The target article provides an important discussion of the potential pitfalls in taking experimentally obtained behaviors and making general inferences about numerical cognition in infants and children. All too often, researchers assume that the practices of counting and differentiation evidenced in experiments directly reflect an underlying cognitive system that is a prototype of mature adult numerical cognition. One cannot assume that the internal systemic properties of number are bootstrapped from experience with real objects. Could it be the case that concepts of number are acquired by pure rationalism in the unconscious mind as children attempt to make sense of the number system and its properties?

The question we must ask is whether the abstract formal properties of number theory are a necessary part of the representation of number, as opposed to being part of a meta-analysis of the number system that is taught in schools. It is one thing for children to behave as if they had a principle of associativity or commutativity; it is another to explicitly represent such a principle in a formal logical mental theory-structure. If the number principles are learned in a natural way or are innate, then where are the boundaries of complexity? Must we require induction of the commutative principle, but not Goedel's theorem? Perhaps the boundaries lie just at the level of figuring out what changes and does not change upon minor rearrangements of configuration.

To reject the idea that number principles are arrived at through mathematical induction, we need an account of how number properties can be acquired without the formal machinery. Notice that we are not talking about bootstrapping as a stepping-stone to a formal representation, but about having the right truth-value structure in numerical reasoning without needing the formal inductive mechanisms described in the target article.

Let us consider whether the formal machinery gets us to the right place in accounting for human numerical cognition. We are told that it follows from the successor function that there is no final number, since the set of all numbers must be infinite. But this is just not true for all cultures. We are now familiar with cultures that have very few numbers, such as the Pirahã (Everett Reference Everett2005; Gordon Reference Gordon2004) and Munduruku (Pica et al. Reference Pica, Lemer, Izard and Dehaene2004), who clearly lack knowledge of numerical principles. Some cultures count by body parts, such as the Yupno of Papua New Guinea, who finish counting at the name of the right testicle (Menninger Reference Menninger1898/1969). More intriguing are the Polynesian cultures whose original language had a counting system that went to a million, but stopped dead there (Beller & Bender Reference Beller and Bender2008). The correct answer to the question: “Is there a largest number?” in these cultures is “Yes.” Surely such cultures had something like a successor function, but if such a function guarantees discrete infinity of the number system, then how did these Polynesians end up with a finite counting system? We must take into account the linguistic basis of counting and how that directly affects the nature of mathematical principles that apply, modulo the culture. Therefore, if a counting system is generative, but not fully recursive, you end up with a number set that is very large but finite. In the same way, if a natural language has a generative grammar that lacks recursion, as might be the case for Pirahã (Everett Reference Everett2005) and possibly Hixkarayana (Derbyshire Reference Derbyshire1979; Geoff Pullum, personal communication), then the number of sentences generated by the language is also very large, but finite.

How, then, might the child figure out that associativity and commutativity hold, absent any formal instruction or complex machinery of mathematical induction? Perhaps if we step back a bit, we can see that these principles are quite similar to the principle of conservation of number (Piaget Reference Piaget1952). The principle of conservation tells us that quantities do not change as a result of irrelevant transformations. The commutative and associative principles for addition tell us that adding two quantities, or numbers, does not change their sum under irrelevant transformation (switching the addends or regrouping). In Carey's (Reference Carey, Inhelder, Caprona and Cornu-Wells1987) reanalysis of conservation phenomena, she points out that conservation does not follow from logical necessity, as Piaget had proposed, and so learning conservation could not therefore follow from logical analysis of the domain. Instead, conservation is an empirical fact about the world and turns crucially on the notion of “relevance” of the transformation. A favorite teaching trick of mine after a class on conservation, is to ask students whether the area bounded by a piece of string is changed upon altering the shape of the string slightly. Most students will say “no” and, of course, will be shown to be wrong as the string is flattened into a shape with bounded area of zero. The point is made more forcefully if you then invite a 3-year-old into the room and they get the correct answer!

If conservation is learned empirically rather than logically, then figuring out that substances or individuals do not change quantity upon being poured into a different container or spatially rearranged, would involve learning about the nature of matter and objects in the physical world. It isn't so much experience with objects in counting situations that bootstraps these number principles, nor that these principles are arrived at autonomously within the unconscious theorem-generating mental number space, but that the indirect process of trying to figure out the nature of objects as objects and individuals within sets might be the catalyst for arriving at these principles in the relatively informal manner that we also acquire knowledge of other domains of folk-science. The inference that these principles apply universally to all quantities or numbers, might be understood at the folk-math level in the same way that we have faith that the sun will rise tomorrow, by everyday Hume-style, deductively invalid, inductive generalization. The formal use of successor functions and mathematical induction/deduction might only come into play when children are taught explicitly in school.