Second, Rips et al.'s proposition is also challenged by the transversal study of different cultures. Indeed, individuals who are living in societies in which the development of a counting system is still at the initial stages show similar abilities as people in Western and other modern societies in tasks requiring to approximate numerosities, but no signs of a true mastery of the concept of natural numbers (see the studies of the Mundurukù by Pica et al. [Reference Pica, Lemer, Izard and Dehaene2004] and those of Pirahà by Gordon [Reference Gordon2004]). For instance, when asked how many dots are presented, the Mundurukù do not use the few numerals of their lexicon in a counting sequence. Faced with five dots, they produce the word “five” only in 28% of the trials. This same numeral is also used when 6, 7, 8, or 9 dots are presented. Similarly, in simple subtraction, they might select 2 dots as the answer for 6–3 dots. In the same way, the Pirahà can be precise when asked to select the same number of batteries as presented in a model if it involves less than 3 batteries, but beyond that point, their answer is approximate. Accordingly, how can we ascertain whether these people know that each number has only one immediate successor and one immediate predecessor if the label connected to a specific numerosity is variable or if the construction of a collection equivalent to another one is only approximate? In another study of a group from Papua New Guinea, Wassmann and Dasen (Reference Wassmann and Dasen1994) compared the addition skills of the different members of the Yupno communities. The Yupno count neither days, nor people. To them, counting is only meaningful for the exchange of the bride price. The old men know the traditional body-part counting system and can use it for solving simple additions. The children go to math classes, where they learn the Tok Pisin number system (close to English) and use it to calculate. Yet, the young men who have not been to school know neither the Tok Pisin, nor the old body-counting system, and cannot solve any addition. Thus, here again, natural numbers seem only accessible to the people of the community who, given their specific environment, have been exposed to, and can master, a counting system.

Third, if we look at the development of math abilities in children who are raised in cultures in which numbers are meaningful and an elaborate counting system exists, it appears that it takes them time to grasp the meaning of natural numbers. If we all had an innate schema of natural numbers, why does it take so long, even after the learning of the counting string, to grasp the meaning of natural numbers? For Rips et al., this delay is explained by the fact that, in addition to this innate schema, children would need to be exposed to the “key information.” However, what is this key information? Why and how would the exposure to this key information be crucial for the development of that math schema? If the failure to be exposed to this key information explains the very limited number comprehension of Mundurukùs or Pirahàs, what is that key information, if not a number system developed and shared by people in that culture?

In summary, the innate schema of natural numbers postulated by Rips et al. does not account for a large body of historical, cultural, and developmental observations. This model, based on the Dedekind and Peano axioms of natural numbers, provides only a description of the mature understanding of natural numbers; it is not helpful in explaining the phylogenetic and ontogenetic construction of natural numbers. Moreover, from the natural selection viewpoint, the competing advantage of such an innate model is not obvious.