Recently, there has been a growing consensus favoring an abstract representation of numerical magnitude. According to Cohen Kadosh & Walsh (CK&W), this assumption might be premature, and they raise the alternative possibility that the default numerical representation is not abstract, but rather, dependent on notational input. CK&W argue that we will be more likely to observe evidence for a non-abstract representation under automatic processing conditions.

We applaud the idea of examining numerical representations under conditions of automatic processing, because then number processing is less affected by intentional strategies. However, based on research using the priming paradigm, the conclusion of CK&W is not supported. The priming paradigm is a very popular method to investigate underlying representations under highly automatic conditions. In a typical number priming paradigm, participants have to react on a target number that is preceded by a prime number while the numerical distance between both is systematically varied. Interestingly, when the numerical distance between prime and target is small (e.g., “1” preceded by “2”), the target is processed faster than when the prime-target distance is large (e.g., “1” preceded by “9”) – that is, the priming distance effect. This effect was first shown by Den Heyer and Briant (Reference Den Heyer and Briant1986) and later replicated under more automatic conditions using short stimulus onset asynchronies between prime and target and visual masking in order to eliminate strategic prime use (e.g., Reynvoet & Brysbaert Reference Reynvoet and Brysbaert1999).

Certain priming studies manipulated the notation of prime and target: Prime and target were either presented in the same notation (e.g., prime “6” – target “7”) or in different notations (e.g., prime “six” – target “7”). Originally, Koechlin et al. (Reference Koechlin, Naccache, Block and Dehaene1999) observed an interaction between notational change and distance priming: no distance priming effect was observed when prime and target were presented in different notations. This led to the proposal of notation dependent representations of quantity in line with the proposal of CK&W. However, the same group of researchers did observe cross-notation distance priming later on (Naccache & Dehaene Reference Naccache and Dehaene2001b), and they argued that the interaction effect obtained previously was a “spurious effect” and that “cross-notation priming is the norm rather than the exception” (Naccache & Dehaene Reference Naccache and Dehaene2001b, pp. 234–35). We ourselves (Reynvoet & Brysbaert Reference Reynvoet and Brysbaert2004; Reynvoet et al. Reference Reynvoet, Brysbaert and Fias2002) confirmed this observation of cross-notation priming in which magnitude processing was highly automatic (i.e., short stimulus-onset asynchronies between prime and target). Furthermore, we also extended this finding by showing cross-notation priming under conditions in which the target had to be named, which implies that the numerical magnitude was completely irrelevant for the task. In addition, we also demonstrated cross-notation priming under conditions in which the formation of intentional links between different notation-dependent representations is discouraged. In a first study, the targets were always presented in the same notation, while the preceding primes, invisible to the subjects, were presented in different notations. Still, a distance priming effect was observed when notation changed (Van Opstal et al. Reference Van Opstal, Reynvoet and Verguts2005a). In a second study, investigating the lexico-semantic system of trilingual speakers, we found distance priming when masked English number word primes (e.g., “three”) were presented before a to-be-named Dutch (e.g., “twee”) or French (e.g., “deux”) number word target (Duyck et al. Reference Duyck, Depestel, Fias and Reynvoet2008). This cross-notation and cross-language priming provide convincing evidence for an abstract numerical representation under automatic conditions.

If behavioral priming supports the idea of an abstract numerical representation, how can we account for the interactions between number notations and magnitude-related effects observed using other paradigms reviewed by CK&W? It should be noted that many of the observed interactions are due to the (near) absence of the magnitude-related effects in particular conditions (e.g., Dehaene & Akhavein Reference Dehaene and Akhavein1995; Fias Reference Fias2001; Hung et al. Reference Hung, Hung, Tzeng and Wu2008; Ito & Hatta Reference Ito and Hatta2004). These null results may be caused, for example, by differences in the strength of the associations between an input notation and the corresponding magnitude and/or spatial representations (see also Brysbaert Reference Brysbaert and Campbell2005). Because of these notation-dependent associations, it could be that a magnitude-related effect is not observed for a particular notation, which results in an interaction when combined with other notations in an omnibus analysis. We believe that a better way to address the issue of notation-dependent representations is to look for positive results, and that is exactly what we did in our cross-notational priming experiments, where we found evidence for cross-notation priming in many experiments, even under highly automatic conditions.

In sum, much of the evidence in favor of a notation-dependent magnitude representation is based on a null effect in a particular condition. We reviewed a series of experiments using cross-notational priming that were based on the presence of an effect, that is, cross-notational distance priming. These results are in favor of an abstract magnitude representation and should be dealt with in any theory on numerical processing.