We agree with Rips et al. that we can best understand the nature of number concepts by studying judgments about numbers (sect. 3.3). Studies of the distance and size effects have long established that our number concepts are not of an abstract logical nature, as is envisioned by the authors; instead, they remain analogue psychological representations of the quantities they reflect. More recently, it has been shown that numbers are habitually spatially coded, and this could help to explain how number concepts are generalized. We briefly review the evidence for spatial coding of numbers before explaining its relevance to the issue at hand.
The notion of a link between our mental representations of number and space dates back to Galton (Reference Galton1880) and has been highlighted by the recent surge of interest in the Spatial-Numerical Association of Response Codes (SNARC) effect. The SNARC effect entails faster left-side responses to small numbers and faster right-side responses to larger numbers, for example, when classifying them by parity or magnitude. Since its discovery by Dehaene et al. (Reference Dehaene, Bossini and Giraux1993), this pervasive association between numbers and space has been shown to affect cognitive processes from attention allocation to movement execution, in a wide range of tasks (Fischer & Mills, submitted). Importantly, the SNARC effect is automatic in nature, as it occurs even when number magnitude is task-irrelevant. Examples of such automaticity include spatial biases when searching for phonemes in a number's name (Fias et al. Reference Fias, Brysbaert, Geypens and d'Ydewalle1996) or when bisecting long strings made of either large or small digits (Fischer Reference Fischer2001). However, a number's spatial association is probably not part of its conceptual representation because its spatial mapping is rather context-dependent and changes flexibly (for a recent review, see Wood et al., in press).
The lesson from SNARC is that, while the direction of the spatial mapping of numbers onto space is flexible, the mapping itself appears to be obligatory. This fundamental constraint of our number comprehension reflects our experience that every increment in numerosity is also an increment in spatial extent, and vice versa for decrements. Such spatial coding of numerosity is more consistent than other inferential transfers from physical to mathematical operations considered by Rips et al. (sect. 4). This reliable correlation between numerosity and spatial extent in addition and subtraction might provide the missing constraint to rule out competing inferences: It suggests that children can derive the successor principle for numbers from the unlimited succession of spatial positions during object manipulation.
Much more than merely a conceptual metaphor, spatial cognition is engrained in our number comprehension, and this is also reflected in the way we first learn to understand numbers by using our fingers, an experience that still shapes the SNARC effect in adulthood (Fischer Reference Fischer2008). Spatial–numerical associations seem to emerge at around 3 years of age, well before formal schooling and reading acquisition; in Western cultures, they consist of counting from left to right, adding from left to right, and subtracting from right to left. Importantly, this spatial preference predicts performance in integer-matching and object-matching tasks with numerosities exceeding 4 (Opfer & Thompson Reference Opfer, Thompson, Sun and Miyaki2006).
Rips et al. suggest that “number concepts may come into play only at a more abstract level – for example, in arithmetic – where the focus is on the numbers themselves rather than on physical objects” (sect. 6, para. 3). It is thus important that the spatial biases observed with single numbers do indeed predict performance at the level of arithmetic. For example, adults point to a number's location farther to the right on a visually presented number line when it is the result of an addition, and farther to the left when it is the result of a subtraction (Pinhas & Fischer, in press). We predict that this spatial bias is also present in children once they have acquired a general number concept.
In conclusion, we think that a closer look at situated and embodied cognition in number acquisition can dispense of innate concepts to account for the development of number comprehension.
