We agree with Clarke and Beck (C&B) that the approximate number system (ANS) genuinely represents numbers of a familiar sort, including natural and rational numbers. We systematically demonstrate our support based on the special case of an empty set – the non-symbolic manifestation of zero. An empty set is particularly interesting because of its perceptual and semantic uniqueness (e.g., Nieder, Reference Nieder2016; Zaks-Ohayon, Pinhas, & Tzelgov, Reference Zaks-Ohayon, Pinhas and Tzelgov2021a, Reference Zaks-Ohayon, Pinhas and Tzelgov2021b), which makes it a special case study in favor of the claim that the ANS represents rational numbers.

C&B reject arguments relating to congruency, confounds, and imprecision that claim the ANS fails to represent numbers. As expected, and in accord with the vast literature on the ANS, their position is supported by data and examples that are based on the use of non-null quantities that, regardless of their specific numerical values, are presented as “something” (i.e., at least one object). This is not the case for an empty set, however, which corresponds to null quantity or “nothing,” and is presented as a form without content. In turn, when a frame containing an array of dots is compared with an empty frame, the perceptual uniqueness of the latter creates built-in, unavoidable confounds between the stimuli's discrete dimension – numerical magnitude – and its continuous non-numerical dimensions. Indeed, five continuous non-numerical stimuli dimensions were reported to interact with the numerical magnitude: the convex hull, the total surface area occupied by the dots, the density of the dots, the dot diameter, and the total dot circumference (for reviews, see Gebuis, Cohen Kadosh, & Gevers, Reference Gebuis, Cohen Kadosh and Gevers2016; Leibovich, Katzin, Harel, & Henik, Reference Leibovich, Katzin, Harel and Henik2017). In comparison with empty sets, each of these continuous non-numerical stimuli dimensions is confounded with the numerical magnitude in a congruent manner. For instance, the total surface area occupied by dots will always be larger in the non-empty set that contains one or more dots than in the empty set that does not, consistent with the non-empty set being numerically larger. Because these confounds are inherent to comparisons between empty and non-empty sets, they cannot be experimentally manipulated or controlled for. Moreover, contrary to the processing of non-empty sets, it would be hard to argue that the ANS represents an empty set imprecisely because if the frame is empty, it contains no objects and there is nothing to enumerate or estimate. In that sense, the perceptual prominence of empty sets presumably leads to a precise numerical evaluation that resembles subitizing (Kaufman, Lord, Reese, & Volkmann, Reference Kaufman, Lord, Reese and Volkmann1949), the quick and accurate identification process for a small number (from 1 to 4) of objects.

Despite the inherent confounds in comparison with empty sets and the distinctive characteristics of null quantity, comprehensive behavioral and neural research on human and nonhuman animals (e.g., Beran, Perdue, & Evans, Reference Beran, Perdue, Evans, Cohen Kadosh and Dowker2015; Biro & Matsuzawa, Reference Biro and Matsuzawa2001; Howard, Avarguès-Weber, Garcia, Greentree, & Dyer, Reference Howard, Avarguès-Weber, Garcia, Greentree and Dyer2018; Merritt & Brannon, Reference Merritt and Brannon2013; Merritt, Rugani, & Brannon, Reference Merritt, Rugani and Brannon2009; Okuyama, Kuki, & Mushiake, Reference Okuyama, Kuki and Mushiake2015; Pepperberg & Gordon, Reference Pepperberg and Gordon2005; Ramirez-Cardenas, Moskaleva, & Nieder, Reference Nieder2016; Zaks-Ohayon et al., Reference Zaks-Ohayon, Pinhas and Tzelgov2021a, Reference Zaks-Ohayon, Pinhas and Tzelgov2021b) has shown that empty sets can be mapped onto the ANS together with non-empty sets. Specifically, comparisons between empty and non-empty sets result in a distance effect (Moyer & Landauer, Reference Moyer and Landauer1967) and an end effect (Banks, Reference Banks and Bower1977; Leth-Steensen & Marley, Reference Leth-Steensen and Marley2000), both of which are considered markers for numeric representation. Accordingly, response latencies decrease with the increase in the numerical value of the non-empty set that is being compared to empty set, and comparisons between empty and non-empty sets are responded to faster than comparisons of non-empty sets, respectively (e.g., Merritt et al., Reference Merritt, Rugani and Brannon2009; Merritt & Brannon, Reference Merritt and Brannon2013; Zaks-Ohayon et al., Reference Zaks-Ohayon, Pinhas and Tzelgov2021a, Reference Zaks-Ohayon, Pinhas and Tzelgov2021b). Furthermore, single-cell recordings from the ventral parietal and prefrontal cortex of monkeys (Macaca fuscata and Macaca mulatta) led to identifying two different types of “number neurons” that were selectively activated in response to empty sets. One was an exclusive type, showing increased activity selective to empty sets and decreased activity to non-empty sets, and the other a continuous type, showing maximum activity in response to empty sets and gradually decreased activity to successively larger non-empty sets (Okuyama et al., Reference Okuyama, Kuki and Mushiake2015; Ramirez-Cardenas et al., Reference Ramirez-Cardenas, Moskaleva and Nieder2016). Number-selective neurons of the second type were also previously reported for non-null quantities (e.g., Nieder, Reference Nieder2013).

Next, we turn to C&B's question of what kind of numbers can be represented by the ANS. Although we have previously shown that, psychologically, the number 0 can be perceived as a natural number (Pinhas et al., Reference Pinhas, Buchman, Lavro, Mesika, Tzelgov and Berger2015; Pinhas & Tzelgov, Reference Pinhas and Tzelgov2012), it is not considered as such in mathematical terms because it is neither a positive nor a negative integer. However, mathematically, zero can be expressed as the ratio a/b, if it serves as the numerator a, and therefore, fits the definition of a rational number, consistent with the type of numbers that are represented by the ANS according to C&B. More generally, when considering what kind(s) of numbers can be represented by the ANS, the inclusion of zero breaks down the orderly relationship that exists between a number's ordinality (i.e., its position in the number sequence) and cardinality (i.e., its numerical value) when only natural numbers are considered (Seife, Reference Seife2000). Accordingly, if only dealing with natural numbers, 1 is the first, 2 is the second, 3 is the third, and so on. However, when 0 is also included, 0 is the first, 1 is the second, 2 is the third, and so on. Thus, the fact that the ANS represents empty sets as zero indicates that the ordinality and cardinality properties of numbers are no longer interchangeable, and presumably sets the stage for more “complex” forms of numbers to be represented by the ANS.

Clearly, further research is still needed to fully characterize the nature of the representations captured by the ANS. However, by highlighting the case of the empty set, we hope to inspire future research focused on other unique numerical concepts that, similar to zero, may reveal something fundamental about the way numerical information is represented.