According to Cohen Kadosh & Walsh (CK&W) a neural representation of quantity is abstract if “neuronal populations that code numerical quantity are insensitive to the form of input in which the numerical information was presented” (sect. 2, para. 1). This definition is extremely shallow; it does not state at what level the hypothesized neural code takes place; and it fails in taking into account that different attributes of stimuli may be coded by separate yet interconnected brain areas, by separate yet interconnected neuronal populations within the same brain area, or by exactly the same neuronal populations, yet with different tuning schemes (deCharms & Zador Reference deCharms and Zador2000). Indeed, in the specific case of the comparison between symbolic and non-symbolic numerical processing, we have proposed that our observation might be compatible with a common quantity code instantiated in the firing of a common set of parietal neurons with different tuning schemes for different input formats (finer tuning schemes for symbolic and broader tuning schemes for non-symbolic stimuli), as also predicted by a powerful computational model of number processing (Dehaene Reference Dehaene, Haggard and Rossetti2007; Piazza et al. Reference Piazza, Pinel, Le Bihan and Dehaene2007; Verguts & Fias Reference Verguts and Fias2004).

On CK&W's view, how should such quantity code be defined – as abstract or non-abstract? At what level(s) does abstractness need to be present for a given representation to be qualified as abstract? What if there are neural populations within parietal cortex that encode the quantity aspect of each notation separately, but that these populations are so deeply interconnected that the spread of the activation is automatic and the resulting larger scale population code appears as invariant to modality? In this case, the abstractness of the coding would almost literally be in the connections. Would this mean that the emerging population code is non-abstract? Despite the absolute centrality of these issues, this discussion is not even approached, and the existence of different possible levels of neural coding are not even considered. To us, this reflects an extremely naïve approach to a question (on what is a neural code) that is central and highly debated in the current neuroscientific literature. As a result, the dichotomy between abstractness versus non-abstractness seems like a very weak heuristic tool for our ability to answer the most meaningful issue, which is to unveil the computational principles underlying processing of numerical quantity in parietal cortex.

We also feel uncomfortable with the more general definition of neural representations in this target article. CK&W define their view of neural representations in extremely vague terms (“patterns of activation within the brain,” sect. 2, para. 2), but in the rest of the article their operational definition of neural representations appears to be grounded on implicit views inspired by the classical cognitive psychology notion of module (Fodor Reference Fodor1983). Despite overtly criticizing the idea of neural module, CK&W actually embrace two important aspects of the classical definition of module: encapsulation (which here implies seriality of information processing), and domain selectivity.

The assumption of encapsulation and seriality is evident in CK&W's strict use of the additive factor method, for which the presence of statistical interactions between quantity-related effects and notation effects is taken as evidence in favor of notation-specific quantity representations. In light of our most recent knowledge of fine brain structure and function, encapsulation and seriality are extremely difficult to maintain. It is a known fact that different and also distant brain regions are massively bidirectionally interconnected and work in parallel (Felleman & Van Essen Reference Felleman and Van Essen1991). Not surprisingly, models that do take into account some degree of parallel processing show that interactions between factors can occur even if the factors affect completely separate processing stages (McClelland Reference McClelland1979; Rumelhart & McClelland Reference Rumelhart and McClelland1986). This can happen, for example, if one factor affects one aspect of the information processing (say, the rate of evidence accumulation), and the other factor affects another aspect (say, the decision threshold). More refined data analysis taking into account precise response distributions and testing alternative (and more neurally plausible) models are therefore necessary before driving conclusions on the basis of scattered reports of (often weak) statistical interactions between factors.

Second, CK&W implicitly link abstractness to domain-selectivity. Thus, the fact that brain regions or individual neurons that are modulated by numerical quantity may also be modulated by other quantity- as well as non-quantity-related parameters, like physical length (Tudusciuc & Nieder Reference Tudusciuc and Nieder2007), color (Shuman & Kanwisher Reference Shuman and Kanwisher2004), or motion direction (Nieder et al. Reference Nieder, Diester and Tudusciuc2006), is taken as undermining the possibility of an abstract representation of quantity. We disagree with this view.

In the first place, selectivity might be a property that arises at some level of neural coding that is not yet probed by most current methods of investigation (but see Tudusciuc & Nieder Reference Tudusciuc and Nieder2007). For example, in the case of numerical quantity, it is possible that numerical quantity-selective representations do exist, but emerge only as the averaged population activity of numerical quantity-sensitive, but not selective neurons. This would be the case, if, for example, some neurons respond both to number and length, others to both number and motion direction, and others to both length and motion direction (Nieder et al. Reference Nieder, Diester and Tudusciuc2006; Pinel et al. Reference Pinel, Piazza, Le Bihan and Dehaene2004; Shuman & Kanwisher Reference Shuman and Kanwisher2004; Tudusciuc & Nieder Reference Tudusciuc and Nieder2007). Under this scenario, at the population level the codes for number, length, and motion direction would be distinct, but neither the single neuron spiking activity nor the functional magnetic resonance imaging (fMRI) signal averaging across all three populations would reveal such selectivity. In this respect, multi-unit recording allowing population coding properties to be more clearly unveiled, and multivoxel pattern analysis of the fMRI signal, capitalizing on very small variations in domain specialization across voxels, might reveal population codes which are indeed selective (Kamitani & Tong Reference Kamitani and Tong2005; Tudusciuc & Nieder Reference Tudusciuc and Nieder2007). Alternatively, because of massive local connectivity, functional selectivity may only be revealed by taking into account long-range connections between distant brain regions. In either case, the issues of selectivity and of abstractness are two orthogonal ones. In conclusion, we think that reasoning in terms of ill-defined dichotomies (selectivity vs. non-selectivity, abstractness vs. non-abstractness) may induce unnecessary over-simplifications and is very unlikely to bring us towards a deeper understanding in this domain.