Cohen Kadosh & Walsh (CK&W) focus on the mental representation of numbers. They do not state it explicitly, but it is clear from their argument that they are interested in the default internal representation of an external stimulus that corresponds to a specific magnitude. It should be distinguished from “working representations” generated to perform specific tasks. This default representation symbolizes magnitude as stored in semantic memory and is sometimes described in terms of the “mental number line” (e.g., Restle Reference Restle1970). The authors suggest that the locus of this representation is in the parietal lobes and propose that it is best probed by automatic processing of numerical information. The authors review the existing data and conclude that it does not allow a conclusion in favor of an abstract representation of numerical information. As an alternative, they propose that numerical processing starts with non-abstract representation (coded by different neuronal populations for different categories/modalities of inputs) in the parietal lobes, which is automatic in the sense of not being affected by task requirements. Abstract representations may emerge later in the prefrontal cortex (PFC) as a result of the task requirements.

Whatever the representation of numerical information in the brain is, it codes the relevant features of the mental number line, such as the increased discriminability between magnitudes farther away from each other, known as the distance effect, and, for a given intra-pair distance, better discrimination for pairs of small numbers, known as the size effect (e.g., Moyer & Landauer Reference Moyer and Landauer1967). The size congruity effect (SiCE) is frequently used as a marker of automatic processing of numerical information (e.g., Tzelgov & Ganor-Stern Reference Tzelgov, Ganor-Stern and Campbell2005). It refers to shorter latencies of physical size decisions when the presented physically larger number is also numerically larger (congruent condition, e.g., 5 3) than when it is numerically smaller (incongruent condition, e.g., 3 5). The SiCE increases with the intra-pair numerical distance (e.g., Tzelgov et al. Reference Tzelgov, Yehene, Kotler and Alon2000), and is larger for numerically smaller pairs (Cohen Kadosh et al. Reference Cohen Kadosh, Tzelgov and Henik2008g, but see Van Opstal et al. 2008; Verguts & Van Opstal Reference Verguts and Van Opstal2008; Verguts et al. Reference Verguts, Fias and Stevens2005). Thus, conceiving magnitudes means activating the mental number line and mapping the specific magnitudes on it.

We do not believe that representation of magnitudes in general, and numbers in particular, can be reduced to the neural activation of specific populations of neurons. Another definition of the representation of magnitudes would be in terms of the relevant neural circuit activated when information in a given domain (e.g., numerical) is processed. Dehaene et al. (Reference Dehaene, Piazza, Pinel and Cohen2003) pointed out that the horizontal intraparietal sulcus (HIPS) is activated by numbers independently of their notation, and proposed it as the neuronal locus of the mental number line. Thus, while specific populations of neurons are the locus of activation of feature detectors for different kinds of stimuli corresponding to magnitudes, it seems to us that the HIPS is a possible candidate for the convergence zone (Damasio Reference Damasio1989) for magnitude information. It follows that even if different populations of neurons fire for different kinds of inputs, magnitude is represented once the HIPS is activated. In this sense, the network resulting from the convergence of the stimulus-specific neuron populations, corresponding to various kinds of quantities and numbers in various notations, whose activation converge in the HIPS, generates what Barsalou (Reference Barsalou1999; Reference Barsalou, Gershoff-Stowe and Rakinson2005) calls a simulator. It simulates the “mental number line,” and while it may be based on perceptual symbols (Barsalou Reference Barsalou1999), it summarizes the magnitudes specifically coded by the various populations of neurons. In this sense, while the automatically probed numerical representations may be non-abstract in the sense of being based on the firing of different population of neurons, they are “abstract” in the sense of summarizing the different inputs by the HIPS. Although, according to CK&W, the existing data do not allow for a clear conclusion in favor of an abstract representation, they do not support the opposite conclusion, either.

Only numbers that are stored in memory as part of the mental number line can be automatically activated, that is, retrieved from memory without intentional effort (Logan Reference Logan1988; Perruchet & Vinter Reference Perruchet and Vinter2002). According to findings from our lab, the set of numbers included on the mental number line is quite limited. It seems that the mental number line represents only single-digit positive integers, that is, it does not include two-digit numbers (Ganor-Stern et al. Reference Ganor-Stern, Tzelgov and Ellenbogen2007), negative numbers (Tzelgov et al. Reference Tzelgov, Ganor-Stern and Maymon-Schreiber2009), or fractions (Kallai & Tzelgov, in press). Thus, the single-digit positive integers may be considered as the “primitives” of numerical cognition, and are automatically accessed (or “retrieved from memory”).

The representations of numbers that are not part of the mental number line are generated on-line, when needed, by intentional operations. They are generated as part of “working representations,” as a result of intentionally applying task-relevant operations on the “primitives” included on the mental number line. The prefrontal cortex is apparently active in this process, as it is in additional operations that involve numbers, such as arithmetic (Ansari Reference Ansari2008). Therefore, neural activity in the prefrontal cortex reflects the working representations heavily loaded by the task requirements. Thus, while we agree with CK&W that probing automatic processing should provide the answer to the question of whether the representation of numerical information is abstract, we emphasize that this should be done by analyzing the numbers that are members of the mental number line. The data that one can accumulate according to these criteria, at this point are not strong enough for the rejection of a hypothesis of abstract representation of numerical information.