Visual and auditory number stimuli inherently correlate with dimensions such as size, density, rate, and so forth, and observers are sometimes biased towards these dimensions: Changing the density of a collection of dots also changes which set observers believe to be more numerous. Leibovich et al., following the footsteps of recent findings reporting such “congruency effects,” argue that number may be entirely or partly dependent on these dimensions – that there is no innate number sense independent from our perception of density, convex hull, size, and so forth.
However, their critique leaves a key question open: At what level of processing do non-numeric dimensions exert their hold on number? There are at least three independent possibilities, and only one of them is consistent with the central claim against an independent number sense.
The first possibility is that number is encoded using low-level visual features, such as orientation, contrast, spatial frequency, and so forth, which are shared with other dimensions, rather than out of its own dedicated feature detectors (e.g., Dakin et al. Reference Dakin, Tibber, Greenwood, Kingdom and Morgan2011 vs. Burr & Ross Reference Burr and Ross2008). For example, consider that face perception strongly depends on a unique mix of low- and high-spatial frequency, and, therefore, changing frequency information also changes which emotion is most strongly perceived (Vuilleumier et al. Reference Vuilleumier, Armony, Driver and Dolan2003). In this same manner, there are now many reasons to suspect that number encoding depends on features such as low-spatial frequency (Dakin et al. Reference Dakin, Tibber, Greenwood, Kingdom and Morgan2011), and that it may even depend on distinct features at different levels of crowding (Anobile et al. Reference Anobile, Cicchini and Burr2014). Thus, manipulating density (i.e., low-spatial frequency information) can result in changes in number perception, not because of number being represented as density, but rather because of their shared dependence on identical low-level features. Congruency effects, therefore, could be interpreted as positive results describing the nature of low-level features used to encode number, not as evidence against its dependence on non-numeric dimensions. At the very least, claims to number's non-independence must first account for the shared low-level features.
The second possibility is that number and non-numeric dimensions compete for the same decision-making component, such as putting a common load on working memory or yielding similar response conflicts (Hurewitz et al. Reference Hurewitz, Gelman and Schnitzer2006; Odic et al. Reference Odic, Lisboa, Eisinger, Olivera, Maiche and Halberda2016; Van Opstal & Verguts Reference Van Opstal and Verguts2013). Once again, consider an analogy: congruency effects found in the Stroop effect do not imply that color perception is dependent on and statistically learned from reading ability, but rather that multiple dimensions can compete for the same response. Because we know that density and area perception tend to be more accurate in adults compared with number, there is plenty of reason to think that these dimensions will win a “horse race” for the same response as number, creating congruency effects without any shared representations (Hurewitz et al. Reference Hurewitz, Gelman and Schnitzer2006). Consistent with this, my colleagues and I have demonstrated that number and time perception only correlate when individual differences in working memory are not controlled for (Odic et al. Reference Odic, Lisboa, Eisinger, Olivera, Maiche and Halberda2016). More recently, we have found that the effect of non-numeric dimensions such as contour length is entirely eliminated when Stroop-like response conflicts are alleviated (Picon & Odic, Reference Picon and Odicin preparation). Together, these results suggest that many demonstrated congruency effects could be response conflicts, and that any claim for dependence between number perception and non-numeric dimensions should first control for these factors.
Finally, the third possibility for the link between number and non-numeric dimensions – and one that is most consistent with the claims of Leibovich et al. – is that number may be (antecedently) represented on the same representational scale as other dimensions, either by being directly represented as, for example, area, or alternatively by being represented on a domain-general, unitless magnitude scale that simply codes for more versus less (Cantrell & Smith Reference Cantrell and Smith2013; Lourenco & Longo Reference Lourenco and Longo2010; Walsh Reference Walsh2003). Although Leibovich et al. suggest that statistical learning eventually separates number from these dimensions, their theory requires that – from birth until some later age – numerical information is represented in one of these two ways. But, as reviewed previously, evidence for shared representations must first control for the possibility of shared encoding or decision-making factors; given that the majority of existing work fails to do so, what is the evidence for shared/unified representations? Perhaps the most convincing case cited by Lebovich et al. is that of Tudusciuc and Nieder (Reference Tudusciuc and Nieder2007), who found neurons in the parietal cortex that respond to both number and length. But a closer inspection of their data reveals that these neurons often code in opposing ways: The same neuron may code for small numbers, but very long lengths, or vice versa, running contrary to the idea of a shared scale and instead consistent with a set of overlapping population coding neurons that play different roles for each dimension.
Another approach at demonstrating shared representations is to simultaneously measure number and the candidate shared dimensions, such as area, length, density, and time; if number shares the scale for these representations, any individual and developmental variability within number should be accounted for by differences in these other dimensions. Recently, my lab followed this logic through and tested 2- to 12-year-old children and adults on these five discrimination tasks. We found that number develops independently from area, length, density, and time, which in turn develop independently from it stretching back to age 2 (Odic Reference Odic2017; see also Odic et al. Reference Odic, Libertus, Feigenson and Halberda2013). Hence, unless the kind of proposed statistical learning proposed leads to complete differentiation by age 2, it is difficult to imagine how these results could be obtained without a significant independence between number and area, length, density, and time perception.
To conclude, Leibovich et al. make a bold claim – that congruency effects are illustrative of number's dependence on non-numeric dimensions – but their critique fails to account for the possibility that these effects stem from shared encoding or decision-making components, not shared representations. Future work exploring number's dependence should carefully disentangle the contributions of other dimensions to encoding and decision making, as these levels are not constitutive of the independent representations of number.
Visual and auditory number stimuli inherently correlate with dimensions such as size, density, rate, and so forth, and observers are sometimes biased towards these dimensions: Changing the density of a collection of dots also changes which set observers believe to be more numerous. Leibovich et al., following the footsteps of recent findings reporting such “congruency effects,” argue that number may be entirely or partly dependent on these dimensions – that there is no innate number sense independent from our perception of density, convex hull, size, and so forth.
However, their critique leaves a key question open: At what level of processing do non-numeric dimensions exert their hold on number? There are at least three independent possibilities, and only one of them is consistent with the central claim against an independent number sense.
The first possibility is that number is encoded using low-level visual features, such as orientation, contrast, spatial frequency, and so forth, which are shared with other dimensions, rather than out of its own dedicated feature detectors (e.g., Dakin et al. Reference Dakin, Tibber, Greenwood, Kingdom and Morgan2011 vs. Burr & Ross Reference Burr and Ross2008). For example, consider that face perception strongly depends on a unique mix of low- and high-spatial frequency, and, therefore, changing frequency information also changes which emotion is most strongly perceived (Vuilleumier et al. Reference Vuilleumier, Armony, Driver and Dolan2003). In this same manner, there are now many reasons to suspect that number encoding depends on features such as low-spatial frequency (Dakin et al. Reference Dakin, Tibber, Greenwood, Kingdom and Morgan2011), and that it may even depend on distinct features at different levels of crowding (Anobile et al. Reference Anobile, Cicchini and Burr2014). Thus, manipulating density (i.e., low-spatial frequency information) can result in changes in number perception, not because of number being represented as density, but rather because of their shared dependence on identical low-level features. Congruency effects, therefore, could be interpreted as positive results describing the nature of low-level features used to encode number, not as evidence against its dependence on non-numeric dimensions. At the very least, claims to number's non-independence must first account for the shared low-level features.
The second possibility is that number and non-numeric dimensions compete for the same decision-making component, such as putting a common load on working memory or yielding similar response conflicts (Hurewitz et al. Reference Hurewitz, Gelman and Schnitzer2006; Odic et al. Reference Odic, Lisboa, Eisinger, Olivera, Maiche and Halberda2016; Van Opstal & Verguts Reference Van Opstal and Verguts2013). Once again, consider an analogy: congruency effects found in the Stroop effect do not imply that color perception is dependent on and statistically learned from reading ability, but rather that multiple dimensions can compete for the same response. Because we know that density and area perception tend to be more accurate in adults compared with number, there is plenty of reason to think that these dimensions will win a “horse race” for the same response as number, creating congruency effects without any shared representations (Hurewitz et al. Reference Hurewitz, Gelman and Schnitzer2006). Consistent with this, my colleagues and I have demonstrated that number and time perception only correlate when individual differences in working memory are not controlled for (Odic et al. Reference Odic, Lisboa, Eisinger, Olivera, Maiche and Halberda2016). More recently, we have found that the effect of non-numeric dimensions such as contour length is entirely eliminated when Stroop-like response conflicts are alleviated (Picon & Odic, Reference Picon and Odicin preparation). Together, these results suggest that many demonstrated congruency effects could be response conflicts, and that any claim for dependence between number perception and non-numeric dimensions should first control for these factors.
Finally, the third possibility for the link between number and non-numeric dimensions – and one that is most consistent with the claims of Leibovich et al. – is that number may be (antecedently) represented on the same representational scale as other dimensions, either by being directly represented as, for example, area, or alternatively by being represented on a domain-general, unitless magnitude scale that simply codes for more versus less (Cantrell & Smith Reference Cantrell and Smith2013; Lourenco & Longo Reference Lourenco and Longo2010; Walsh Reference Walsh2003). Although Leibovich et al. suggest that statistical learning eventually separates number from these dimensions, their theory requires that – from birth until some later age – numerical information is represented in one of these two ways. But, as reviewed previously, evidence for shared representations must first control for the possibility of shared encoding or decision-making factors; given that the majority of existing work fails to do so, what is the evidence for shared/unified representations? Perhaps the most convincing case cited by Lebovich et al. is that of Tudusciuc and Nieder (Reference Tudusciuc and Nieder2007), who found neurons in the parietal cortex that respond to both number and length. But a closer inspection of their data reveals that these neurons often code in opposing ways: The same neuron may code for small numbers, but very long lengths, or vice versa, running contrary to the idea of a shared scale and instead consistent with a set of overlapping population coding neurons that play different roles for each dimension.
Another approach at demonstrating shared representations is to simultaneously measure number and the candidate shared dimensions, such as area, length, density, and time; if number shares the scale for these representations, any individual and developmental variability within number should be accounted for by differences in these other dimensions. Recently, my lab followed this logic through and tested 2- to 12-year-old children and adults on these five discrimination tasks. We found that number develops independently from area, length, density, and time, which in turn develop independently from it stretching back to age 2 (Odic Reference Odic2017; see also Odic et al. Reference Odic, Libertus, Feigenson and Halberda2013). Hence, unless the kind of proposed statistical learning proposed leads to complete differentiation by age 2, it is difficult to imagine how these results could be obtained without a significant independence between number and area, length, density, and time perception.
To conclude, Leibovich et al. make a bold claim – that congruency effects are illustrative of number's dependence on non-numeric dimensions – but their critique fails to account for the possibility that these effects stem from shared encoding or decision-making components, not shared representations. Future work exploring number's dependence should carefully disentangle the contributions of other dimensions to encoding and decision making, as these levels are not constitutive of the independent representations of number.