Clarke and Beck (C&B) describe the human numerical system as unitary. They review effects of various perceptual properties (e.g., size or density) on judgments of numerosity in section 3: “congruency” and section 4: “confounds.” However, C&B argue that these effects are inconsequential because of the unitary nature of the numerical system. We disagree. We suggest that converging behavioral and neuroimaging evidence has shown that the number system is not unitary but diversified. First, behavioral evidence has consistently reported involvement and influence of basic continuous properties on the processing of numbers (for a review, see Leibovich, Katzin, Harel, & Henik, Reference Leibovich, Katzin, Harel and Henik2017). Second, neuroimaging evidence has highlighted distinct brain areas associated with various numerical and quantification tasks. Taken together, this evidence obligates re-examination of the premise that the number system is unitary. In turn, this provides interesting insights into the processing of rational numbers.
Throughout this review, the authors draw an analogy between the approximate number system (ANS) and the visual system. They state, “the visual system is often viewed as unified by its function, despite comprising relatively autonomous sub-modules performing dedicated tasks at various levels of visual analysis.” For example, the visual system is characterized by two co-existing visual pathways: (1) the primary, evolutionarily younger geniculo-striate system, and (2) an evolutionarily older retino-tectal system. Although these two systems have somewhat different roles, they are connected and have mutual effects on vision (Henik, Rafal, & Rhodes, Reference Henik, Rafal and Rhodes1994). The relatively less known evolutionarily older system deals mainly with spatial aspects of vision and connects to the parietal lobes, including the intraparietal sulcus (IPS), a key brain region linked to number processing.
The IPS and adjacent brain structures are involved in basic number processing, but also action (i.e., reaching and grasping). Walsh (Reference Walsh2003) suggested that important computational demands of an action system (reaching and grasping) are the basis for the involvement of the parietal lobes in comparative judgment tasks. Namely, the activity of the parietal lobe reflects computational demands of the brain dorsal system (that starts at the visual areas of the occipital lobe and connects to the parietal lobe) involved in perception for action (Goodale, Reference Goodale and Gazzaniga2000). However, it might be the other way around. Specifically, routines and brain structures underlying comparative judgments that are needed for action might have evolved from a single system that originally supported computing magnitudes (e.g., size). In line with this notion, for the dorsal brain system to develop, it was evolutionarily critical to first be able to compute amount or size and size differences. A neurocognitive system that handles this aspect of cognition (i.e., the evaluation of size or amount) might have been foundational for the development of the occipito-parietal dorsal brain system (i.e., the system that supports perception for action). Critically, this same system (i.e., the evaluation of size or amount) was also foundational for the development or advancement of the numerical system. Accordingly, we have suggested the coexistence of two systems (Henik, Gliksman, Kallai, & Leibovich, Reference Henik, Gliksman, Kallai and Leibovich2017); an older system that underpins the evaluation of size or amounts of substance and a number system that is discrete in nature and supports the evaluation of precise numerical quantities (Leibovich, Ashkenazi, Rubinsten, & Henik, Reference Leibovich, Ashkenazi, Rubinsten and Henik2013).
Recent meta-analyses of neuroimaging studies that evaluate the neural correlates of number processing across formats and non-numerical magnitude processing support this idea. Specifically, Sokolowski, Fias, Ononye, and Ansari (Reference Sokolowski, Fias, Ononye and Ansari2017) show the set of brain regions supporting symbolic and non-symbolic number processing highly overlap with the brain regions supporting non-numerical magnitude processing (e.g., size, length, and luminance). However, symbolic and non-symbolic number processing are also associated with additional, format-specific regions lateralized within the parietal lobes (with symbolic on the left and non-symbolic on the right). This meta-analytic data go against the idea that a single system supports all of numerical cognition, instead suggesting that numerical cognition is supported by diversified systems, one of which is a general magnitude system. Most of the empirical studies included in the meta-analysis use active tasks that involve decision making and motor response (i.e., perception and action), which are known to be associated with the IPS. A recent functional magnetic resonance imaging (fMRI) adaptation study (https://psyarxiv.com/xw2fq/) highlights that symbols, quantities, and physical size have overlapping but also distinct brain regions in the parietal lobes, and quantities and size are quite similar in terms of the patterns of activation whereas symbols are distinct. This reveals overlapping and distinct brain regions supporting numerical and non-numerical magnitude processing in the absence of active tasks. Other recent data from Zimmermann (Reference Zimmermann2018) provide direct evidence that different mechanisms account for the perception of visual numerosity. Specifically, Zimmermann shows that low numbers are sensed directly as a primary visual attribute, but the estimation of high numbers depends on the area/size over which the objects are spread. Hence, subsystems within the two systems proposed above may support computations of particular quantities and amounts.
The proposal that numerical cognition is supported by diversified systems sheds new light on the authors' discussion of rational numbers. C&B conceptualize rational numbers as a representation of numerical ratios among positive integers. They suggest that the ANS first represents natural numbers of concrete pluralities and only then derives ratios therefrom. Within the framework of a diversified numerical system, the magnitude and number systems may operate in parallel to extract the necessary information. Within such a composite system, proportions could be extracted in the way suggested by the authors (based on the number system), or more directly by the magnitude system, or in an orchestrated operation of the two systems.
In summary, we posit the idea that numerical cognition is supported by diversified systems, rather than a unified system. Such a divergent system aligns more closely to the structure of the visual system, is better supported by empirical data in the field of numerical cognition, and provides a more adequate explanation for the way the human mind processes rational numbers.
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Clarke and Beck (C&B) describe the human numerical system as unitary. They review effects of various perceptual properties (e.g., size or density) on judgments of numerosity in section 3: “congruency” and section 4: “confounds.” However, C&B argue that these effects are inconsequential because of the unitary nature of the numerical system. We disagree. We suggest that converging behavioral and neuroimaging evidence has shown that the number system is not unitary but diversified. First, behavioral evidence has consistently reported involvement and influence of basic continuous properties on the processing of numbers (for a review, see Leibovich, Katzin, Harel, & Henik, Reference Leibovich, Katzin, Harel and Henik2017). Second, neuroimaging evidence has highlighted distinct brain areas associated with various numerical and quantification tasks. Taken together, this evidence obligates re-examination of the premise that the number system is unitary. In turn, this provides interesting insights into the processing of rational numbers.
Throughout this review, the authors draw an analogy between the approximate number system (ANS) and the visual system. They state, “the visual system is often viewed as unified by its function, despite comprising relatively autonomous sub-modules performing dedicated tasks at various levels of visual analysis.” For example, the visual system is characterized by two co-existing visual pathways: (1) the primary, evolutionarily younger geniculo-striate system, and (2) an evolutionarily older retino-tectal system. Although these two systems have somewhat different roles, they are connected and have mutual effects on vision (Henik, Rafal, & Rhodes, Reference Henik, Rafal and Rhodes1994). The relatively less known evolutionarily older system deals mainly with spatial aspects of vision and connects to the parietal lobes, including the intraparietal sulcus (IPS), a key brain region linked to number processing.
The IPS and adjacent brain structures are involved in basic number processing, but also action (i.e., reaching and grasping). Walsh (Reference Walsh2003) suggested that important computational demands of an action system (reaching and grasping) are the basis for the involvement of the parietal lobes in comparative judgment tasks. Namely, the activity of the parietal lobe reflects computational demands of the brain dorsal system (that starts at the visual areas of the occipital lobe and connects to the parietal lobe) involved in perception for action (Goodale, Reference Goodale and Gazzaniga2000). However, it might be the other way around. Specifically, routines and brain structures underlying comparative judgments that are needed for action might have evolved from a single system that originally supported computing magnitudes (e.g., size). In line with this notion, for the dorsal brain system to develop, it was evolutionarily critical to first be able to compute amount or size and size differences. A neurocognitive system that handles this aspect of cognition (i.e., the evaluation of size or amount) might have been foundational for the development of the occipito-parietal dorsal brain system (i.e., the system that supports perception for action). Critically, this same system (i.e., the evaluation of size or amount) was also foundational for the development or advancement of the numerical system. Accordingly, we have suggested the coexistence of two systems (Henik, Gliksman, Kallai, & Leibovich, Reference Henik, Gliksman, Kallai and Leibovich2017); an older system that underpins the evaluation of size or amounts of substance and a number system that is discrete in nature and supports the evaluation of precise numerical quantities (Leibovich, Ashkenazi, Rubinsten, & Henik, Reference Leibovich, Ashkenazi, Rubinsten and Henik2013).
Recent meta-analyses of neuroimaging studies that evaluate the neural correlates of number processing across formats and non-numerical magnitude processing support this idea. Specifically, Sokolowski, Fias, Ononye, and Ansari (Reference Sokolowski, Fias, Ononye and Ansari2017) show the set of brain regions supporting symbolic and non-symbolic number processing highly overlap with the brain regions supporting non-numerical magnitude processing (e.g., size, length, and luminance). However, symbolic and non-symbolic number processing are also associated with additional, format-specific regions lateralized within the parietal lobes (with symbolic on the left and non-symbolic on the right). This meta-analytic data go against the idea that a single system supports all of numerical cognition, instead suggesting that numerical cognition is supported by diversified systems, one of which is a general magnitude system. Most of the empirical studies included in the meta-analysis use active tasks that involve decision making and motor response (i.e., perception and action), which are known to be associated with the IPS. A recent functional magnetic resonance imaging (fMRI) adaptation study (https://psyarxiv.com/xw2fq/) highlights that symbols, quantities, and physical size have overlapping but also distinct brain regions in the parietal lobes, and quantities and size are quite similar in terms of the patterns of activation whereas symbols are distinct. This reveals overlapping and distinct brain regions supporting numerical and non-numerical magnitude processing in the absence of active tasks. Other recent data from Zimmermann (Reference Zimmermann2018) provide direct evidence that different mechanisms account for the perception of visual numerosity. Specifically, Zimmermann shows that low numbers are sensed directly as a primary visual attribute, but the estimation of high numbers depends on the area/size over which the objects are spread. Hence, subsystems within the two systems proposed above may support computations of particular quantities and amounts.
The proposal that numerical cognition is supported by diversified systems sheds new light on the authors' discussion of rational numbers. C&B conceptualize rational numbers as a representation of numerical ratios among positive integers. They suggest that the ANS first represents natural numbers of concrete pluralities and only then derives ratios therefrom. Within the framework of a diversified numerical system, the magnitude and number systems may operate in parallel to extract the necessary information. Within such a composite system, proportions could be extracted in the way suggested by the authors (based on the number system), or more directly by the magnitude system, or in an orchestrated operation of the two systems.
In summary, we posit the idea that numerical cognition is supported by diversified systems, rather than a unified system. Such a divergent system aligns more closely to the structure of the visual system, is better supported by empirical data in the field of numerical cognition, and provides a more adequate explanation for the way the human mind processes rational numbers.
Conflict of interest
There is no conflict of interest.