The review of Cohen Kadosh & Walsh (CK&W) in the target article demonstrates that numerical representations are modulated by task demands. The authors propose the existence of two cognitive codes for numbers, that is, (1) an automatic non-abstract representation, which is notation and modality dependent, and (2) an intentional abstract representation, which is notation and modality independent. We agree with this hypothesis. However, we think that a dual-code model of number representation needs to take into account further aspects about the differences in the nature of the two representations and should theoretically distinguish between an asemantic representation of the symbolic stimulus (e.g., Arabic numeral) and the representation of its associated meaning (e.g., magnitude information). The controversy about abstract magnitude representation in the parietal lobes relates, therefore, directly to the ongoing debate about symbol grounding.

Interestingly, CK&W's model of number processing seems to be inspired by the recently suggested Language and Situated Simulation (LASS) theory of Barsalou et al. (Reference Barsalou, Santos, Simmons, Wilson, De Vega, Glenberg and Graesser2008) and their proposal of dual codes in embodied representations of conceptual knowledge. Indeed, both models hold that automatic and intentional stimulus processing results in different types of cognitive representations and assume that a deeper deliberate processing is central to the semantic representation of abstract concepts. We believe that both notions provide an interesting framework that contributes directly to the question of whether magnitude representations are abstract or non-abstract. However, we want to point out some substantial theoretical differences between the current model of number processing and dual-code models on conceptual knowledge in psycholinguistic research.

First, most dual-code models of semantic processing view the faster emerging automatic representation as an asemantic stimulus coding that does not go beyond a perceptual, linguistic, or phonological representation of the presented word or number symbol (see, e.g., Barsalou et al. Reference Barsalou, Santos, Simmons, Wilson, De Vega, Glenberg and Graesser2008; Mahon & Caramazza Reference Mahon and Caramazza2008). The target article shows that effects of magnitude processing are modulated by the notation of the number symbol if and only if the task does not require any semantic processing of the numerical magnitude information. Taking a closer look at studies on automatic processing that provide evidence in favor of non-abstract number representations, it becomes clear that effects of number meaning are smaller or even nonexistent if the number stimulus is perceptually more complex (e.g., number words vs. digits: Cohen Kadosh Reference Cohen Kadosh2008a; or Japanese Kanji numbers vs. Kana numbers: Ito & Hata Reference Ito and Hatta2003) or printed in an unfamiliar notation (e.g., Indian vs. Arabic digits: Ganor-Stern & Tzelgov Reference Ganor-Stern and Tzelgov2008). Moreover, CK&W argue that because automatic processing is unaffected by intentional strategies, indirect number tasks provide an unbiased inside look into the nature of the representation of numerical magnitude information. However, an alternative reason for the presence of notation-dependent effects in these tasks might be that automatic processing of number words and unfamiliar symbols is likely to be restricted to a superficial stimulus coding without activation of the underlying numerical magnitude information. We consequently assume, in line with dual-code models of conceptual knowledge (Barsalou et al. Reference Barsalou, Santos, Simmons, Wilson, De Vega, Glenberg and Graesser2008), that under conditions of automatic processing, complex and unfamiliar number symbols will be merely transcoded and represented asemantically in a perceptual, linguistic, or phonological format. It is therefore questionable whether the automatic non-abstract number representation proposed by CK&W can be understood as a semantic representation of numerical magnitude information. Alternatively, the transition from an automatic to an intentional numerical representation might be better conceptualized by different levels of semantic processing and the translation of a perceptual representation of the number symbol into a semantic representation of the associated magnitude.

Second, a central aspect of the LASS theory of Barsalou et al. (Reference Barsalou, Santos, Simmons, Wilson, De Vega, Glenberg and Graesser2008) is the assumption that abstract symbols such as words and numbers become meaningful only when they are somehow mapped to concrete bodily experiences (the so-called embodied cognition hypothesis; for reviews see, e.g., Fischer & Zwaan Reference Fischer and Zwaan2008). Following this view, intentional number processing consists of an activation of correlated information in brain areas that originally evolved to process non-symbolic stimuli, that is, perceptual and motor areas. This embodied mechanism of conceptual knowledge representation has been described as perceptual and motor resonance (Rueschemeyer et al., in press a) and represents a bidirectional coupling of semantic representations with processes of action planning and motor control. Interestingly, empirical evidence for such a sensorimotor grounding of symbol meaning has not only been reported by studies on word processing (Glenberg & Kaschak Reference Glenberg and Kaschak2002; Lindemann et al. Reference Lindemann, Stenneken, van, H and Bekkering2006; Rueschemeyer et al., in press b; Zwaan & Taylor Reference Zwaan and Taylor2006), but has also been recently shown for the processing of Arabic digits in numerical tasks (Andres et al. Reference Andres, Davare, Pesenti, Olivier and Seron2004; Fischer Reference Fischer2008; Lindemann et al. Reference Lindemann, Abolafia, Girardi and Bekkering2007). One might therefore speculate that the abstraction of number meaning consists of an association between the numerical information and magnitude-related motor codes – an assumption that is in line with the idea that magnitude representations for numbers and actions share common cognitive codes within a generalized system for size and quantity (Walsh Reference Walsh2003).

Since the notion of embodied cognition is one of the most controversially debated hypotheses of recent psycholinguistic dual-code models of conceptual processing, the principle of motor resonance should also be addressed in the context of number processing models. In conclusion, we think that the idea of an action-based number semantics might provide new insights into the nature of numerical magnitude representations, and might thereby fruitfully contribute to future theoretical developments in mathematical cognition research.