The authors (hereafter, C&B) make an interesting argument for the consideration that the approximate number system (ANS) represents number rather than numerosity. The clarity in disentangling what is represented versus how it is represented is also a valuable contribution to the field. However, there are some issues that the argument leaves unresolved despite being critical to the nature of the ANS. A relevant one is the distinction between the representation and processing of non-symbolic and symbolic number. This is a well-established distinction in the numerical cognition literature, whose dismissal – for example, by reducing it to a matter of how number is represented – would be overlooking its depth and implications. Characterizing the ANS as a “primitive and prelinguistic capacity” shared across many species is a clear signal that C&B focus their case on the processing of non-symbolic number. However, the evidence cited mixes data from non-symbolic and symbolic studies (e.g., Henik & Tzelgov, Reference Henik and Tzelgov1982), blurring an otherwise clear and engaging argument. Although C&B distinguish in their exposition between ANS representations and precise number concepts, this distinction is intended to separate ANS representations from the advanced number constructions studied by mathematics, and it does not address the non-symbolic/symbolic contrast.
Myer and Landauer's (Reference Moyer and Landauer1967) study on single-digit number comparison may be considered an essential piece of evidence suggesting that number symbols are to some extent represented by the ANS. These authors showed that young adults' error rates and response times in comparing two digits decrease with increasing numerical distance between them. However, the consideration of these two types of numerical processing, non-symbolic and symbolic, brings an extra layer of complexity to number processing. For instance, symbolic number processing introduces unit-decade-compatibility effects which are meaningless in non-symbolic processing. Nuerk, Weger, and Willmes (Reference Nuerk, Weger and Willmes2001) presented this concept and proved that the comparison of multi-digit numbers is strongly affected by a competition between the numbers and the digits that compose them: Comparing 42 and 57 is easier than comparing 47 and 62 because in the former case the larger number coincides with the number with the larger decade and unit, whereas in the latter case the larger number has the larger decade but the smaller unit. Effects such as this one are specific to symbolic number representations, as the comparison of a set of 42 blue dots versus one of 57 yellow dots will likely lead to a similar outcome to that of a set of 47 blue dots versus one of 62 yellow dots (both comparisons engage the ANS and the yellow:blue ratios in each case are about 1.3). Altogether, the ANS seems able to represent number symbols, but this representation would be limited to single-digit numbers (see also Nuerk, Moeller, Klein, Willmes, & Fischer, Reference Nuerk, Moeller, Klein, Willmes and Fischer2011).
When it comes to rational numbers, the distinction between non-symbolic and symbolic processing becomes even more complex, and the research scarcer. Non-symbolic processing in this case refers to the capacity of perceiving and using ratios, whereas symbolic processing brings to the table fractions and decimals. Fractions are visually depicted as two natural numbers separated by a line. There is plenty of evidence that human adults perceive the magnitudes of natural numbers in an automatic manner (i.e., even if it is not relevant for the task, see Henik & Tzelgov, Reference Henik and Tzelgov1982). The magnitudes of fractions, however, seem to be activated only when they are relevant for the task (Bonato, Fabbri, Umiltà, & Zorzi, Reference Bonato, Fabbri, Umiltà and Zorzi2007; Gabriel, Szucs, & Content, Reference Gabriel, Szucs and Content2013; Kallai & Tzelgov, Reference Kallai and Tzelgov2012). Children's intuitive reasoning with fractions show important congruency effects (erroneously judging a fraction as larger than another if its components are larger, e.g., concluding that 2/3 < 4/9 because 2 < 3 and 4 < 9; see e.g., Gómez & Dartnell, Reference Gómez and Dartnell2019; Ni & Zhou, Reference Ni and Zhou2005; Van Hoof, Lijnen, Verschaffel, & Van Dooren, Reference Van Hoof, Lijnen, Verschaffel and Van Dooren2013). It is difficult to ascribe fraction comparison performance to the ANS, however. Although some studies have reported distance effects in response times in fraction comparison tasks, these times are too large to license conclusions about the mental representations of fractions (e.g., Schneider & Siegler, Reference Schneider and Siegler2010) or the task is to simple to actually engage fraction representations (e.g., Bonato et al., Reference Bonato, Fabbri, Umiltà and Zorzi2007). Nonetheless, adults who are highly mathematically competent also show congruency effects in their response times but also distance effects (Morales, Dartnell, & Gómez, Reference Morales, Dartnell and Gómez2020; Obersteiner, Van Dooren, Van Hoof, & Verschaffel, Reference Obersteiner, Van Dooren, Van Hoof and Verschaffel2013), showing that the discussion is far from over (see also Binzak & Hubbard, Reference Binzak and Hubbard2020, for positive evidence for ANS involvement in fraction comparison).
Ratios (or non-symbolic rational numbers) as a unifying percept of the number sense would be a very compelling theory (e.g., with natural numbers represented as ratios with respect to one). It would be consistent with the measuring function of numbers, common to both natural and rational numbers. As C&B note, natural number counting is not essentially tied to one as the unit, as counting can occur by pairs, tens, or dozens. But, even in this scenario, ratios are a limited aspect of rational numbers and, similarly, non-symbolic numbers are a limited aspect of natural numbers. Although C&B are convincing about the ANS representing non-symbolic numbers and ratios, the case about symbolic ones is less successful. In this regard, it is worth asking to what extent we can restrict our concept of number in order to call numbers to ANS representations. I suggest that the non-symbolic/symbolic distinction is, in this sense, a key one. If the ANS is not convincingly involved in processing of symbolic numbers (naturals and rationals), it would be more parsimonious to claim that it represents ratios rather than rational numbers.
The authors (hereafter, C&B) make an interesting argument for the consideration that the approximate number system (ANS) represents number rather than numerosity. The clarity in disentangling what is represented versus how it is represented is also a valuable contribution to the field. However, there are some issues that the argument leaves unresolved despite being critical to the nature of the ANS. A relevant one is the distinction between the representation and processing of non-symbolic and symbolic number. This is a well-established distinction in the numerical cognition literature, whose dismissal – for example, by reducing it to a matter of how number is represented – would be overlooking its depth and implications. Characterizing the ANS as a “primitive and prelinguistic capacity” shared across many species is a clear signal that C&B focus their case on the processing of non-symbolic number. However, the evidence cited mixes data from non-symbolic and symbolic studies (e.g., Henik & Tzelgov, Reference Henik and Tzelgov1982), blurring an otherwise clear and engaging argument. Although C&B distinguish in their exposition between ANS representations and precise number concepts, this distinction is intended to separate ANS representations from the advanced number constructions studied by mathematics, and it does not address the non-symbolic/symbolic contrast.
Myer and Landauer's (Reference Moyer and Landauer1967) study on single-digit number comparison may be considered an essential piece of evidence suggesting that number symbols are to some extent represented by the ANS. These authors showed that young adults' error rates and response times in comparing two digits decrease with increasing numerical distance between them. However, the consideration of these two types of numerical processing, non-symbolic and symbolic, brings an extra layer of complexity to number processing. For instance, symbolic number processing introduces unit-decade-compatibility effects which are meaningless in non-symbolic processing. Nuerk, Weger, and Willmes (Reference Nuerk, Weger and Willmes2001) presented this concept and proved that the comparison of multi-digit numbers is strongly affected by a competition between the numbers and the digits that compose them: Comparing 42 and 57 is easier than comparing 47 and 62 because in the former case the larger number coincides with the number with the larger decade and unit, whereas in the latter case the larger number has the larger decade but the smaller unit. Effects such as this one are specific to symbolic number representations, as the comparison of a set of 42 blue dots versus one of 57 yellow dots will likely lead to a similar outcome to that of a set of 47 blue dots versus one of 62 yellow dots (both comparisons engage the ANS and the yellow:blue ratios in each case are about 1.3). Altogether, the ANS seems able to represent number symbols, but this representation would be limited to single-digit numbers (see also Nuerk, Moeller, Klein, Willmes, & Fischer, Reference Nuerk, Moeller, Klein, Willmes and Fischer2011).
When it comes to rational numbers, the distinction between non-symbolic and symbolic processing becomes even more complex, and the research scarcer. Non-symbolic processing in this case refers to the capacity of perceiving and using ratios, whereas symbolic processing brings to the table fractions and decimals. Fractions are visually depicted as two natural numbers separated by a line. There is plenty of evidence that human adults perceive the magnitudes of natural numbers in an automatic manner (i.e., even if it is not relevant for the task, see Henik & Tzelgov, Reference Henik and Tzelgov1982). The magnitudes of fractions, however, seem to be activated only when they are relevant for the task (Bonato, Fabbri, Umiltà, & Zorzi, Reference Bonato, Fabbri, Umiltà and Zorzi2007; Gabriel, Szucs, & Content, Reference Gabriel, Szucs and Content2013; Kallai & Tzelgov, Reference Kallai and Tzelgov2012). Children's intuitive reasoning with fractions show important congruency effects (erroneously judging a fraction as larger than another if its components are larger, e.g., concluding that 2/3 < 4/9 because 2 < 3 and 4 < 9; see e.g., Gómez & Dartnell, Reference Gómez and Dartnell2019; Ni & Zhou, Reference Ni and Zhou2005; Van Hoof, Lijnen, Verschaffel, & Van Dooren, Reference Van Hoof, Lijnen, Verschaffel and Van Dooren2013). It is difficult to ascribe fraction comparison performance to the ANS, however. Although some studies have reported distance effects in response times in fraction comparison tasks, these times are too large to license conclusions about the mental representations of fractions (e.g., Schneider & Siegler, Reference Schneider and Siegler2010) or the task is to simple to actually engage fraction representations (e.g., Bonato et al., Reference Bonato, Fabbri, Umiltà and Zorzi2007). Nonetheless, adults who are highly mathematically competent also show congruency effects in their response times but also distance effects (Morales, Dartnell, & Gómez, Reference Morales, Dartnell and Gómez2020; Obersteiner, Van Dooren, Van Hoof, & Verschaffel, Reference Obersteiner, Van Dooren, Van Hoof and Verschaffel2013), showing that the discussion is far from over (see also Binzak & Hubbard, Reference Binzak and Hubbard2020, for positive evidence for ANS involvement in fraction comparison).
Ratios (or non-symbolic rational numbers) as a unifying percept of the number sense would be a very compelling theory (e.g., with natural numbers represented as ratios with respect to one). It would be consistent with the measuring function of numbers, common to both natural and rational numbers. As C&B note, natural number counting is not essentially tied to one as the unit, as counting can occur by pairs, tens, or dozens. But, even in this scenario, ratios are a limited aspect of rational numbers and, similarly, non-symbolic numbers are a limited aspect of natural numbers. Although C&B are convincing about the ANS representing non-symbolic numbers and ratios, the case about symbolic ones is less successful. In this regard, it is worth asking to what extent we can restrict our concept of number in order to call numbers to ANS representations. I suggest that the non-symbolic/symbolic distinction is, in this sense, a key one. If the ANS is not convincingly involved in processing of symbolic numbers (naturals and rationals), it would be more parsimonious to claim that it represents ratios rather than rational numbers.
Financial support
The author gratefully acknowledges the support of the grant ANID/PIA/Basal FB0003.
Conflict of interest
The author declares no conflict of interest.