Some maintain that we do not have sufficient evidence to establish the existence of the approximate number system (ANS) – that experimental results fail to convince that there is sensitivity (in line with Weber's Law) to the “numerosity” of collections of individuals rather than certain potential confounds such as the total area of the disparate region covered by those individuals. Clarke and Beck (C&B) effectively refute such scepticism. They point to the existence of cross-modal studies, in which, for example, the “numerosity” of a collection of dots is compared to that of a collection of tones, and ask the pointed question – what can the alleged confound be in such a case? They also draw attention to the dumbbell effect, which provides strong evidence that the sensitivity of the ANS is to the discretely varying size of a collection of individuals – a second-order property of a given scenario – rather than a continuously varying magnitude, such as that of the area covered by those individuals. Such results (alongside myriad others) leave no grounds for reasonable doubt that the ANS exists.
But what does this “Number Sense” represent? C&B suggest that the ANS represents, well… numbers – and more specifically, rational numbers. They also hope to show that appeal to facts about imprecision in the representational capacity of the ANS does not preclude such an answer, or support an alternative one, such as Burge's (Reference Burge2010) view that the cognitive system in question represents the “pure magnitudes” theorized by Eudoxus in antiquity.
Peacocke (Reference Peacocke, Coliva, Munz and Moyal-Sharrock2015) has clarified that Eudoxus' pure magnitudes are extensive, meaning that they can be added to one another: If we take an object with mass m1, and combine it with an object with mass m2, the result is an “object” with mass m1 + m2. Intensities, by contrast, cannot be added. Carey (Reference Carey2009) discusses density, which comes in degrees. We can say how dense something is (in comparison with other things), and even measure this quantitatively. Nevertheless, the density of an “object” that results from combining two objects with densities d1 and d2 cannot be assumed to be the sum d1 + d2 – it depends on the relative sizes of the two objects that are combined! (The reason, of course, is that density is ultimately a relation between two extensities, the mass of an object and its volume.)
C&B argue that the ANS represents rational numbers, and that this suggestion has ecological validity, because it is useful to an organism to represent, for example, probabilities (which are often – although not always – determined by certain ratios). Now, rational numbers are extensive magnitudes: it makes sense to ask how much ½ + ¾ is. But, as far as I can see, C&B cite no evidence that suggests additivity here. Take the (wonderful!) lollipop experiment they discuss: Infants can succeed in choosing a jar with a greater chance that a lollipop randomly selected from it will be of their preferred flavour; yet this only requires that they represent the ratios of their preferred flavour to the other flavour (or to the total). Ratios, however, are intensities: We can compare them; but it makes no sense to ask what 1:2 + 3:4 is. (Indeed, “one is to two plus three is to four” is ungrammatical.) Perhaps, the conclusion that rational numbers are represented (rather than ratios) is premature.
C&B are also keen to distinguish the question of what the ANS represents from that of how it does so: but care is required in practice to do so. Their view appears to be that the ANS does not represent numbers in the abstract, as objects; rather, it attributes number properties to collections of individuals – in its approximative way. But, if the ANS attributes a numerical size to a collection of objects, we can surely ask what property exactly it represents that collection as having – and it seems we can distinguish the views that it attributes being (roughly) such and such size (which is, in fact, numerical, being a size of a collection) and that it attributes being (roughly) so numerous.
What would answer the question? Presumably, something about the processing sensitivities of the ANS – although no theorist should embrace the strong sensitivity principle C&B articulate, for the reasons they give. And C&B are surely right that the ANS does not represent magnitudes that are indeterminate in kind between species that vary continuously and species that vary discretely – there are no such magnitudes (even if there are “pure” continuous magnitudes that are, for instance, neither spatial distances nor temporal durations). Yet it might represent numerical sizes without representing them as varying discretely. Arguably, this would be so if the only computations performed on the representations were well-defined on continuously varying magnitudes as well, such as comparison and addition/subtraction. (A system that also exhibited sensitivity to whether there is a one–one correspondence between two collections might be said to represent certain magnitudes as cardinal numbers; and one that displayed a sensitivity to the immediate successor relation might be taken to represent some magnitudes as natural numbers, if these are taken to be things related to zero by the ancestral of that relation.) Is this only a question of how the (numerical) magnitudes are represented?
In any case, it seems there is a difference between attributing the properties of being eight in number and being roughly eight in number: If the collection to which the property is attributed has nine items in it, the second attribution is correct, whereas the first is not. Therefore, this distinction would appear to concern what is represented, not how it is represented. Perhaps, C&B will say this shows instead only that cognitive episodes involving the ANS have accuracy conditions, which admit of degrees, rather than veridicality or truth conditions, which do not – and that it is indeterminate what (i.e., which property) is represented by the ANS?
Some maintain that we do not have sufficient evidence to establish the existence of the approximate number system (ANS) – that experimental results fail to convince that there is sensitivity (in line with Weber's Law) to the “numerosity” of collections of individuals rather than certain potential confounds such as the total area of the disparate region covered by those individuals. Clarke and Beck (C&B) effectively refute such scepticism. They point to the existence of cross-modal studies, in which, for example, the “numerosity” of a collection of dots is compared to that of a collection of tones, and ask the pointed question – what can the alleged confound be in such a case? They also draw attention to the dumbbell effect, which provides strong evidence that the sensitivity of the ANS is to the discretely varying size of a collection of individuals – a second-order property of a given scenario – rather than a continuously varying magnitude, such as that of the area covered by those individuals. Such results (alongside myriad others) leave no grounds for reasonable doubt that the ANS exists.
But what does this “Number Sense” represent? C&B suggest that the ANS represents, well… numbers – and more specifically, rational numbers. They also hope to show that appeal to facts about imprecision in the representational capacity of the ANS does not preclude such an answer, or support an alternative one, such as Burge's (Reference Burge2010) view that the cognitive system in question represents the “pure magnitudes” theorized by Eudoxus in antiquity.
Peacocke (Reference Peacocke, Coliva, Munz and Moyal-Sharrock2015) has clarified that Eudoxus' pure magnitudes are extensive, meaning that they can be added to one another: If we take an object with mass m1, and combine it with an object with mass m2, the result is an “object” with mass m1 + m2. Intensities, by contrast, cannot be added. Carey (Reference Carey2009) discusses density, which comes in degrees. We can say how dense something is (in comparison with other things), and even measure this quantitatively. Nevertheless, the density of an “object” that results from combining two objects with densities d1 and d2 cannot be assumed to be the sum d1 + d2 – it depends on the relative sizes of the two objects that are combined! (The reason, of course, is that density is ultimately a relation between two extensities, the mass of an object and its volume.)
C&B argue that the ANS represents rational numbers, and that this suggestion has ecological validity, because it is useful to an organism to represent, for example, probabilities (which are often – although not always – determined by certain ratios). Now, rational numbers are extensive magnitudes: it makes sense to ask how much ½ + ¾ is. But, as far as I can see, C&B cite no evidence that suggests additivity here. Take the (wonderful!) lollipop experiment they discuss: Infants can succeed in choosing a jar with a greater chance that a lollipop randomly selected from it will be of their preferred flavour; yet this only requires that they represent the ratios of their preferred flavour to the other flavour (or to the total). Ratios, however, are intensities: We can compare them; but it makes no sense to ask what 1:2 + 3:4 is. (Indeed, “one is to two plus three is to four” is ungrammatical.) Perhaps, the conclusion that rational numbers are represented (rather than ratios) is premature.
C&B are also keen to distinguish the question of what the ANS represents from that of how it does so: but care is required in practice to do so. Their view appears to be that the ANS does not represent numbers in the abstract, as objects; rather, it attributes number properties to collections of individuals – in its approximative way. But, if the ANS attributes a numerical size to a collection of objects, we can surely ask what property exactly it represents that collection as having – and it seems we can distinguish the views that it attributes being (roughly) such and such size (which is, in fact, numerical, being a size of a collection) and that it attributes being (roughly) so numerous.
What would answer the question? Presumably, something about the processing sensitivities of the ANS – although no theorist should embrace the strong sensitivity principle C&B articulate, for the reasons they give. And C&B are surely right that the ANS does not represent magnitudes that are indeterminate in kind between species that vary continuously and species that vary discretely – there are no such magnitudes (even if there are “pure” continuous magnitudes that are, for instance, neither spatial distances nor temporal durations). Yet it might represent numerical sizes without representing them as varying discretely. Arguably, this would be so if the only computations performed on the representations were well-defined on continuously varying magnitudes as well, such as comparison and addition/subtraction. (A system that also exhibited sensitivity to whether there is a one–one correspondence between two collections might be said to represent certain magnitudes as cardinal numbers; and one that displayed a sensitivity to the immediate successor relation might be taken to represent some magnitudes as natural numbers, if these are taken to be things related to zero by the ancestral of that relation.) Is this only a question of how the (numerical) magnitudes are represented?
In any case, it seems there is a difference between attributing the properties of being eight in number and being roughly eight in number: If the collection to which the property is attributed has nine items in it, the second attribution is correct, whereas the first is not. Therefore, this distinction would appear to concern what is represented, not how it is represented. Perhaps, C&B will say this shows instead only that cognitive episodes involving the ANS have accuracy conditions, which admit of degrees, rather than veridicality or truth conditions, which do not – and that it is indeterminate what (i.e., which property) is represented by the ANS?
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