We agree that the approximate number system (ANS) represents number and aim to clarify theoretical arguments that are entangled in questions about terminology. What do we mean by area, number, numerosity, and representation?
Although the authors are right that the “argument from congruency” and the “argument from confounds” ultimately fail, some evidence bolstering those arguments is shaky in the first place. When we use physical area to infer the relative contributions of continuous and discrete stimulus properties to quantity judgments, we're neglecting a history of psychophysical evidence that perceived and physical area differ (Barth, Reference Barth2008). Empirical support for this idea came from three experiments in which cumulative area judgments were driven by perceived, not physical, area. Some “arguments from congruency” depend on interpretations based on physical area (e.g., Hurewitz, Gelman, & Schnitzer, Reference Hurewitz, Gelman and Schnitzer2006; Rousselle & Noël, Reference Rousselle and Noël2008). Yet quantity judgments can yield apparent congruency effects that disappear when perceived area is considered instead of physical area (Barth, Reference Barth2008). Incorporating perceived area won't resolve controversies surrounding discrete versus continuous quantity (see Aulet & Lourenco, Reference Aulet and Lourenco2021; Savelkouls & Cordes, Reference Savelkouls and Cordes2020; Yousif & Keil, Reference Yousif and Keil2020). Nevertheless, to identify processes underlying quantity judgments, subjective magnitudes should be explored as potential behavior cues. Otherwise, we'll get the wrong idea about whether number influences area or vice versa, or both, or neither.
We also have to be clear on the terms “number” and “numerosity.” We were surprised at the authors' lengthy condemnation of “numerosity.” In our usage, “numerosity” refers to a property of a stimulus, not a representation. An array of dots (or string of sounds) has a numerosity. That numerosity is larger when the elements are more numerous. If we've used the phrase “numerosity representation,” we weren't referring to woolly “number-like properties” (Burge, Reference Burge2010). We meant “mental representation of number that refers to the numerosity of a stimulus.” It's not a hedge – it's shorthand.
Do other psychologists share our understanding of what “numerosity” means, in which case the target article is simply wrong that our language implies “an assumption that, strictly speaking, the ANS represents numerosities, not numbers” (Clarke & Beck [C&B], sect. 6, para. 4)? Or, are psychologists' uses of “numerosity” inconsistent? We think C&B (and Burge) are wrong about what “numerosity” means to researchers, but either way they've done a service in exposing this confusion, and the field had better get clear about what it does mean.
That said, dropping “numerosity” for “number” isn't the answer. “Number” is ambiguous, and ambiguity breeds confusion. “Number” can refer to number words and Arabic numerals (i.e., symbols for natural numbers) or a property of stimuli (i.e., numerosity) or mathematical entities. For psychologists, it is useful to have a term that unambiguously refers to the number of items in a stimulus. “Numerosity” allows psychologists to discuss discrete quantity without endorsing commitments about how it is represented in the mind.
The target article itself suffers from terminological confusion, over “number” and, at times, “representation.” Use of “number” when the authors appear to intend “natural number” frequently obscures their meaning. (We spent considerable time decoding what was meant by each instance of “number”!) And C&B seem to answer claims about what is made explicit by a representational system with arguments about the contents of representations within that system. For example, Carey (Reference Carey2009) argued that the ANS as a representational system cannot grant natural number concepts to an organism. But this is not a critique of the idea that ANS representations have true numerical content! Carey (Reference Carey2009) is clear that the ANS represents number: “that analog magnitude representations constitute one system of number representations deployed by human adults has been established beyond any reasonable doubt” (p. 131) and “analog magnitudes are explicit symbols of approximate cardinal values of sets” (p. 135). Carey's argument doesn't attempt “to undermine the hypothesis that the ANS represents number” (C&B, sect. 5, para. 1).
Furthermore, C&B wave away the question of “modes of presentation,” arguing that the same property under different modes of presentation is still the same property. Therefore, they argue, the word “number” should suffice to describe that property. For psychologists, however, mode of presentation is not an afterthought. How do different representations of identical aspects of the world map on to each other in the mind? Which modes of presentation subserve word learning, computational tasks, and behavior?
When we ask a question like “where do human number concepts come from,” we see that the use of a single word like “number” elides questions of interest. The ANS as a representational system does not encode exactness or the successor function, essential components of natural number. This limitation is important in querying what roles the ANS can play in learning. We concur that ANS representations don't serve as the conceptual source of precise number concepts (Carey & Barner, Reference Carey and Barner2019), and empirical evidence indicates that children don't learn number word meanings via mappings to ANS representations (Carey, Shusterman, Haward, & Distefano, Reference Carey, Shusterman, Haward and Distefano2017). The fact that the ANS encodes some aspects of number (e.g., its second-order character), but not others (e.g., exactness), highlights the importance of using more specific terminology to clarify which aspects of number, and which properties of relevant representations, are under discussion. The authors' push to use the term “number” promiscuously has a muddying effect rather than a clarifying one.
Eronen and Bringmann (Reference Eronen and Bringmann2021) argue that theory development in psychology suffers, in part, from “the relative lack of robust phenomena that impose constraints on possible theories” and “problems of validity of psychological constructs.” Numerical cognition is rich in robust phenomena, and construct validity is coming along. But we have an enduring terminology tangle. Carey (Reference Carey2009) wrote: “It then becomes a merely terminological matter whether one wants to use the term ‘number’ only for natural number or for the integers or for the integers plus the rationals plus the reals (in which case there is no core cognition of number) and adopt some other term for the quantificational content of core cognition systems” (p. 297). Maybe it's not so “merely” after all.
We agree that the approximate number system (ANS) represents number and aim to clarify theoretical arguments that are entangled in questions about terminology. What do we mean by area, number, numerosity, and representation?
Although the authors are right that the “argument from congruency” and the “argument from confounds” ultimately fail, some evidence bolstering those arguments is shaky in the first place. When we use physical area to infer the relative contributions of continuous and discrete stimulus properties to quantity judgments, we're neglecting a history of psychophysical evidence that perceived and physical area differ (Barth, Reference Barth2008). Empirical support for this idea came from three experiments in which cumulative area judgments were driven by perceived, not physical, area. Some “arguments from congruency” depend on interpretations based on physical area (e.g., Hurewitz, Gelman, & Schnitzer, Reference Hurewitz, Gelman and Schnitzer2006; Rousselle & Noël, Reference Rousselle and Noël2008). Yet quantity judgments can yield apparent congruency effects that disappear when perceived area is considered instead of physical area (Barth, Reference Barth2008). Incorporating perceived area won't resolve controversies surrounding discrete versus continuous quantity (see Aulet & Lourenco, Reference Aulet and Lourenco2021; Savelkouls & Cordes, Reference Savelkouls and Cordes2020; Yousif & Keil, Reference Yousif and Keil2020). Nevertheless, to identify processes underlying quantity judgments, subjective magnitudes should be explored as potential behavior cues. Otherwise, we'll get the wrong idea about whether number influences area or vice versa, or both, or neither.
We also have to be clear on the terms “number” and “numerosity.” We were surprised at the authors' lengthy condemnation of “numerosity.” In our usage, “numerosity” refers to a property of a stimulus, not a representation. An array of dots (or string of sounds) has a numerosity. That numerosity is larger when the elements are more numerous. If we've used the phrase “numerosity representation,” we weren't referring to woolly “number-like properties” (Burge, Reference Burge2010). We meant “mental representation of number that refers to the numerosity of a stimulus.” It's not a hedge – it's shorthand.
Do other psychologists share our understanding of what “numerosity” means, in which case the target article is simply wrong that our language implies “an assumption that, strictly speaking, the ANS represents numerosities, not numbers” (Clarke & Beck [C&B], sect. 6, para. 4)? Or, are psychologists' uses of “numerosity” inconsistent? We think C&B (and Burge) are wrong about what “numerosity” means to researchers, but either way they've done a service in exposing this confusion, and the field had better get clear about what it does mean.
That said, dropping “numerosity” for “number” isn't the answer. “Number” is ambiguous, and ambiguity breeds confusion. “Number” can refer to number words and Arabic numerals (i.e., symbols for natural numbers) or a property of stimuli (i.e., numerosity) or mathematical entities. For psychologists, it is useful to have a term that unambiguously refers to the number of items in a stimulus. “Numerosity” allows psychologists to discuss discrete quantity without endorsing commitments about how it is represented in the mind.
The target article itself suffers from terminological confusion, over “number” and, at times, “representation.” Use of “number” when the authors appear to intend “natural number” frequently obscures their meaning. (We spent considerable time decoding what was meant by each instance of “number”!) And C&B seem to answer claims about what is made explicit by a representational system with arguments about the contents of representations within that system. For example, Carey (Reference Carey2009) argued that the ANS as a representational system cannot grant natural number concepts to an organism. But this is not a critique of the idea that ANS representations have true numerical content! Carey (Reference Carey2009) is clear that the ANS represents number: “that analog magnitude representations constitute one system of number representations deployed by human adults has been established beyond any reasonable doubt” (p. 131) and “analog magnitudes are explicit symbols of approximate cardinal values of sets” (p. 135). Carey's argument doesn't attempt “to undermine the hypothesis that the ANS represents number” (C&B, sect. 5, para. 1).
Furthermore, C&B wave away the question of “modes of presentation,” arguing that the same property under different modes of presentation is still the same property. Therefore, they argue, the word “number” should suffice to describe that property. For psychologists, however, mode of presentation is not an afterthought. How do different representations of identical aspects of the world map on to each other in the mind? Which modes of presentation subserve word learning, computational tasks, and behavior?
When we ask a question like “where do human number concepts come from,” we see that the use of a single word like “number” elides questions of interest. The ANS as a representational system does not encode exactness or the successor function, essential components of natural number. This limitation is important in querying what roles the ANS can play in learning. We concur that ANS representations don't serve as the conceptual source of precise number concepts (Carey & Barner, Reference Carey and Barner2019), and empirical evidence indicates that children don't learn number word meanings via mappings to ANS representations (Carey, Shusterman, Haward, & Distefano, Reference Carey, Shusterman, Haward and Distefano2017). The fact that the ANS encodes some aspects of number (e.g., its second-order character), but not others (e.g., exactness), highlights the importance of using more specific terminology to clarify which aspects of number, and which properties of relevant representations, are under discussion. The authors' push to use the term “number” promiscuously has a muddying effect rather than a clarifying one.
Eronen and Bringmann (Reference Eronen and Bringmann2021) argue that theory development in psychology suffers, in part, from “the relative lack of robust phenomena that impose constraints on possible theories” and “problems of validity of psychological constructs.” Numerical cognition is rich in robust phenomena, and construct validity is coming along. But we have an enduring terminology tangle. Carey (Reference Carey2009) wrote: “It then becomes a merely terminological matter whether one wants to use the term ‘number’ only for natural number or for the integers or for the integers plus the rationals plus the reals (in which case there is no core cognition of number) and adopt some other term for the quantificational content of core cognition systems” (p. 297). Maybe it's not so “merely” after all.
Financial support
This work was supported by the National Science Foundation (HB: NSF DRL-1920445; AS: NSF DRK12-2010547)
Conflict of interest
None.