In their very attractive theory on navigation in a three-dimensional world, Jeffery et al. propose that our representations of the environment are based on cognitive maps, which separately integrate horizontal and vertical dimensions of space. However, knowledge of our environment is also built through our online motor behaviours, which improve the processing of other associated dimensions to flexibly adapt future responses. From this perspective, it has been suggested that a common metrics mechanism associated with sensorimotor experience is in charge of processing different magnitude dimensions, such as space, time, and numbers (Walsh Reference Walsh2003). My goal in this commentary is to suggest that a cognitive map based only on three-dimensional spaces might be an incomplete picture if abstract semantic dimensions are not fully considered. Accordingly, the representation of number magnitudes can also interact with the representation of the horizontal and vertical dimensions of space during performed actions and contribute to a global map of the three-dimensional world representation.

Evidence for a link among physical space, actions, and numbers comes from studies that show the association between small numbers with left and lower parts of space and large numbers with right and upper parts (Hubbard et al. Reference Hubbard, Piazza, Pinel and Dehaene2005; Umiltà et al. Reference Umiltà, Priftis and Zorzi2009). For instance, Dehaene et al. (Reference Dehaene, Bossini and Giraux1993) found that small number processing can prime hand movement to the left part of space, and large numbers can prime movement to the right part. Schwarz and Keus (Reference Schwarz and Keus2004) revealed an association between the vertical dimension and eye movements – that is, downward and upward saccades were initiated more quickly in response to small and large numbers, respectively. Interestingly, Loetscher et al. (Reference Loetscher, Bockisch, Nicholls and Brugger2010) found that during a random number generation task, the leftward and downward adjustment of eye locations predicted that the to-be-spoken number would be smaller than the last one. Conversely, a prediction of large numbers was made through the right and upward location of eyes. Altogether, these number-space associations occur in an automatic way, and magnitude representation could be represented as a number map that integrates spatial dimensions and actions (Schwarz & Keus Reference Schwarz and Keus2004; Walsh Reference Walsh2003). At a neurophysiological level, number magnitudes and three-dimensional Cartesian coordinates of external space are processed in a common parietal area (Hubbard et al. Reference Hubbard, Piazza, Pinel and Dehaene2005). It is probable that daily life experience, such as adding more objects to a pile (for the vertical dimension), or cultural factors, such as the direction of writing (for the horizontal dimension), could partially be in charge of the link between numbers and spatially directed actions (Gevers et al. Reference Gevers, Lammertyn, Notebaert, Verguts and Fias2006). Another factor could also come from the fact that during infancy, counting strategies often involve finger movements, which in turn reinforce the number-space association through sensorimotor experience (Butterworth Reference Butterworth1999; Michaux et al. Reference Michaux, Pesenti, Badets, Di Luca and Andres2010).

Based on sensorimotor accounts, Walsh's ATOM (“A Theory of Magnitude”; Walsh Reference Walsh2003) proposes that all magnitude dimensions (e.g., space, time, numbers, and lengths) are processed inside a common metrics mechanism located in the parietal cortex (Bueti & Walsh Reference Bueti and Walsh2009). Note that neurons in the parietal cortex of different animal species, such as cats and macaque monkeys, can also respond to number processing (Nieder & Miller Reference Nieder and Miller2004; Thompson et al. Reference Thompson, Mayers, Robertson and Patterson1970). Importantly, the core assumption of ATOM is that we represent space and time through environment-directed actions. As stated by Walsh (Reference Walsh2003), “the inferior parietal cortex reflects the common need for space, time and quantity information to be used in the sensorimotor transformations that are the main goal of these areas of cortex” (p. 483). It is worth noting that the parietal region is also strongly involved in route-based navigation in primates, especially in the integration of self-movement information (Sato et al. Reference Sato, Sakata, Tanaka and Taira2006). From an evolutionary viewpoint, one may wonder why such common metrics mechanisms exist. The most probable, straightforward answer is that the brains of human and nonhuman animals are mainly shaped for anticipating upcoming events in the environment (Corballis Reference Corballis2013; Hommel et al. Reference Hommel, Müsseler, Aschersleben and Prinz2001; Suddendorf & Corballis Reference Suddendorf and Corballis2007). On this view, we have recently revealed that the mere processing of number magnitudes is automatically associated with a general sensorimotor and anticipative mechanism (Badets et al. Reference Badets, Koch and Toussaint2013). In this theory, distal references in the environment provide information of different magnitudes that is simulated in an anticipative way to flexibly adapt future numerical- and motor-based responses (see Badets & Pesenti [Reference Badets and Pesenti2011] for this “anticipated-magnitude code”).

Such anticipative mechanisms for sensorimotor adaptations are well documented in the literature on motor control (Hommel et al. Reference Hommel, Müsseler, Aschersleben and Prinz2001). For example, envisage a person who wishes to move a bottle of wine from the table to the upper part of the kitchen shelf. During this motor sequence, there is (1) the processing of a horizontal dimension for the reach-to-grasp movement of the hand towards the bottle on the table, and subsequently, (2) the processing of a vertical dimension for the placement of the bottle on the upper part of the shelf. According to Jeffery and colleagues, both dimensions (i.e., horizontal and vertical) should be processed and represented separately. However, data on similar paradigms revealed that when the final placement was high on the shelf (vertical goal), the reach-to-grasp position of the hand (horizontal goal) was situated in the lower part of the object (Cohen & Rosenbaum Reference Cohen and Rosenbaum2004). This finding indicates that the metric encoding the vertical goal is concomitantly anticipated and accurately represented during enactment of the horizontal goal.

In summary, based on several lines of evidence that a common metrics mechanism located in the parietal region of the brain is in charge of the processing of space, numbers, and actions, I propose that different magnitude dimensions are most likely intertwined with the horizontality and verticality of space during environment-directed actions. Semantic knowledge, such as the meaning of numbers, is represented inside this common scale and can refine the representation of physical space. In other words, semantic sides of three-dimensional space representation could be anticipatorily activated in a sensorimotor mechanism, which could give us the capacity to adapt different behaviours for potential environmental constraints.