Children learn about numbers in the cultural context of a number system as this is conceptualized by adults. For the latter, Rips et al. comment, “Activities such as estimating the number of objects in a collection … may proceed without drawing on natural number concepts. Number concepts may come into play only at a more abstract level – for example, in arithmetic” (sect. 6, para. 3). In brief, there are different logics for counting versus natural numbers, despite overlap in number names. In pre-literate societies, counting number systems are used for enumeration, but seldom for computation (Hallpike Reference Hallpike1979); only with the natural numbers do we have a conceptual structure for computation. Even counting number systems are non-isomorphic across different cultures.

A counting number is based on two primitive concepts: (1) individuation, whereby an object can be distinguished from the collection of objects to which it belongs, and (2) matching, whereby an individuated object in a particular collection can be matched with an individuated object in another collection. The logic of matching determines an equivalence relation – call it “same size” – over an ensemble, E, of object collections as follows. For collections A and B in E, match an individuated object from A with an individuated object from B and then remove these individuated objects from their respective collections. Next, recursively apply this matching procedure to the modified collections until either A or B has no more objects. If both collections are emptied simultaneously, the collections A and B have the same size. Same size exhaustively partitions E into disjoint subensembles, each of which contains all (and only) collections of the same size, since it is an equivalence relation.

Next, identify a fixed reference collection – say, the twiggle collection. Let the counting number TWIGGLE be the name for the subensemble of E (if any) containing a collection of the same size as the twiggle collection. Apply the counting number TWIGGLE to any collection in this subensemble. This labeling procedure is not determined by the collections making up E; hence, we may use the counting number, TWIGGLE, without reference to E.

The Iqwaye of Papua New Guinea (Mimica Reference Mimica1988) used this matching method for a counting number to represent the quantity of males needed to raid a neighboring group. A special string of shells was matched with the collection of males to determine whether they were enough for a raid. The Iqwaye used matching in this context despite having counting numbers for enumeration. Our quantity, dozen, is similarly just a counting number designating a quantity. Its distinction from the natural numbers is shown by the sentence “I want a half dozen cookies” versus the nonsense sentence “I want a half 12 cookies.” The natural number, 12, only provides the reference size for defining the counting number, dozen.

These examples illustrate that sequentiality is not necessary when defining counting numbers. Neither the string of shells nor dozen involves a sequence of counting numbers. Sequentiality derives from the between concept (“Is there a collection with size between the counting numbers X and Y?”). Recursive use of between stops with sequential counting numbers, beginning with the counting number “one,” defined as the name for the size of a collection with a single, individuated object.

The logic of sequentiality leads to finite collections of counting numbers typified by “one, two, three, many” counting systems, where many has the meaning “a collection size larger than any counting number.” Lest we think that counting systems of this kind are impoverished natural number systems, we need only consider the word “infinite” in Alfred Tennyson's Timbuctoo: “Where are the infinite ways.” Here infinity refers to a size beyond the counting numbers, not the cardinality of the natural numbers. Thus, in our “one, two, three … infinity” counting system, common usage of infinity is equivalent to many, and we only have finitely many (named) counting numbers.

Just as the logic underlying a counting number does not entail sequentiality, the logic underlying sequentiality does not entail the incorporation of a successor function, s (n)=n+1, applied invariantly to all counting numbers. Consider the Paiela (Papua New Guinea) counting numbers, which are as follows:

The number names identify sequential body part pairs, with a modifier added to designate when the pair refers to the other side of the body. Thus, “one” is “pair of little fingers” and “twenty-seven” is “pair of little fingers on the other side.” We can interpret the number names of the Paiela as referring to 1st pair, 2nd pair, and so on, up to the 13th pair; that is, they count sequentially by two's rather than by one's. Note that “fourteen” is “nose” even though we have a pair of nostrils, and “twenty-eight” is “clenched hands” even though we have a pair of hands. Their choices for these numbers show that the Paiela counting system is conceptually closed and not the open system entailed by the successor function.

I suggest that children initially learn the primitive concepts for the counting numbers through experience. Alone, this leads neither to a particular counting number system, nor to the natural numbers. For these, the developing child has to work out patterning in number names – and patterning in computations with the natural numbers for cultures in which the logic of the natural numbers is also part of the adult's cultural repertoire – through the child's interaction with culture-bearing adults. A child goes beyond experientially obtained primitive concepts through enculturation, the process by which cultural knowledge (in the sense of systems of ideas and concepts) is transmitted from one generation to the next.