I wish to challenge two points in Cohen Kadosh & Walsh's (CK&W) argument: First, the definition of abstraction is too restricted; second, the authors' distinction between representations and operations is too clear-cut. I challenge their argument from the viewpoint of cognitive developmental psychology, using a famous experimental design by Jean Piaget as an example: the “conservation of number task” (Piaget Reference Piaget1952; Reference Piaget and Mussen1984).
CK&W discuss the abstraction of number representations only in relationship to the form of the input in which the numerical information was presented; namely, specific verbal or nonverbal means of denoting numbers (following Dehaene's definition). In doing so, they miss directly discussing the question of cognitive abstraction of number representations in relation to space per se. However, numbers in the human brain – particularly in the parietal lobes – also must be abstracted from space. This question is especially relevant in developmental studies on number conservation. Remember that, in the conservation-of-number task (Piaget Reference Piaget1952; Reference Piaget and Mussen1984), when shown two rows containing the same number of objects, but of different lengths (after the objects in one of the rows have been spread apart), the child has to determine whether the two rows have the same number of objects. Up to the age of 7 years, children erroneously respond that there are more objects in the longer row, reflecting the use of the misleading visuospatial “length-equals-number” strategy that they fail to inhibit (Houdé Reference Houdé1997; Reference Houdé2000; Houdé & Guichart Reference Houdé and Guichart2001). Moreover, adult brains don't fully overcome this spatial bias; hence, they require an executive prefrontal network to overcome it (Daurignac et al. Reference Daurignac, Houdé and Jouvent2006; Leroux et al. Reference Leroux, Joliot, Dubal, Mazoyer, Tzourio-Mazoyer and Houdé2006; Reference Leroux, Spiess, Zago, Rossi, Lubin, Turbelin, Mazoyer, Tzourio-Mazoyer, Houdé and Joliot2009).
Even if, as stated by CK&W, number representations are primarily non-abstract and are supported by different neuronal populations residing in the parietal cortex, the activation of number representations may nevertheless require that individuals inhibit irrelevant visuospatial cues. In my mind, this executive process corresponds to an abstraction process from space (or perception of space) to numbers. The authors suggest, however, that such intentional prefrontal processes (executive functions, cognitive control, selective attention, and so on) are “important for some numerical operations, but not representations” (sect. 9, para. 1, emphasis theirs). I disagree. During cognitive development (Houdé Reference Houdé2000), as well as in the dynamic large-scale-network cognition which characterizes adult brains (Fuster Reference Fuster2003), logico-mathematical operations and representations are not so easily dissociable. The Piagetian concept of number conservation is a good example. It is both a numerical operation and a numerical representation, that is, the child's or adult's ability to represent or “keep in mind” (conserve) numbers independently of any irrelevant visuospatial cues (e.g., the unequal lengths of the two rows in Piaget's task). This number representation requires “at its heart” an inhibitory operation. Brain imaging (fMRI) results from our laboratory have clearly shown that this number-conservation ability is sustained, both by posterior (especially the bilateral intraparietal sulcus [IPS] and superior parietal gyrus), anterior-cingulate-cortex (ACC), and prefrontal activations (Leroux et al. Reference Leroux, Spiess, Zago, Rossi, Lubin, Turbelin, Mazoyer, Tzourio-Mazoyer, Houdé and Joliot2009).
Other examples of this kind of abstraction process are available in the literature on the neural foundations of logical and mathematical cognition (Houdé & Tzourio-Mazoyer Reference Houdé and Tzourio-Mazoyer2003). In the field of deductive reasoning, it has been shown that, after error-inhibition training, a clear activation of the cerebral activation pattern occurred, which shifted from the posterior part of the brain when individuals relied on an erroneous visuospatial strategy to the prefrontal part when they accessed abstract logic (Houdé Reference Houdé2008; Houdé et al. Reference Houdé, Zago, Mellet, Moutier, Pineau, Mazoyer and Tzourio-Mazoyer2000; see also Prado & Noveck Reference Prado and Noveck2007).
In conclusion, I think that another way to avoid current orthodoxy in the field of numerical cognition is to consider, beyond the question of input notations, an alternative idea of abstraction; that is, abstraction through inhibition in children and adults.
