We commend Rips et al.'s initiative to put to the fore how a conceptual understanding of the natural numbers is achieved by our cognitive system. This is a necessary step towards the integration of scientific progress in the now largely separate domains of basic number representations and more complex forms of numerical cognition. In this respect, it is a crucial question whether basic number representations constitute the basis for the development of a complete knowledge of the natural number system.

We do not agree, however, with the authors' conclusion that magnitude representations cannot be the precursor of understanding the properties of the natural number system. Our main point is that the authors are selective in regarding the magnitude representations they envisage. The authors' argument is exclusively built on analog magnitude representations. Although the biological reality of analog coding has been demonstrated (Roitman et al. Reference Roitman, Brannon and Platt2007), there are reasons to believe that its functional importance in numerical cognition is limited. In a behavioral priming experiment, we have shown that in a naming task, dot displays evoke a priming pattern that is consistent with an analog magnitude code, but that Arabic digit primes do not (Roggeman et al. Reference Roggeman, Verguts and Fias2007). Moreover, analog magnitude coding falls short in explaining some behavioral effects observed in number processing tasks (Verguts et al. Reference Verguts, Fias and Stevens2005). In a computational modeling study, we have proposed that analog magnitude coding is primarily an intermediate processing step that is necessary to transform visual input into another type of magnitude representation: namely, place coding (Verguts & Fias Reference Verguts and Fias2004).

Place coding is another type of basic magnitude representation that is neglected by Rips et al., although it contains characteristics that make it well-suited as a precursor for full knowledge of the concept of natural numbers. In place coding, a number is represented not in an analog way, but rather, by demarcating a position on a numerical continuum. This is accomplished by neurons that are tuned to a particular numerosity (Nieder et al. Reference Nieder, Freedman and Miller2002). For instance, a neuron tuned to numerosity 3 responds maximally to three objects, less strongly to two or four objects, and even more weakly to one or five objects. Neural place coding has been demonstrated in monkeys (Nieder et al. Reference Nieder, Freedman and Miller2002), children (Cantlon et al. Reference Cantlon, Brannon, Carter and Pelphrey2006), and adults (Piazza et al. Reference Piazza, Izard, Pinel, Le Bihan and Dehaene2004), and applies to the coding of symbolic and non-symbolic numbers (Piazza et al. Reference Piazza, Pinel, Le Bihan and Dehaene2007).

A first important property is that, because place-coding neurons code number independently from continuous variables that are correlated with numerosity (Nieder et al. Reference Nieder, Freedman and Miller2002), continuous magnitude is kept isolated from place coding, thereby allowing the identification of a unique first element. Second, number tuning is not perfect, such that when a given number is presented, not only this number but also neighboring numbers become activated; it naturally follows that predecessor and successor elements are determined.

More required properties are obtained when symbolic numbers are introduced. First, as pointed out by Rips et al., for a magnitude representation system to serve as a precursor for the concept of number, it is necessary that the magnitude representation system can code number in a precise way. Although the place coding system is only approximate for nonsymbolic stimuli (Nieder et al. Reference Nieder, Freedman and Miller2002; Piazza et al. Reference Piazza, Izard, Pinel, Le Bihan and Dehaene2004), Verguts and Fias (Reference Verguts and Fias2004) have pointed out that the use of language, or symbols more generally, leads to a strong enhancement of the representational accuracy of place coding. We simulated the learning of the numerical meaning of symbols by simply associating number symbols with nonsymbolic number inputs (collections of dots) that were already trained to be processed by place-coding neurons. After training, the behavior of the place-coding neurons was investigated when presented with symbolic input alone. It was found that the place-coding neurons encoded symbolic numbers much more precisely than nonsymbolic numbers, although, importantly, precision was not perfect such that the predecessor/successor properties were present as well (e.g., a neuron coding for symbolic number 3 was also slightly active for its predecessor 2 and successor 4). Recent neurophysiological studies in humans (Piazza et al. Reference Piazza, Pinel, Le Bihan and Dehaene2007) and also in macaque monkeys (Diester et al. Reference Diester and Nieder2007) confirm the predictions of the model. In a subsequent modeling study, we showed that the resulting system explains adult human behavior in a variety of number processing tasks (Verguts et al. Reference Verguts, Fias and Stevens2005). Hence, a key contribution of language to a mature conceptual understanding of natural numbers is that it makes the basic number representations more precise.

Second, Rips et al. note that some form of mathematical induction must be assumed, which essentially entails that properties learned in the context of some numbers can be generalized to others. If larger numbers are represented as combinations based on the building blocks 1 to 9, such generalization is in principle possible: Verguts and Fias (Reference Verguts and Fias2006) showed how a neural network can discover and apply such generalizations in the context of number naming. Empirically, it has been found that even two-digit numbers appear to be represented by such building block combinations, rather than holistically (Nuerk et al. Reference Nuerk, Weger and Willmes2001; Ratinckx et al. Reference Ratinckx, Brysbaert and Fias2005, Verguts & De Moor Reference Verguts and De Moor2005).

Finally, we believe that language also facilitates linking the place-coding system to other types of information that may further constrain it towards the principles of the natural number system. For instance, numbers are frequently displayed from left to right (as on rulers, keyboards, etc.). There is now good evidence that the coding system becomes systematically associated with this spatial information through development. For instance, small numbers are preferentially responded to with the left hand, and large numbers with the right hand (for review, see Fias & Fischer Reference Fias, Fischer and Campbell2005). Such number-space associations make the deduction that natural numbers represent alternative systems such as modular arithmetic, unlikely.

Although much work remains to be done to capture the empirical and theoretical details of the developmental trajectory from the place-coding system to a complete conceptual understanding of natural numbers, we argue that the place-coding system needs to be considered as a serious candidate to serve as a precursor for the natural numbers.