Leibovich et al. breathe new life into an age-old question, which is whether discrete quantity representation (i.e., number) is on a par with, or a derivative of, the representation of continuous physical magnitudes (e.g., length, area, volume, loudness, pitch, color). The authors rightly point out that recent theorizing has favored a view (the “number sense” theory) in which number representation is innate and distinct from the representation of other magnitudes. However, their argument in favor of an alternative view (the “sense of magnitude” theory) falls short, having neither sufficient empirical support, nor logical soundness. These shortcomings stem from the authors' legitimate and veracious claim that because number out in the world naturally correlates with a host of non-numerical quantitative dimensions (e.g., a set of “four apples” has roughly twice the volume, twice the surface area, twice the weight, and twice the “redness,” as a set of “two apples,” not to mention other variables such as convex hull [perimeter of the set], density of the array, and amount of time it takes to visually scan the display), that it is “impossible to create two sets of items that differ in numerosity only” (sect. 3, para. 1). The implication here is that unless a researcher rules out every possible alternative in their study, they cannot be certain that responses were based on number per se, making it impossible to find pure evidence in support of the independence of number processing. The conclusion the authors draw from this – that lack of evidence for one theory (“number sense”) means the alternative theory is true – however, is fallacious (ad ignorantiam).
Although Leibovich et al. are right that it is mathematically impossible to rule out all continuous variables simultaneously, I argue that this fact does not undermine efforts to control some dimensions while examining numerical judgments. Logically speaking, finding that anti-correlated features influence performance (as in Hurewitz et al. Reference Hurewitz, Gelman and Schnitzer2006, for example) does not necessarily mean that subjects' judgments are not also influenced by number. The interference may stem from the dimensions being represented together (as proposed in “sense of magnitude” theory), or they may stem from other sources (e.g., attention, response competition, etc.). Likewise, just because one cannot control all possible continuous dimensions, it does not follow that subjects therefore must be using the uncontrolled dimension(s). There are many examples in the literature and cited in this article (and at least two important studies left uncited – McCrink & Wynn [Reference McCrink and Wynn2004] and Izard et al. [Reference Izard, Sann, Spelke and Steri2009]) in which the dimensions left to vary are clearly not driving performance because they predict different (and unobserved) patterns of performance than do judgments made on number. For example, through careful stimulus design Izard et al. (Reference Izard, Sann, Spelke and Steri2009) were able to rule out infants' use of “intensive parameters” (e.g., item size, density) and “extensive parameters” (summed luminance, total surface area of the array). Specifically, infants' use of intensive parameters (which were equated across numerosities) predicted equal looking to all of the test images, while their use of extensive parameters predicted the same direction of preference across familiarization conditions. However, neither of these patterns was obtained – infants looked longer at the numerically matching display in both familiarization conditions. Thus, despite not having perfect control, the study design had strong discriminant validity, allowing the authors to conclude which of the three cues (numerosity, intensive quantities, and extensive quantities) were driving the observed looking time pattern. Indeed, the deliberate control and manipulation of competing variables allowed Izard et al. to provide what may be the strongest evidence to date for the “number sense” theory.
A related point is that contrary to the authors' claims, the “number sense” theory may actually be more parsimonious than the “magnitude sense” proposal. The authors seem to treat all continuous dimensions the same, as if they are equivalent (equally salient, equally informative, equally accessible, equally accurate, etc.). Given the specific study they are critiquing, the authors can claim subjects are using surface area or volume or density or perimeter, rather than number. Importantly though, because the specific dimensions being controlled (or not) vary from experiment to experiment, the explanation for what subjects are doing instead of number across this constellation of studies is necessarily complex. Therefore, it may be argued that appealing to number is really the more economical approach.
My final point is more general. I applaud the authors for tackling this critical issue. They are absolutely right that determining the nature of these systems is necessary for understanding the basis of human quantitative reasoning. The development of these systems must be more fully determined if we want to understand how early emerging abilities affect the early learning of symbolic mathematics. However, I do not agree that simply appealing to “continuous magnitudes” as if the various continuous dimensions are interchangeable is any better than ruling out only a subset of them and claiming that subjects were definitely using number! We know very little about whether various dimensions are treated similarly or differently and how the performance profile for each dimension changes over development. Perhaps a good course of action would be to step back and determine whether the “continuous magnitudes” actually hang together within the same system before deciding whether number does. Of course, this may not even be possible; continuous magnitudes tend to correlate not only with number, but also with each other, making it just as challenging to isolate surface area, for example, from all other dimensions, as it is to isolate number. Nonetheless, I would argue that the endeavor is still very much worth the effort. With clever manipulation of the variables available in a given stimulus set, researchers may be able to discover not only whether discrete and continuous quantities stem from the same or different cognitive sources, but also how they interact and influence each other, as they surely must do. Such insights will no doubt help us on our way to understanding how intuitive number and magnitude sense contributes to the crucial acquisition and development of symbolic mathematical skills and knowledge.
Leibovich et al. breathe new life into an age-old question, which is whether discrete quantity representation (i.e., number) is on a par with, or a derivative of, the representation of continuous physical magnitudes (e.g., length, area, volume, loudness, pitch, color). The authors rightly point out that recent theorizing has favored a view (the “number sense” theory) in which number representation is innate and distinct from the representation of other magnitudes. However, their argument in favor of an alternative view (the “sense of magnitude” theory) falls short, having neither sufficient empirical support, nor logical soundness. These shortcomings stem from the authors' legitimate and veracious claim that because number out in the world naturally correlates with a host of non-numerical quantitative dimensions (e.g., a set of “four apples” has roughly twice the volume, twice the surface area, twice the weight, and twice the “redness,” as a set of “two apples,” not to mention other variables such as convex hull [perimeter of the set], density of the array, and amount of time it takes to visually scan the display), that it is “impossible to create two sets of items that differ in numerosity only” (sect. 3, para. 1). The implication here is that unless a researcher rules out every possible alternative in their study, they cannot be certain that responses were based on number per se, making it impossible to find pure evidence in support of the independence of number processing. The conclusion the authors draw from this – that lack of evidence for one theory (“number sense”) means the alternative theory is true – however, is fallacious (ad ignorantiam).
Although Leibovich et al. are right that it is mathematically impossible to rule out all continuous variables simultaneously, I argue that this fact does not undermine efforts to control some dimensions while examining numerical judgments. Logically speaking, finding that anti-correlated features influence performance (as in Hurewitz et al. Reference Hurewitz, Gelman and Schnitzer2006, for example) does not necessarily mean that subjects' judgments are not also influenced by number. The interference may stem from the dimensions being represented together (as proposed in “sense of magnitude” theory), or they may stem from other sources (e.g., attention, response competition, etc.). Likewise, just because one cannot control all possible continuous dimensions, it does not follow that subjects therefore must be using the uncontrolled dimension(s). There are many examples in the literature and cited in this article (and at least two important studies left uncited – McCrink & Wynn [Reference McCrink and Wynn2004] and Izard et al. [Reference Izard, Sann, Spelke and Steri2009]) in which the dimensions left to vary are clearly not driving performance because they predict different (and unobserved) patterns of performance than do judgments made on number. For example, through careful stimulus design Izard et al. (Reference Izard, Sann, Spelke and Steri2009) were able to rule out infants' use of “intensive parameters” (e.g., item size, density) and “extensive parameters” (summed luminance, total surface area of the array). Specifically, infants' use of intensive parameters (which were equated across numerosities) predicted equal looking to all of the test images, while their use of extensive parameters predicted the same direction of preference across familiarization conditions. However, neither of these patterns was obtained – infants looked longer at the numerically matching display in both familiarization conditions. Thus, despite not having perfect control, the study design had strong discriminant validity, allowing the authors to conclude which of the three cues (numerosity, intensive quantities, and extensive quantities) were driving the observed looking time pattern. Indeed, the deliberate control and manipulation of competing variables allowed Izard et al. to provide what may be the strongest evidence to date for the “number sense” theory.
A related point is that contrary to the authors' claims, the “number sense” theory may actually be more parsimonious than the “magnitude sense” proposal. The authors seem to treat all continuous dimensions the same, as if they are equivalent (equally salient, equally informative, equally accessible, equally accurate, etc.). Given the specific study they are critiquing, the authors can claim subjects are using surface area or volume or density or perimeter, rather than number. Importantly though, because the specific dimensions being controlled (or not) vary from experiment to experiment, the explanation for what subjects are doing instead of number across this constellation of studies is necessarily complex. Therefore, it may be argued that appealing to number is really the more economical approach.
My final point is more general. I applaud the authors for tackling this critical issue. They are absolutely right that determining the nature of these systems is necessary for understanding the basis of human quantitative reasoning. The development of these systems must be more fully determined if we want to understand how early emerging abilities affect the early learning of symbolic mathematics. However, I do not agree that simply appealing to “continuous magnitudes” as if the various continuous dimensions are interchangeable is any better than ruling out only a subset of them and claiming that subjects were definitely using number! We know very little about whether various dimensions are treated similarly or differently and how the performance profile for each dimension changes over development. Perhaps a good course of action would be to step back and determine whether the “continuous magnitudes” actually hang together within the same system before deciding whether number does. Of course, this may not even be possible; continuous magnitudes tend to correlate not only with number, but also with each other, making it just as challenging to isolate surface area, for example, from all other dimensions, as it is to isolate number. Nonetheless, I would argue that the endeavor is still very much worth the effort. With clever manipulation of the variables available in a given stimulus set, researchers may be able to discover not only whether discrete and continuous quantities stem from the same or different cognitive sources, but also how they interact and influence each other, as they surely must do. Such insights will no doubt help us on our way to understanding how intuitive number and magnitude sense contributes to the crucial acquisition and development of symbolic mathematical skills and knowledge.