Cohen Kadosh & Walsh (CK&W) propose a new dual-code model of numerical representation that posits a psychological and neural distinction between fast, automatic notation-dependent representations and slower, intentional notation-independent representations. In their model, notation- and modality-specific (non-abstract) representations are psychologically more primary than abstract representations. We argue that this proposal is limited in its psychological and neurobiological perspective on numerical abstraction, and that the evidence they offer is either neutral on the issue of whether numbers are represented abstractly, or equally compatible with existing models of number representation.

A central limitation of CK&W's proposal is the coarse manner in which it surveys the theoretical landscape of numerical abstraction. At the psychological level, numerical abstraction can refer to notation independence, modality independence, or the representation of number independently of dimensions such as time, space, size, and color. For instance, the capacity to recognize that a group of three elephants is equal in number to a group of three umbrellas, but that both are fewer in number than a series of ten gunshots, is a feat of numerical abstraction. We know from scores of behavioral studies that human infants and nonhuman animals smoothly represent non-symbolic numerical values across modalities and dimensions (e.g., Cantlon & Brannon Reference Cantlon and Brannon2006; Church & Meck Reference Church, Meck, Roitblat, Bever and Terrace1984; Hauser et al. Reference Hauser, Dehaene, Dehaene-Lambertz and Patalano2002; Jordan & Brannon Reference Jordan and Brannon2006; Jordan et al. Reference Jordan, Brannon, Logothetis and Ghazanfar2005; Reference Jordan, MacLean and Brannon2008; Kobayashi et al. Reference Kobayashi, Hiraki and Hasegawa2005; Nieder et al. Reference Nieder, Diester and Tudusciuc2006; Starkey et al. Reference Starkey, Spelke and Gelman1983; Wood & Spelke Reference Wood and Spelke2005). Importantly, infants and nonhuman animals exhibit these abstract numerical representations in the absence of symbolic language, and they do so spontaneously. Thus, numerical representations can be abstract in the absence of discrete symbolic representations or explicit task demands. CK&W's claim that “numerical representation is primarily non-abstract” (target article, Abstract) and that intentional processing is required to achieve notation- and modality-independent representations of numerical values is at odds with the demonstrated existence of this non-symbolic form of numerical abstraction.

Abstract non-symbolic numerical representations are important to any theory of numerical representation because they are hypothesized to provide the evolutionary and developmental foundation upon which symbolic numerical representations are psychologically constructed (e.g., Carey Reference Carey2004; Gallistel & Gelman Reference Gallistel and Gelman2000). In other words, current developmental and evolutionary theories propose that numerical representations are abstract before they are symbolic. Therefore, CK&W's proposal needs to either (1) provide a theoretical account of the alleged developmental disappearance of automatic numerical abstraction in human children, or (2) make the case that preverbal infants and nonhuman animals spontaneously engage in intentional processing to represent numerical values across modalities and dimensions.

A second theoretical limitation of CK&W's proposal is that their neurobiological definition of numerical abstraction risks reductio ad absurdum. That is, the stipulation that numerical abstraction requires identical responses in identical neurons is potentially impossible to satisfy. Yet, even if it were possible to satisfy that criterion, it is not clear whether it is the appropriate criterion for establishing numerical abstraction. As the authors review, regions of the intraparietal sulcus (IPS) respond during numerical processing across notations, modalities, and dimensions. The mounting evidence that numerical representations across notations, modalities, and dimensions are “distributed but overlapping” in the IPS is neutral on the issue of whether the underlying representations are abstract. Instead, such evidence suggests that different numerical forms invoke both shared and separate neural processes. CK&W's conclusion that the neurobiological data weigh more heavily in favor of notation-dependent neural processes is therefore merely an assertion at this stage.

Other empirical evidence that CK&W cite in favor of their account does not do the theoretical work the authors are asking of it. The authors review both behavioral and neurobiological evidence purportedly revealing notation-specific interactions in numerical tasks. However, many of the notation-specific interactions they review hinge on generic differences in performance level. Specifically, if a single psychological process is involved in judging numerical values from two different numerical notations (e.g., numerical judgments of Arabic numerals and arrays of dots), yet the judgment is easier for one of the two notations (e.g., because the input mode is more rapid, reliable, or fluent), a notation-specific interaction may emerge simply because performance on the easier notation hit ceiling accuracy or floor speed. Such interactions, though cited by CK&W, do not invite the theoretical implications that CK&W draw. Instead, notation- or modality-specific interactions that arise under these circumstances reflect a quantitative difference in performance between notations or modalities. Note that this argument may also apply to neurobiological findings under circumstances in which floor or ceiling response levels are achieved. While bearing this issue in mind, we encourage CK&W to re-evaluate the relevance of the following studies to their argument for notation- and modality-dependent number representations: Dehaene and Akhavein (Reference Dehaene and Akhavein1995), Droit-Volet et al. (Reference Droit-Volet, Clement and Fayol2008), Ganor-Stern and Tzelgov (Reference Ganor-Stern and Tzelgov2008), Hung et al. (Reference Hung, Hung, Tzeng and Wu2008), and Ito and Hatta (Reference Ito and Hatta2003). These studies (and likely others) report interactions that do not necessarily support a notation- or modality-dependent account of numerical representation.

Importantly, any notation- or modality-dependent interaction that survives inspection for a generic performance effect likely can be accounted for by the two-system view of approximate and exact numerical representation proposed by Dehaene et al. (Reference Dehaene, Spelke, Pinel, Stanescu and Tsivkin1999). In the pre-existing two-system proposal, notation-specific interactions may arise from an interplay between the exact and approximate numerical codes. Unfortunately, CK&W have not distinguished the empirical predictions of their dual-code view from the existing two-system view.

In short, although we applaud CK&W for highlighting some of the many remaining puzzles about the nature of numerical abstraction in the mind and brain, the solutions they offer do not adequately account for the data. Moreover, the open questions surrounding the cognitive and neural basis of numerical abstraction do not warrant a restructuring of the field of numerical cognition. Robust evidence demonstrates that with or without language, number is represented abstractly – independently of perceptual features, dimensions, modality, and notation. In fact, this is the very definition of “number.”