Cohen Kadosh & Walsh (CK&W) provide a compelling argument for challenging the currently accepted view – intuitively appealing and convenient – that the nature of numerical representation is inherently abstract. They lucidly explain why it is implausible that number has a single specific representation center given that classic cases for external world attributes such as color and motion lack a single locus of representation. In reviewing relevant behavioral, neuroimaging, and single-neuron (monkey) studies, they convincingly show that, despite the fact that many reports appear to support the abstract representation view, the evidence is incomplete and presents serious methodological and theoretical problems (e.g., null results, paradigm insensitivity, task specificity factors). Indeed, the data seem to support the opposite view. However, contrary to what the authors' defend, from this it doesn't follow that “numbers are not abstract” (sect. 5's title). The authors' definition and usage of crucial concepts such as “abstraction,” “representation,” and “number,” lacks conceptual clarity and creates unnecessary confusion and discontinuities with the understanding of other realms of cognition. Here, I only address the first one – abstraction.

CK&W provide a narrow and confusing characterization of abstraction. First, they adopt a behavioral definition (sect. 2, para. 1) that applies exclusively to numbers (and, as an extension, they operationalize the notion of abstract representation in terms of neuronal populations' insensitiveness to the form and notation of the input in which numerical information is presented). This operational definition of abstraction is specific but unnecessarily restrictive, making its extension to other non-numerical areas of cognition hard, if not impossible. How are we to relate the authors' arguments with other, presumably supra-modal conceptual domains? Second, the authors' terms “abstract” and “non-abstract” as defined in the narrow domain of neural sensitiveness, become unfortunate misnomers that generate confusion. The only hint the authors give of the nature of (not specifically numerical) abstraction, is a reference to Barsalou (sect. 10, para. 1): “abstraction is a skill that supports goal achievement in particular situations” (Barsalou Reference Barsalou2003, p. 1184). But, this is vague and incomplete. That work deals with concepts that refer to entities or state of affairs “about the world” (e.g., chair), where even the abstract concept of truth is analyzed in that manner (p. 1185). What can then be said about the passage from the abstracted cardinal numbers to, say, transfinite or infinitesimal numbers (i.e., actual infinity), cases in which there is no fact about the world that corresponds to the concept (Núñez Reference Núñez and Hersh2006; in press; Lakoff & Núñez Reference Lakoff and Núñez2000)? We do need the term “abstraction” in order to correctly handle such concepts. And we do need the authors' arguments to clearly address the fact that neuronal populations are sensitive to form of input. What is required is a fruitful theoretical proposal of supra-modal entity processing (e.g., numbers) that is able to encompass the authors' arguments while leaving room for fruitful distinctions between abstract and non-abstract concepts in a way that is clearly extendable, without discontinuities, to other forms of everyday and mathematical abstraction.

In addressing abstraction, CK&W make a distinction between automatic and intentional processing (of numerical information with different notations). By automatic, they mean a process that does not need monitoring to be executed (sect. 5, para. 3). They argue that by adopting such a distinction, we can yield a better characterization of abstract and non-abstract representations (sect. 11, para. 4). And they go on to build a parallel between this distinction and Barsalou's linguistic and situated simulation (imagery) systems. But, again, this is narrow and problematic. The authors' arguments are mainly concerned with the side of cognition that deals with perception and comprehension. When we look at the productive side of cognition we can see that the automatic/intentional (and linguistic/imagery) distinction can be really troublesome. Consider a person who, during an everyday conversation, says, “that was way back in my childhood,” and while doing so points with her thumb backwards over the shoulder. Such speech-gesture coproduction (1) is not monitored (McNeill Reference McNeill1992) and, therefore, is “automatic,” and (2) it is inherently abstract because the uttered word “back” and the backwards pointing don't refer to anything in the real world, but to the past, which is metaphorically conceived as being spatially behind ego (Núñez Reference Núñez, Núñez and Freeman1999; Núñez & Sweetser Reference Núñez and Sweetser2006). Moreover, this fast spontaneous speech-gesture coproduction is, both, linguistic and imagery-driven. When trying to understand the productive side of cognition involving abstract concepts (numbers included) and their neural instantiations, the authors' distinctions break down. More conceptual clarity is needed.

But, looking at the productive side of cognition can actually be quite supportive of the CK&W's claims. For instance, the authors argue that “it is entirely possible that similar behavioural effects can be subserved by different areas, or neuronal populations in a single brain area” (sect. 5, para. 1). Regarding our space-time mapping example, it is known that humans can spontaneously conceptualize temporal events as being spatial locations along the sagittal axis, with events in the future as being in front of ego and events in the past as being behind. And, neurally, this can be instantiated through the recruitment of neural populations in the ventral intra-parietal area (VIP) and the polysensory zone (PZ), among others (Núñez et al. Reference Núñez, Huang and Sereno2007). However, the linguistic and spontaneous gestural production of the Aymara of the Andes shows a striking counterexample: They conceive future events as being behind them and past events as being in front of them (Núñez & Sweetser Reference Núñez and Sweetser2006; AAAS 2006). These two forms of spatial conceptions of time are internally consistent but mutually inconsistent, having, presumably, different neural realization. Similarly, numbers and arithmetic can be realized conceptually through different metaphorical mappings such as object collection and motion along a path (Lakoff & Núñez Reference Lakoff and Núñez2000). These distinct mappings can have different neural instantiations while supporting the same inferential organization (e.g., in both cases “two plus three is five.”) Both conceptions characterize a cascade of isomorphic entailments, which are presumably, as the authors claim, subserved by different neural populations in the brain.