The concept “seven” applies just as easily to seven elephants, as seven mice, as seven apples. Numbers, as concepts, are abstract entities. “Seven” can be used to describe sets that differ vastly in their perceptual makeup because the symbol “seven” is dissociated from the sensory input (Dehaene, Reference Dehaene1992). In making the case for non-symbolic number as a genuine dimension, Clarke and Beck (C&B) argue that the approximate number system (ANS), a perceptual system, is similarly abstract. Thus, whether elephants, mice, or apples, the ANS represents number, irrespective of their physical differences. In support of this perspective, C&B argue that (1) number is independent of other magnitudes; and (2) number, unlike other magnitudes, exhibits a “second-order” property. Together, these arguments form the prerequisites for rational number.

Here, we argue that number is not (perceptually) independent of other magnitudes, nor is it unique by comparison. Moreover, we suggest that the appeal to a second-order character for number fundamentally confuses the distinction between percept and concept, such that number, as a percept (and similar to other magnitudes) is not second order.

C&B argue that congruency effects do not dispute number as an abstract entity because the interactions between number and other magnitudes reflect post-perceptual processes such as a response-stage conflict. However, other evidence, not cited by C&B, provides more definitive evidence for the non-independence of number from other magnitudes. For example, by making clever use of Müller-Lyer (Dormal, Larigaldie, Lefèvre, Pesenti, & Andres, Reference Dormal, Larigaldie, Lefèvre, Pesenti and Andres2018) and Ebbinghaus (Picon, Dramkin, & Odic, Reference Picon, Dramkin and Odic2019) illusions, researchers have demonstrated that number perception is influenced by illusory changes in non-numerical magnitudes such as length and density. Importantly, such effects were observed with estimation tasks in which participants estimated number, ruling out a response-stage conflict typical of magnitude comparison tasks, in which the response applies to both task-relevant and task-irrelevant magnitudes.

Our own research suggests that number and area are perceived holistically as integral dimensions (Aulet & Lourenco, Reference Aulet and Lourenco2021a). In Aulet and Lourenco, we found that perceived similarity for dot arrays, which varied parametrically in number and cumulative area, was best modeled by Euclidean, as opposed to city-block, distance within the stimulus space (Garner, Reference Garner, Moskowitz, Scharf and Stevens1974; Shepard, Reference Shepard1964), comparable to classically integral dimensions (e.g., brightness and saturation) but different from separable dimensions (e.g., shape and color). Importantly, results replicated across tasks and could not be explained by effects of confounding magnitudes or non-magnitude image similarity. In other words, perceived number may not be fully abstracted from co-occurring area but, instead, appears to be perceptually interdependent with it.

Relatedly, C&B claim that number is unique compared to other magnitudes – in terms of ratio and the second-order character. Others have similarly argued that number is uniquely salient (Cicchini, Anobile, & Burr, Reference Cicchini, Anobile and Burr2016; Ferrigno, Jara-Ettinger, Piantadosi, & Cantlon, Reference Ferrigno, Jara-Ettinger, Piantadosi and Cantlon2017). Recent evidence from our lab, however, goes against the uniqueness claim. For example, we found that when perceptual discriminability between number and cumulative area was matched, area biased children's number judgments more than the reverse (Aulet & Lourenco, Reference Aulet and Lourenco2021b) and children sorted visual stimuli according to area, not number (Aulet & Lourenco, Reference Aulet and Lourenco2021c), suggesting greater intrinsic salience for non-numerical magnitude, and consistent with others who have argued against the uniqueness of number (Leibovich, Katzin, Harel, & Henik, Reference Leibovich, Katzin, Harel and Henik2017; Newcombe, Levine, & Mixs, Reference Newcombe, Levine and Mix2015). Similarly, Testolin, Dolfi, Rochus, and Zorzi (Reference Testolin, Dolfi, Rochus and Zorzi2020) found that the internal encoding of “mature” computational networks, trained to discriminate stimuli according to number, treated total perimeter and convex hull as comparable to number. These effects were even more striking in “young” networks where the internal encoding was primarily driven by convex hull, not number.

C&B also posit that number, unlike other magnitudes, has a second-order character, such that the estimation of number requires stipulating what is being enumerated. For example, among a collection of shoes, number could apply to individual shoes or, alternatively, pairs of shoes. According to C&B, the representation of number is not set unless a relevant unit is specified (Burge, Reference Burge2010; Frege, Reference Frege1884). That is, a group of objects has no inherent number absent this stipulation. For the shoe example, the numerical value is n if considering individual shoes, but n/2 if considering pairs of shoes.

We agree that, when reasoning in this way, number exhibits a second-order character. C&B, however, apply this logic to the perception of number, which we would argue conflates the percept with the concept (Halberda, Reference Halberda2019). They describe “dumbbell” studies in which participants underestimate individual dots that are connected by lines to form dumbbells (e.g., Franconeri, Bemis, & Alvarez, Reference Franconeri, Bemis and Alvarez2009). According to C&B, this effect suggests a second-order character for perceived number because participants' number perception changes in the absence of changes to non-numerical properties (besides connectedness). However, if number perception was genuinely second order, then it should be just as easy to continue perceiving the number of dots, instead of being biased toward the number of dumbbells. But this is not the case! Number percepts are not as flexible as number concepts. Number perception is constrained by physical (e.g., spatial individuation) and Gestalt principles (e.g., common motion; Wynn, Bloom, & Chiang, Reference Wynn, Bloom and Chiang2002). Similar constraints (e.g., color grouping) apply to the “ratio” experiments with infants (e.g., McCrink & Wynn, Reference McCrink and Wynn2007) described by C&B. Moreover, number is perceived in accordance with these principles when arrays are passively, or even unconsciously, viewed (DeWind, Park, Woldorff, & Brannon, Reference DeWind, Park, Woldorff and Brannon2019; Fornaciai & Park, Reference Fornaciai and Park2021; Lucero et al., Reference Lucero, Brookshire, Sava-Segal, Bottini, Goldin-Meadow, Vogel and Casasanto2020; Van Rinsveld et al., Reference Van Rinsveld, Guillaume, Kohler, Schiltz, Gevers and Content2020). Accordingly, we suggest that number perception, similar to the perception of other magnitudes, is a first-order property. We can conceive and count individual shoes, or the pairs of shoes they make up, but we perceive individual shoes. We can conceive and count dumbbells, or their individual component dots, but we perceive dumbbells.

In summary, although we agree with C&B's description of the ANS as a perceptual system, we would argue that perceived number is not abstract, as it is to a conceptual system with access to symbolic representations such as number words. We have argued that perceived number may not be independent of other magnitudes and it does not appear to exhibit a unique status, including second-order character – calling into question the existence of an ANS that represents rational number.