A crucial part of Cohen Kadosh & Walsh's (CK&W) argument against abstract representations concerns the fact that different effects (e.g., distance, SNARC, compatibility effects) are often not quantitatively the same for different number forms like notation (e.g., Arabic vs. verbal), format (e.g., symbolic vs. non-symbolic), and modality (e.g., visual vs. auditory modality). For concreteness, we will here focus on the distance effect. The authors note that distance between two numbers in a comparison task interacts with notation: Cohen Kadosh (Reference Cohen Kadosh2008a), for example, reports that the distance effect is larger for numbers in Arabic notation than in verbal notation (notation–distance interaction). However, the fact that a common abstract coding system is accessed by different notations does not mean that the representation of these notations should be exactly equal. In a neural system, obtaining exactly the same representations for different notations would be possible only if the input pathways to the common coding system for the different notations are exactly the same, which is clearly impossible. If there is a common coding system, but some divergence between the input pathways to it, the activation pattern on the common system will be (at least slightly) different for different notations, and any effects downstream from the common representation will be influenced.

This also holds when number formats are different, in particular when comparing symbolic (Arabic, verbal) with non-symbolic (collections of objects) number formats. Computational modeling has suggested that there is a common abstract coding system for symbolic and non-symbolic formats (Dehaene & Changeux Reference Dehaene and Changeux1993; Verguts & Fias Reference Verguts and Fias2004), but that the input pathways for the two formats are different, with one format (non-symbolic number) being much more noisy and passing via an extra representational processing stage (Santens et al., in press). Because of this extra stage, there can again be different behavioral signatures for the two formats (e.g., format–distance interaction; Roggeman et al. Reference Roggeman, Verguts and Fias2007), even when they eventually converge on a common coding system. Finally, given that different modalities are processed by different sensory input systems, a similar argument holds for modality–distance interactions.

Having argued that the evidence against an abstract coding system of number is not convincing, we now turn to some positive evidence in favor of such a system. CK&W argue that it may exist, but only in limited circumstances, in particular when participants are required as part of the task demands to treat different number notations (or formats, or modalities) as similar, or when they (for whatever reason) intentionally choose to treat different notations as similar. To evaluate this claim, we look at number priming studies in which (1) prime and target numbers are presented in different notations (so numbers in the prime notation are not included in the task demands); and (2) it is explicitly demonstrated that primes are not consciously processed (so it is excluded that participants would intentionally create a cross-notation number system). In such studies, we can then check whether there is a cross-notation prime-target distance effect, which is commonly regarded as evidence for access of prime and target to a common semantic coding system (e.g., Reynvoet et al. Reference Reynvoet, Brysbaert and Fias2002). At least two studies fulfill both criteria. Both studies used a number comparison task with a fixed standard for comparison. Reynvoet and Ratinckx (Reference Reynvoet and Ratinckx2004) used Arabic and verbal primes, but Arabic target numbers only, so verbal numbers were not included in the task demands. In their Experiment 2b, primes were shown to be not consciously perceived, so participants could not have intentionally included verbal numbers in their number set. When prime and target were both presented in the left hemifield (right hemisphere), there was no cross-notation priming effect; however, the effect was present in the right hemifield (left hemisphere). Hence, at least in the left hemisphere, there seems to be a common coding system, even when it is not required by task demands and outside participants' intentions. Second, in the study of Van Opstal et al. (Reference Van Opstal, Reynvoet and Verguts2005a), primes were presented in both notations (verbal and Arabic), and targets were presented in one notation only (either verbal or Arabic, varied across participants), so participants were again performing the comparison task on one notation only. Primes were demonstrated to be unconscious, so they were also aware of one notation only. A prime-target distance effect was present in this case (reported in Van Opstal et al. Reference Van Opstal, Reynvoet and Verguts2005b). Further detailed analysis of this effect showed a significant cross-notation prime-target distance effect when primes were Arabic and targets were verbal (F(1, 21)=8.59, p<.01). The prime-target distance effect was not significant when primes were verbal and targets were Arabic (F(1, 21)=1.64, p=.21), perhaps because verbal stimuli less easily survive visual masking (cf. Reynvoet & Ratinckx Reference Reynvoet and Ratinckx2004). Whatever the reason, these priming studies demonstrate cross-notational, semantic priming even under circumstances in which participants do not consciously perceive the prime notation during the whole experiment.

To sum up, interactions of an experimental factor like distance with notation, format, and modality are not informative with respect to the existence of an abstract coding system; cross-notation priming effects are, and they demonstrate that such a system exists.