Rips et al. paint a picture of math knowledge as divorced from the object tracking and simple enumeration abilities of young children. Something is right here, and something is wrong. What is right is that arithmetic is a domain of relations across numbers, numbers (at least on many stories) are abstract objects, and the truths of mathematics are not dependent on anything in the world, like objects or approximation abilities. But what is wrong with this picture is that neither humans over the course of history nor children over the course of development initially understand mathematics as a domain concerning abstract entities divorced from the world. The ability to describe relationships across real-world collections of objects was presumably the historical impetus for discovering arithmetic, it is how children begin engaging in mathematics, and for most humans, it remains the major forum for applying arithmetic knowledge. A fortiori, a bridging mechanism between mathematical truths about numbers (i.e., abstract objects) and collections (i.e., things in the world) is required if such knowledge is to be made relevant to real-world calculation.
We suggest that the concept set and ordered relations across sets can serve as a bridging mechanism (another possibility is plural variables). Rips and colleagues do not list set representations among the early capacities potentially relevant for math knowledge. We suggest that the concept set is required and that this notion cannot come from object tracking, the approximate number system, or language.
Conceiving of a set requires representing the hierarchical relationship between individual items and the larger structure into which they are bound (note that this is distinct from perceptual grouping or recoding, both processes that destroy the individuals comprising the group). Whereas individuals exist in the world, sets are abstractions that exist in the mind; it takes a mind to maintain multiple hierarchically ordered construals.
The object tracking system (and working memory) will not provide the concept set. This system will not return a representation of one-ness, two-ness, or three-ness when it is engaged in tracking one, two, or three objects. This is because the objects are completely separate individuals in the world; maintaining this separateness was one of the main motivators for positing a tracking system in the first place.
Nor will the approximate number system (ANS), which produces analog magnitude representations, provide the concept set. The representation of 1 in the ANS is a continuous distribution of activation (e.g., Gallistel and Gelman [Reference Gallistel and Gelman2000] describe this system as instantiating the real numbers, not the integers). Also, the representation for 8 in the ANS (the 8-distribution) is in no way made up of 8 individual distributions of 1. Further, on some models, the entities (in the world) that enter into this representation are not even individuated (in the mind) prior to creating this estimate (rather, they are Gaussian activations passed through a continuous normalizing filter; Dehaene & Changeux Reference Dehaene and Changeux1993). With no discrete representation of 1, no compositionality, and, on some models, no individuation prior to enumeration, the ANS cannot provide the concept set.
Nor can language do the job. “set” is a notion with content. The syntax provides algorithms for manipulating content. Although the language faculty might provide relevant computational possibilities, including embedding, agreement, and recursion, these are all content-free syntactic operations. One must maintain a distinction between the computation that binds any two individuals into a set and the concept of a set of two individuals. The former might reside in the language faculty, but not the latter, and it is the latter that is needed to bridge math to the world.
What then is needed to connect set with number? If we begin with object tracking, which will provide us with representations of discrete and separate individuals, one might take these separate individuals (e.g., two separate object files) and bind them into a set of individuals. Rather than tracking, for example, Joe and Jenny, one could then index the set of Joe and Jenny. Our recent work suggests that 14-month-old infants make this distinction between tracking individuals and tracking sets of individuals (Feigenson & Halberda Reference Feigenson and Halberda2004; in press). Importantly, this work provides evidence that infants maintain representations of the individuals comprising the set, thereby distinguishing set representations from demonstrations of perceptual grouping and tracking (e.g., Wynn et al. Reference Wynn, Bloom and Chiang2002), in which there is no evidence that infants retain representations of a group's components.
Next, the learner must abstract away from any particular instance of tracking, for example, a set of two particular individuals, to represent the more general case of tracking a set of any two individuals. Presently, we know of no evidence that infants construct this more general representation, but we see no computational limitation that makes this abstraction impossible. Last, the learner must order set representations with, for example, 1, 2, and 3 individuals according to the successor function. With this ordering, set representations become bona fide representations of one-ness, two-ness, and three-ness (modulo Rips et al.'s comments concerning mod10). This ordering appears to occur in the course of learning to enumerate collections.
The set-binding computation we discuss may require that items be represented in parallel prior to binding. Limits on parallel individuation or on working memory would therefore limit these set representations to smaller numbers of items. Our guess is that precise large numbers will be known only through a meta-language, or schemas along the lines Rips et al. suggest. Large numbers may derive their semantic content via the functional role they play in relations with other numbers, but ultimately they make contact with the smaller cardinalities, for which set representations are also available.
We suggest that the concept set will be a requirement for any theorist who tries to build an acquisition story from early enumeration abilities to natural number. Furthermore, any story that divorces natural number from these early abilities will require the concept set (or some other bridge), applicable to real-world collections, in order to connect purely mathematical knowledge to the real world.
