Whereas number magnitude was initially assumed to be represented abstractly as a function of powers of ten (McCloskey et al. Reference McCloskey, Caramazza and Basili1985), an analog representation figuring numerical magnitudes by overlapping activations on an oriented and compressed mental line was later proposed (Dehaene Reference Dehaene1992). These views have often been mixed ill-advisedly, and it is now common to read that the parietal cortex hosts “an abstract numerical representation taking the form of an oriented number line” (e.g., Cohen Kadosh et al. Reference Cohen Kadosh, Henik, Rubinsten, Mohr, Dori, Van de Ven, Zorzi, Hendler, Goebel and Linden2005; Dehaene et al. Reference Dehaene, Dehaene-Lambertz and Cohen1998a). This, however, is a double nonsense: by definition, analog representations cannot be abstract; by essence, abstract representations cannot be oriented or compressed. This conceptual drift from an analog to an abstract number line occurred because many authors in the field used loose definitions of abstract/analog representations leading to several profound mistakes. Abstract representations capture the ideational content of knowledge irrespective of the original input modality. Being language-independent, they are formalized in a propositional code specifying the meaning of assertions, thanks to a logical system called the predicate calculus; importantly, being amodal (i.e., not tied to any modality), they possess no isomorphic properties influencing performance (Anderson & Bower Reference Anderson and Bower1973; Pylyshyn Reference Pylyshyn1973). In contrast, analog representations share with the reality they represent a first- or second-order isomorphism and are displayed on mental media preserving physical properties (e.g., spatial distances for visuo-spatial representations; Kosslyn et al. Reference Kosslyn, Ball and Reiser1978). Because the processes acting on analog representations are sensitive to these properties, they can be traced in behavioral performance. Using criteria from seminal papers in cognitive (neuro)psychology, we highlight three important mistakes compromising CK&W's argumentation.

A first mistake consists in attributing the properties of the content of a representation (representation of [object/concept X with properties P]) to its format ([representation of object/concept X] with properties P; Pylyshyn Reference Pylyshyn and Savage1978; Reference Pylyshyn1981). Format and content being independent, content properties can be represented in modality-neutral and modality-specific formats (Caramazza et al. Reference Caramazza, Hillis, Rapp and Romani1990). Stating that a numerical representation is abstract if behavioral effects depend only on magnitude is thus erroneous. Contrary to what CK&W write, McCloskey's semantic representation was not abstract because of its quantitative content, but because of its propositional format. Dehaene's magnitude representation on an analog medium provides the a contrario argument. Another common mistake concerns the inferences drawn from the (in)dependence between semantic representations and the various modalities/notations of access. Although abstract representations are amodal, a representation accessed through several modalities/notations is not necessarily abstract. Number semantics could be accessed from and give rise to similar effects in verbal, Arabic, or non-symbolic inputs while being non-abstract (e.g., Dehaene's analog representation). Conversely, modality/notation-specific effects do not necessarily sign non-abstract representations, because abstract representations could be accessed differently by modality-specific presemantic systems (Riddoch et al. Reference Riddoch, Humphreys, Coltheart and Funnell1988). By simply defining non-abstract representations as “sensitive to input modalities” (see sect. 2, para. 1), CK&W fail to provide a constraining framework. For example, finding that, after habituation, magnitude processing in one notation becomes more vulnerable to virtual lesions of the parietal cortex than in other notations, does not contradict the abstract view, as habituation could have modified the connectivity between notation-specific systems and abstract representations prior to the lesion. Trying to consider representations and processes separately constitutes CK&W's third mistake: Whatever their format, representations cannot be conceived as mental entities distinct from the processes acting upon them (Anderson Reference Anderson1976; Palmer Reference Palmer and Estes1978), and processes are totally determined by the format of the representations upon which they act (Shepard & Podgorny Reference Shepard, Podgorny and Estes1978). Therefore, any proposal of non-abstract representations must come along with a set of compatible processes clearly specified both at the functional and anatomical levels.

To properly assess the format of numerical representations, we make the following recommendations. First, a data type (a way of organizing information in memory; Simon Reference Simon and Savage1978), and the primitive operations that can be performed on it, must be specified and supported by unambiguous behavioral effects. Rightward biases on the number line in neglect patients (Zorzi et al. Reference Zorzi, Priftis and Umiltà2002) is an instance of how intrinsic properties of representations may show through behavioral effects, but other frequently cited effects are not: The distance effect can be accounted by various types of representations (e.g., abstract semantic labels: Banks Reference Banks and Bower1977; distributed representations: Verguts & Fias Reference Verguts and Fias2004), the congruity effect by a power-of-10-based propositional representation, and the SNARC effect by associations with abstract codes (Gevers et al. Reference Gevers, Verguts, Reynvoet, Caessens and Fias2006c). Second, activations must be demonstrated in brain areas whose general cognitive role and pattern of activity are compatible with the assumed type of representation. For example, non-abstract representations tied to space predict numerical processes isomorphic to spatial orientation, with overlapping activations of number and space processing areas. Finally, a lesional approach must demonstrate that these areas are necessary to perform correctly the tasks relying on this representation. Indeed, non-abstract representations could simply be automatic by-products of mental activity and not its medium. CK&W's suggestion to rely on Stroop-like interference to infer the nature of numerical representations is thus irrelevant, because automaticity does not guarantee that the activated representation is functionally required.

The format of representations in memory is a major question in cognitive neurosciences. It was at the heart of the first architectures of numerical cognition, and CK&W should be commended for drawing our attention again to this critical but messy issue. Assessing the existence of abstract representations is a tricky if not impossible enterprise, because it boils down to demonstrating the absence of effects (e.g., of modality). Efforts should therefore rather be expended on testing possible candidates for non-abstract representations. However, although we believe that non-abstract numerical representations may have a true functional role, we think that the definitions and evidence provided by CK&W do not allow them to infer that numerical representations are non-abstract. We recommend using terms like “abstract/analog,” “(a)modal,” “supramodal,” or “modality-(in)dependent,” and so forth, with great care, because each of these conveys a specific meaning. Far from being a rhetorical question, this issue has strong theoretical and empirical implications. It is now time to overcome mistakes that hamper research on numerical representations and their relationship with the functioning of the human brain.