Can magnitude really be considered the core abstract defining feature of numbers as cultural products? We argue here that developmental and other evidence is against this notion. We agree with Cohen-Kadosh & Walsh (CK&W) that numerical representation is primarily not format-independent. However, we propose that a clear distinction must be made between an evolutionarily grounded sense of approximate magnitude (Dehaene Reference Dehaene1997) and a different, culturally acquired, abstract number concept. We propose that what the field requires are criteria to define the abstractness of the number representations as used by humans in mathematics.

In support of CK&W, data of our own suggest that numerical coding is dependent on surface format. Szűcs and Csépe (Reference Szűcs and Csépe2004) presented two numbers (N1 and N2) using consecutive visual presentation. Participants were asked to add the numbers and to decide whether or not the proposed sum was correct. N1 could be presented as an Arabic digit or as a visual word, or acoustically as a heard number word. N2 was always an Arabic digit. The modality of N1 had a systematic effect on the amplitude of event-related brain potentials measured at N2. These data suggest that N1 was not translated into a common abstract number representation. Rather, N1 was retrieved from modality-specific stores when needed. Alternatively, both abstract and modality-specific stores may be involved in coding numbers.

Similarly, the functional magnetic resonance imaging (fMRI) evidence to date does not necessarily provide clear evidence for a core common magnitude representation. Current imaging studies may map the comparison process rather than overlapping representations. For example, the most frequently used marker of the putative magnitude representation is the numerical distance effect. Phenomenologically similar distance effects with various stimulus materials do not in themselves imply a single underlying neural representation. Specifically, Pinel et al. (Reference Pinel, Piazza, Le Bihan and Dehaene2004) have shown that comparing numbers and physical size results in overlapping distance effects in terms of brain activity. Hence, they assume a shared representation for numerical and physical magnitude. But it may be the processes operating on these representations that overlap in terms of brain activity. Number magnitude and physical size may be represented in non-overlapping brain areas, with overlapping brain areas involved in the process of size comparison rather than the representation of abstract magnitude. Indeed, fMRI distance effects are rarely constrained to the intraparietal sulcus (Szűcs et al. Reference Szűcs, Soltész, Jármi and Csépe2007). Hence, a single abstract representation is difficult to define in terms of a simple anatomical hypothesis.

As CK&W note, neural adaptation studies do not provide unambiguous evidence either. Neural adaptation studies supporting an abstract magnitude representation are based on repetition priming and do not require participants to carry out comparisons. This research design excludes confounds related to comparative activity. However, a problem inherent to adaptation paradigms still remains. As Naccache and Dehaene (Reference Naccache and Dehaene2001a, p. 967) state, this is a “general problem of potential strategical or attentional changes elicited by the awareness of repetition.” Hence, “ideally experimental designs based on the priming method should prevent subjects from becoming aware of the presence of repeated versus non-repeated trials” (p. 967). In other words, in number adaptation paradigms participants may direct attention to the numerical dimension, even if they are instructed not to. Hence, adaptation results will be confounded by simple change detection effects. Such confounds are especially likely when participants are instructed explicitly to “pay attention to the quantity conveyed by the stimuli” (Piazza et al. Reference Piazza, Pinel, Le Bihan and Dehaene2007, p. 303) or when symbolic numbers are used (Cohen Kadosh et al. Reference Cohen Kadosh, Cohen Kadosh, Kaas, Henik and Goebel2007b). A published exception is the study of Naccache and Dehaene (Reference Naccache and Dehaene2001a), which used symbolic stimuli and combined the adaptation paradigm with masked priming, so that participants were not aware of any stimulus repetitions. However, this result to date remains unreplicated.

Developmentally, it is important to ask whether a sense of magnitude is actually important in order to carry out successful arithmetic. So far, developmental studies have demonstrated that magnitude comparison performance does not correlate with early arithmetic skills beyond 3–4 years of age (Brannon & Van de Walle Reference Brannon and Van de Walle2001; Mix Reference Mix1999; Mix et al. Reference Mix, Huttenlocher and Levine1996; Rouselle et al. Reference Rousselle, Palmers and Noël2004). This raises the possibility that magnitude discrimination skills do not play a crucial role in the development of the cultural number concept.

For example, Halberda et al. (Reference Halberda, Mazzocco and Feigenson2008, p. 666) claimed that magnitude comparison skill in ninth grade “retrospectively predicted” symbolic maths performance. Hence, they concluded that the magnitude representation “may have a causal role in determining individual maths achievement.” However, this inference is problematic, as magnitude comparison was measured at age 14, whereas arithmetic performance was measured between ages 5 and 11. Hence, it is equally possible that better mathematics skills caused better magnitude performance. This conclusion is actually supported by the data from Amazonian Indians. These data showed that Indians have marginally worse magnitude discrimination skills than adults educated in Europe (Pica et al. Reference Pica, Lemer, Izard and Dehaene2004). It seems unlikely that Amazonian Indians are worse in symbolic maths than Europeans because their number sense is genetically limited. A more plausible explanation would be that Europeans' number discrimination skills are better because they are educated. In fact, if we assumed that there was no difference between Indians and Europeans (given that the difference was marginal), the conclusion would again be that number discrimination skills have no relationship to mathematical skills (as the latter would be expected to be better in educated Europeans).

Therefore, both methodogical and theoretical problems surround the notion of an abstract number representation. Furthermore, a distinction may be needed between an evolutionarily grounded approximate sense of magnitude (Dehaene Reference Dehaene1997), whether abstract or not, and a distinct, culturally acquired abstract number concept. CK&W cite Barsalou (2003), “abstraction is simply a skill that supports goal achievement in particular situations” (p. 1184). Given that the goal of mathematics education is to teach children an abstract number concept, the real question would seem to be whether this putative abstract core number representation bears any relation at all to the analogue magnitude representation.