In their article Cohen Kadosh & Walsh (CK&W) review an extensive psychological and cognitive neuroscience literature, with the aim of showing that, contrary to the dominant view put forward by McCloskey (Reference McCloskey1992) and Dehaene et al. (Reference Dehaene, Dehaene-Lambertz and Cohen1998a), numeric representations are primarily not abstract. The process of abstraction of numbers has a long story (Schmandt-Besserat Reference Schmandt-Besserat, Wilson and Keil1999), starting circa 8000 b.c. from a one-to-one correspondence between three-dimensional tokens (e.g., an ovoid) and units (e.g., a jar of oil), and finally developing, through successive stages, into the current “abstract” numerical representation, which is discussed by CK&W. What CK&W mean by “abstract” is that the relevant representation conveys information only related to the size of the numbers involved, independent of the particular number notation (symbolic: number words or Arabic digits; non-symbolic: e.g., dot patterns), and of the sensory modality of presentation of the stimuli (auditory, visual, somatosensory) (Libertus et al. Reference Libertus, Woldorff and Brannon2007). Over and above its semantic nature, the specific format of such an “abstract” numerical representation has been regarded as primarily propositional by McCloskey (Reference McCloskey1992), while Dehaene et al. (Reference Dehaene, Dehaene-Lambertz and Cohen1998a) have emphasized its spatial characteristics.
CK&W's review, more than focusing on what is promised by their title (“Numerical Representation: Abstract or not Abstract?”), presents evidence drawn from a variety of experimental paradigms to the effect that, under specific experimental conditions and task requirements, different numerical representations may be generated. The conclusion here is that “multiple” representations versus a “single” or “shared” numerical representation exist in the brain. More specifically, CK&W suggest that a shared numerical representation does not exist as default, but may result by connecting on-line multiple representations only when intentional processing of numbers occurs. This shared versus multiple numerical representation distinction, however, speaks little as to the abstract/non-abstract dichotomy, unless the relevant representational formats are specified. In particular, as CK&W note, it is generally assumed that the occurrence of a behavioral effect, as for example the SNARC effect, across different notations (Nuerk et al. Reference Nuerk, Wood and Willmes2005), speaks in favor of a unique numerical “abstract” representation. The fact that the SNARC effect is independent of notation or modality may well suggest that it comes from the activation of a unique “supramodal” representation – but this says little about its abstractness.
For example, in the experiments by Bächtold et al. (Reference Bächtold, Baumuller and Brugger1998), the observation that the SNARC effect is preserved when numbers are conceived as distances on a “ruler,” but reversed when they are conceived as hours on a “clock-face,” suggests the existence of distinct numerical representations. These results indicate that the involved representation is primarily “spatial,” rather than “abstract,” as shown by the observation of a SNARC (“ruler” condition), and of a reversed-SNARC (“clock-face” condition) effect, as well as flexible enough to be modulated by the experimental instructions (i.e., ruler, mental number line-related, vs. clock-face). Accordingly, Bächtold et al. (Reference Bächtold, Baumuller and Brugger1998) discuss their results in terms of spatial stimulus-response factors, which may be conceived as analogical processes, rather than with reference to “abstract” versus “non-abstract” codes.
Other studies suggest the existence of “supramodal,” “non-abstract” numeric representations, independent of the mental number line, and affecting spatial processes. De Hevia et al. (Reference de Hevia, Girelli and Vallar2006), using a manual line bisection paradigm, found that neurologically unimpaired participants displace the subjective midpoint of lines and of unfilled spaces, flanked by two different Arabic numbers (e.g., 2–9, 9—2) towards the larger digit, independent of its left-sided or right-sided position. This effect is not modulated by numerical distance, making unlikely an interpretation in terms of mapping onto a mental number line. These findings may be accounted for by the hypothesis of a “cognitive illusion,” largely independent of the allocation of attentional resources, whereby visually presented larger digits are associated with an expansion of the closer portion of space, be it a segment or an empty space. This “position of the larger number” effect reflects the processing of relative numerical magnitude, since the bisection of horizontal strings composed by larger or smaller digits (i.e., absolute numerical magnitude) does not modulate bisection according to a mental number line effect, namely, with leftward/rightward errors associated with smaller/larger numbers, as found by Fischer (Reference Fischer2001). In a later study, de Hevia et al. (Reference de Hevia, Girelli, Bricolo and Vallar2008) extended these findings that larger numbers are associated with an expansion of spatial extent, using a task whereby participants were required to reproduce the perceived length of an empty space, delimited by pairs of smaller/larger digits. De Hevia and Spelke (Reference de Hevia and Spelke2009) have recently replicated and extended these findings, using not only pairs of visual digits flanking a line, but also non-symbolic visual displays, for example, with nine smaller circles denoting a larger numerical magnitude, and two larger circles a smaller magnitude. With non-symbolic displays, adult participants, young school (7-years-old), and preschool (5-years-old) children show a line bisection bias towards the larger numerical magnitude. With symbolic cues (digits), the deviation towards the side of the larger digit is present in adult participants, but not in 7-year-old children. Taken together, these findings suggest the existence of a “supramodal” (i.e., independent of the particular format of the stimulus) representation of numerical magnitude – unrelated to the mental number line – which modulates spatial representations. With reference to the operational definition adopted by CK&W, this representation is “abstract,” being accessed by different notations. We would however qualify this representation – present as early as in 5-year-old children – as “supramodal” or, more precisely, “notation-independent,” and closely related to, or possibly overlapping with, spatial representations.
In conclusion, CK&W's review is definitely successful in showing that multiple numerical representations exist. The empirical data we have briefly reviewed here, however, indicate that the “abstract”/“not-abstract” dichotomy is too general to capture the variety of possible supramodal numerical representations, to which spatial codes provide a main format.
