Hoerl & McCormack (H&M) describe two systems that supposedly differ in the processing of temporal information and the underlying representation of time. We endorse this notion and propose that the dual-systems approach is not restricted to the dimension of time. The basic assumption is that time, as well as space and number, is part of a generalized magnitude system (Walsh Reference Walsh2003). Therefore, if we adopt the view of a generalized system for magnitude processing and, at the same time, accept the proposed dual-systems approach to account for the domain of temporal cognition, then the two systems should also apply to other domains of magnitude processing. In the following, we give examples of processes in numerical cognition that might correspond to those processes that H&M ascribe to the temporal updating system, an intermediate phase, and the temporal reasoning system.

The development of basic and advanced numerical knowledge in humans is assumed to rely on an evolutionarily ancient innate system dedicated to extracting and representing approximate numerical magnitude information (Amalric & Dehaene Reference Amalric and Dehaene2016; Feigenson et al. Reference Feigenson, Dehaene and Spelke2004; Piazza Reference Piazza2010; Starr et al. Reference Starr, Libertus and Brannon2013). Recent meta-analyses support this view by showing a significant association between approximate numerical magnitude processing skills and symbolic math performance (Chen & Li Reference Chen and Li2014; Fazio et al. Reference Fazio, Bailey, Thompson and Siegler2014; Schneider et al. Reference Schneider, Beeres, Coban, Merz, Susan Schmidt, Stricker and De Smedt2017). We suggest that sensitivity for approximate number might be interpreted analogously to the elapsed-time sensitivity in temporal cognition, which has been described as a mechanism of the temporal updating system. According to theoretical models on the development of numerical concepts, children learn the counting sequence by rote before they understand the numerical meaning of number words (Fuson Reference Fuson1988; Krajewski & Schneider Reference Krajewski and Schneider2009a; Reference Krajewski and Schneider2009b). Consistently, empirical evidence suggests that children are able to recite the counting sequence before they know the exact cardinal meanings of all numerals in that sequence (Le Corre et al. Reference Le Corre, Van de Walle, Brannon and Carey2006). In our opinion, children's early ability to recite the counting sequence is similar to their sensitivity for recurring event sequences. By analogy with mechanisms that H&M ascribe to the more primitive updating system in the temporal domain, we suggest that both the processing of approximate numerical magnitudes and the ability to reproduce the counting sequence do not necessarily involve a mature and flexible concept of number.

At a further stage of numerical development, children assumingly become aware that number words are linked to quantities. Children at this stage rightly decide which of two number words (e.g., “five” vs. “three”) represents more or less, possibly without being able to represent the exact difference between these numbers (Krajewski & Schneider Reference Krajewski and Schneider2009a; Reference Krajewski and Schneider2009b). This might be analogous to what H&M have described as an intermediate phase for the temporal domain; young children discriminate past and future points in time relative to the present without having access to the specific temporal relations between them.

The insight that differences between numbers also consist in numbers is seen as an important step toward complex reasoning about numerical magnitudes (Fuson Reference Fuson1988; Krajewski & Schneider Reference Krajewski and Schneider2009a; Reference Krajewski and Schneider2009b). It enables children to work with specific distances between numbers, which corresponds to the key characteristic of temporal reasoning as proposed by H&M. They suggest that temporal reasoning operates on the basis of a temporal concept that includes unique addresses for different points in time as well as the temporal-causal relations between these points. Furthermore, they argue that temporal reasoning relies on a spatial representation of time in the form of a timeline. Similarly, numerical reasoning is assumed to rely on the flexible manipulation of numerical magnitudes on a mental number line (Dehaene Reference Dehaene1992; Siegler & Braithwaite Reference Siegler and Braithwaite2017). Apparently, flexible reasoning about both temporal events and numerical magnitudes requires a concept of the underlying principle that allows any event (or number) to be logically related to another.

From our point of view, the outlined similarities in children's processing of time and number endorse the application of the proposed dual-systems approach to numerical cognition. It has to be noted, though, that the theoretical models of numerical development cited above assume fine-grained competence levels that build upon each other. In contrast, the dual-systems approach distinguishes only two cognitive systems. However, H&M also acknowledge that premature forms of the temporal reasoning system exist, which they describe as part of an intermediate phase in development. Further specification of the proposed two systems and their associated mechanisms is needed to understand their interplay and possible transitional stages throughout development. Our proposition that H&M's approach might apply to other dimensions of magnitude processing certainly requires validation by future investigations. In our view, systematic comparisons of children's competencies across dimensions and across development will provide a valuable contribution to the debate about the nature of the cognitive system, or the systems, required for the processing of magnitude information.