We support Cohen Kadosh & Walsh's (CK&W) well-argued warning against sliding into the dogma of assuming a uniform abstract representation of number in the intraparietal sulcus, regardless of modality of input or purpose of the task. This question is especially pertinent now, given the last three decades of research into visual reasoning in mathematics and the recent application of work in embodied cognition to mathematics representation and reasoning. Our interest is from a cognitive science and philosophical, as opposed to a neuroscientific, perspective.

In general, we find the notion of abstraction in the field to be not well-defined. In the article, we are given an initial operational definition from Dehaene, where the presence of an abstract representation, the “behavior depends only on the size of the numbers involved, not on the specific verbal or non-verbal means of denoting them” (Dehaene et al. Reference Dehaene, Dehaene-Lambertz and Cohen1998a, p. 356). But towards the end of the article (sect. 10, para. 1), CK&W cite approvingly Barsalou's comment that “abstraction is simply a skill that supports goal achievement” (Barsalou Reference Barsalou2003, p. 1184). Is this in fact consistent with the earlier operational definition? Barsalou's own characterisations in Barsalou (Reference Barsalou, Gershoff-Stowe and Rakinson2005) of the use of the term abstract include a sense in which perception is in itself typically abstract, inasmuch as it involves categorical judgements out of the “confusion of experience.” As this addresses the central issue of the target article, a clearer account of what is at stake here is required.

We do not think that the binary distinction between abstract and not abstract is the right way to conceptualise the problem. In identifying the digit 2 over a range of typefaces, or ignoring the colour of a number of dots in the visual field, it is natural to consider that there is abstraction from the token that is physically present. Certainly the mathematical notion of abstraction allows many levels of abstraction, and we believe that mathematical cognition in general is built in part on some version of arithmetical representations. (This notion of abstraction is also referred to in Barsalou [2005], allowing that there is a question of the degree of abstractness in this case.)

CK&W's analysis in terms of automatic and intentional codes is clearer, and seems potentially of greater explanatory power than deciding on the question of abstractness. Here we note the comments on this topic in Wilson (Reference Wilson2002), on the apparent paradox that in more elaborate tasks, the more automated approach actually allows finer control of the activity than do more “controlled” strategies. The suggestion is that the automation builds up internal representations, thereby providing a more efficient way to deal with some of the regularities of the problem at hand. Thus, with practice in arithmetical calculation, more persistent representations will be formed, and on this view these may well be more abstract.

The distinction between automatic and intentional processing raises the question of how automatic compares to the so-called innate arithmetic of Lakoff and Núñez (Reference Lakoff and Núñez2000). A series of screen experiments on babies (not mentioned by CK&W) suggest that babies are born with basic addition and subtraction skills on small numbers, and indeed appear to recognise sameness of number when objects are replaced with an identical number of different objects (Simon et al. Reference Simon, Hespos and Rochat1995). Presumably, the processing here is automatic, but we are curious as to how these results fit with the authors' account of numerical representation.

Lakoff and Núñez (Reference Lakoff and Núñez2000) emphasise the metaphorical process in mathematical development. They identify four “grounding metaphors” for arithmetic (metaphors in which the source domain is a familiar, environmentally grounded domain) that all abstract into a common target domain. These metaphors are all principally visual. The sense of abstraction here is close to the sense in which the term is used in informatics generally, where structure common to some data or mathematical entities can be reasoned about while ignoring properties that differ from instance to instance. Similarly, multi-modal representations of mathematics such as diagrammatic or algebraic reasoning are assumed to abstract to a common domain. It may therefore be relevant to consider work on the cognitive status of metaphor processes.

In Lakoff (Reference Lakoff and Gibbs2008), it is claimed that mirror neurons are multi-modal, that is, the same mirror neurons fire whether we imagine, or perform/perceive certain actions. The neural theory is based on the premise that if we see two domains simultaneously enough times, then connections between the nodes that process the domains are strengthened and we build a strong association. It is not clear to us if there is substantial disagreement between Lakoff's views, as applied to arithmetic, and the views of CK&W. The target article naturally does not address the issue, of interest to us, of how arithmetical judgements that quantify over all numbers are represented (e.g., associativity of addition). This can be expected to relate to the representation of particular numbers, but remains, as far as we know, an open question.