If Rips et al.'s target article is right, children's understanding of natural number cannot be derived from numerical abilities that they displayed as infants. Two crucial elements that are missing in the article are a basis for the concept of successor and a schema for the starting value of one. I suggest here that ideas from the field of memory may help to fill in these gaps. Specifically, I propose that everyone with a functioning episodic memory possesses a pre-symbolic analogue of counting, which could mediate the acquisition of number concepts.

To start in the memory laboratory, consider an experiment in which the items to be enumerated are not simultaneously displayed in front of the subject, but are dispersed over time. A standard experiment might use 50 different items (e.g., words or pictures), presented in a list totaling 150 study trials. Different stimuli are presented different numbers of times, haphazardly interleaved with presentations of the other items. After the list has ended, subjects are shown each item and asked how many times it appeared. People are surprisingly good at making such frequency judgments. Telling subjects to attend to presentation frequency ahead of time has little or no effect on performance, and 4-year-olds judge frequency about as well as adults (Hasher & Chromiak Reference Hasher and Chromiak1977). “Memory strength” does not appear to be the basis of this ability, because one cannot trick subjects into saying that a once-occurring item appeared two times merely by presenting it recently (Wells Reference Wells1974) or for a long study duration (Hintzman Reference Hintzman2004). It is not plausible that subjects are counting, in the strict sense of the term, because this would require maintaining running totals for many different items, and the usual study instructions give subjects no reason to count.

How can we understand such findings? Hasher and Zacks (Reference Hasher and Zacks1979) proposed that humans are specially prepared by evolution to acquire frequency information. A different hypothesis (Hintzman Reference Hintzman2004) is that the ability is not innate, as such, but derives directly from the operation of more basic memory processes, which yield recursive representations of repeated experience.

I assume that everyone with a functioning episodic memory has three basic abilities: (1) to maintain a complex mental state, (2) to encode a representation of that state into memory, and (3) to retrieve the representation reflexively, when given a matching retrieval cue. I also assume that the mental state that results from retrieval has two parts: one that refers to the present moment (i.e., the encounter with the cue) and one that refers to the past (the reminding of an earlier experience). This distinction between present and past is a fundamental aspect of the rememberer's subjective experience. Schematically, if A represents the subject's mental state upon first encountering a particular stimulus, then the mental state upon encountering the stimulus for the second time is A+R(A).

A crucial consequence of the second encounter with the stimulus is that the corresponding experience, A+R(A), is itself encoded in memory. Now if the same stimulus is encountered yet again, the person is reminded of having been reminded on the previous trial, so the resulting mental state is A+R[A+R(A)], which is itself encoded in memory, and so on. In this notation, the R outside brackets is the current reminding, and the R inside brackets is the reminding that is remembered from the previous trial. It is important to understand that according to the hypothesis, the memory representation itself is recursive, not just the process that generates the representation. Each new mental state – and therefore each new encoding – is a compound that may include (or refer to) previous mental states. It appears that assumptions (1) to (3) stated above will automatically generate recursive representations. A theorist who adopts those assumptions but does not want recursive representations would have to include a restriction that specifically rules them out.

I assume that on a frequency judgment test, the subject retrieves the test item's latest memory representation and maps its depth of embedding or recursion onto the number scale (Hintzman Reference Hintzman2004). There may be no limit to the number of levels of reminding that a memory can represent. Frequency judgments become less and less accurate in an absolute sense as frequency goes higher, but there is no evidence that they ever level off. Certainly, memory mechanisms are subject to several kinds of “noise,” so if there is a memory limit, it may be more akin to a limit of resolution imposed on the entire representation than to a strict limit on the number of levels of recursion that can be encoded in a memory trace.

Of course, a child seldom encounters a series of events as uniform as a list of items that includes exact repetitions. The child does, however, have many novel or first-time experiences, and many experiences that are repetitive or routine. I assume that most experiences of the latter type include remindings of similar experiences from the past. So, if the hypothesis of recursive reminding is applicable to list learning experiments, it may also be applicable to a child's everyday life. The only precondition is a functioning episodic memory. With iterative application of encoding and retrieval, recursive representations will essentially build themselves.

How could this help the child acquire the concepts of one and successor? Let us suppose that all remindings possess a common element, R, which is tacitly understood to reference the past. The mental states A+R(A) and B+R(B), are similar because they include this common element. By way of contrast, the mental states A and B are similar in that the element of reminding is absent. Children know what it means to remember, and what it means to not remember, before the age of three (Fivush & Hudson Reference Fivush and Hudson1990). On the basis of this generalization, a child could form two schemas: X for a novel or initial experience, and X+R(X) for a repeated experience. Once formed, these schemas could be mapped onto one (or first) and greater than one. In a like manner, if less easily, schemas for X+R(X) and X+R[X+R(X)] could be differentiated and mapped onto two (or second) and greater than two, and so on. Limitations on representations and the processes that act on them make precise mappings less and less accurate as one moves up the scale, but even where errors are unavoidable, rough mappings can be learned.

If everyone with a normal memory knows what it means to remember, then they should know that the process is the same, regardless of the represented level of recursion. Consider three memory states: (a) A, (b) A+R(A), and (c) A+R[A+R(A)]. The rememberer understands that the process that leads from (b) to (c) is the same as the process that led from (a) to (b) on the earlier occasion. An appreciation of this identity could be generalized into a schema for the concept of successor. Thus, the common element, R, in remindings may help the child acquire one and successor, both concepts that Rips et al. identify as fundamental to number.

It may be especially easy to associate the “one-two-three” rhyme with number concepts through recursive reminding, because counting and reminding are both discrete and sequential (stepwise) processes. To count objects in a spatial display, one must focus attention on the objects in a particular, often arbitrary, order. This demand is absent when time imposes the order.

An obvious consequence of this hypothesis is that a child will not come to understand natural number, as described in the target article, until the child has developed a normal episodic memory. However, even rats may have the capacity for recursive reminding (see Capaldi Reference Capaldi1994). The human understanding of number presumably requires coordination of this capacity with the use of linguistic symbols.