Clarke and Beck (C&B) vigorously defend the thesis that the approximate number system (ANS) represents number, which they take to include the natural numbers and the rational numbers (fractions). Although they present compelling responses to some (but not all – see below) objections to their view, the evidence that they present seems consistent with the much weaker thesis that the ANS represents different types of cardinality comparisons. My challenge to C&B is to explain why we need anything more than cardinality comparisons to account for the operation of the ANS.
To explain the simplest form of cardinality comparisons, we can begin with the concept of equinumerosity. Informally speaking, two sets are equinumerous when they have the same number of members. In mathematical logic, equinumerosity is standardly understood in terms of there being a 1:1 mapping (a bijection) between the two sets. This concept does not, of course, involve any reference to number or numbers, which is why it is the foundation for the influential approach to understanding numbers in the philosophy of mathematics known as logicism. But there is no need for fancy mathematical machinery to put this concept to work – simply pairing each apple with exactly one orange and each orange with exactly one apple will establish that a set of apples and a set of oranges are equinumerous.
Much of the experimental evidence cited in support of an ANS takes the form of demonstrated sensitivity to situations where two sets or collections are not equinumerous. Therefore, for example, Xu & Spelke (Reference Xu and Spelke2000) showed that infants habituated to a display with 16 dots dishabituate to displays with a number of dots differing from 16 by a sufficiently large ratio (to a 32-dot array, but not one with 24 dots, e.g.). C&B want to say that this is an example of infants representing (imprecisely) the numbers 32 and 16. But that seems to be using a sledgehammer to crack a nut. Why not simply say that the infants are sensitive to the non-equinumerosity of the two arrays when the ratio between them is sufficiently great? “Equinumerosity” is a fancy word, but a simple concept (think about pairing up the apples and oranges). “Number,” in contrast, is a simple word but a (very) fancy concept.
Non-equinumerosity is the simplest form of cardinality comparison, but when non-equinumerosity holds between two sets, one set will always have more members than the other. Representation of the property — has more members than – seems to be what is shown by Barth et al. (Reference Barth, La Mont, Lipton and Spelke2005) studies of preschoolers, and by any study showing that nonhuman animals reliably select sets with larger numbers of food items.
There are (at least) two good reasons for preferring comparative cardinalities to numbers in explicating the ANS. The first is parsimony. To represent comparative cardinalities is to represent a relational property between two sets. To represent numbers is to represent abstract objects that stand in certain arithmetical relations to each other – representations that can be manipulated according to well understood rules and operations. C&B correctly point out that what they term the strong sensitivity principle is misplaced. It is perfectly possible to represent something without representing all of its essential properties. However, because C&B readily concede that there is no evidence that the ANS is sensitive to the successor function or to basic arithmetical operations it seems a good idea to look for representational abilities that are independent of such functions and operations. After all, it does seem impossible to represent something without representing at least some of its essential properties, and if one takes away the functions and operations that define the number system, and allows numbers to be represented as imprecise, then no essential properties of the number system are left to be represented.
The second reason has to do with the performance profile of the ANS, which C&B are at pains to emphasize. The ANS appears to conform to Weber's Law. Weber's Law is a law governing discriminability. Typically, it is used to characterize the perception of just-noticeable differences in psychophysics. Such differences are, by their very nature, relational and comparative. Therefore, one would expect the representational currency of any system to be relational and comparative. Comparative cardinalities fit this description better than numbers. What the ANS does is represent comparative cardinalities such as – is equinumerous to –, – has more members than –, has fewer members than –, rather than absolute properties such as – has (approximately) 16 members – or – has (approximately) 32 members –. By the same token, the auditory system represents properties such as – is the same volume as – and – is louder than –, rather than absolute properties such as – has a volume of 55 decibels.
A final observation. Comparative cardinalities are not numerical magnitudes (or what C&B call “recherché alternatives to numbers”). They are of course related to numerical magnitudes, but that does not mean that they can only be represented by representing numerical magnitudes. A seed-eating bird can represent that one container has more seeds in it than another without representing the first as having 252 and the second as 57, even approximately. By analogy, you or I can represent the sound of a lawn-mower as louder than the sound of distant thunder without representing the first as 90 decibels and the second as 62.
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Clarke and Beck (C&B) vigorously defend the thesis that the approximate number system (ANS) represents number, which they take to include the natural numbers and the rational numbers (fractions). Although they present compelling responses to some (but not all – see below) objections to their view, the evidence that they present seems consistent with the much weaker thesis that the ANS represents different types of cardinality comparisons. My challenge to C&B is to explain why we need anything more than cardinality comparisons to account for the operation of the ANS.
To explain the simplest form of cardinality comparisons, we can begin with the concept of equinumerosity. Informally speaking, two sets are equinumerous when they have the same number of members. In mathematical logic, equinumerosity is standardly understood in terms of there being a 1:1 mapping (a bijection) between the two sets. This concept does not, of course, involve any reference to number or numbers, which is why it is the foundation for the influential approach to understanding numbers in the philosophy of mathematics known as logicism. But there is no need for fancy mathematical machinery to put this concept to work – simply pairing each apple with exactly one orange and each orange with exactly one apple will establish that a set of apples and a set of oranges are equinumerous.
Much of the experimental evidence cited in support of an ANS takes the form of demonstrated sensitivity to situations where two sets or collections are not equinumerous. Therefore, for example, Xu & Spelke (Reference Xu and Spelke2000) showed that infants habituated to a display with 16 dots dishabituate to displays with a number of dots differing from 16 by a sufficiently large ratio (to a 32-dot array, but not one with 24 dots, e.g.). C&B want to say that this is an example of infants representing (imprecisely) the numbers 32 and 16. But that seems to be using a sledgehammer to crack a nut. Why not simply say that the infants are sensitive to the non-equinumerosity of the two arrays when the ratio between them is sufficiently great? “Equinumerosity” is a fancy word, but a simple concept (think about pairing up the apples and oranges). “Number,” in contrast, is a simple word but a (very) fancy concept.
Non-equinumerosity is the simplest form of cardinality comparison, but when non-equinumerosity holds between two sets, one set will always have more members than the other. Representation of the property — has more members than – seems to be what is shown by Barth et al. (Reference Barth, La Mont, Lipton and Spelke2005) studies of preschoolers, and by any study showing that nonhuman animals reliably select sets with larger numbers of food items.
There are (at least) two good reasons for preferring comparative cardinalities to numbers in explicating the ANS. The first is parsimony. To represent comparative cardinalities is to represent a relational property between two sets. To represent numbers is to represent abstract objects that stand in certain arithmetical relations to each other – representations that can be manipulated according to well understood rules and operations. C&B correctly point out that what they term the strong sensitivity principle is misplaced. It is perfectly possible to represent something without representing all of its essential properties. However, because C&B readily concede that there is no evidence that the ANS is sensitive to the successor function or to basic arithmetical operations it seems a good idea to look for representational abilities that are independent of such functions and operations. After all, it does seem impossible to represent something without representing at least some of its essential properties, and if one takes away the functions and operations that define the number system, and allows numbers to be represented as imprecise, then no essential properties of the number system are left to be represented.
The second reason has to do with the performance profile of the ANS, which C&B are at pains to emphasize. The ANS appears to conform to Weber's Law. Weber's Law is a law governing discriminability. Typically, it is used to characterize the perception of just-noticeable differences in psychophysics. Such differences are, by their very nature, relational and comparative. Therefore, one would expect the representational currency of any system to be relational and comparative. Comparative cardinalities fit this description better than numbers. What the ANS does is represent comparative cardinalities such as – is equinumerous to –, – has more members than –, has fewer members than –, rather than absolute properties such as – has (approximately) 16 members – or – has (approximately) 32 members –. By the same token, the auditory system represents properties such as – is the same volume as – and – is louder than –, rather than absolute properties such as – has a volume of 55 decibels.
A final observation. Comparative cardinalities are not numerical magnitudes (or what C&B call “recherché alternatives to numbers”). They are of course related to numerical magnitudes, but that does not mean that they can only be represented by representing numerical magnitudes. A seed-eating bird can represent that one container has more seeds in it than another without representing the first as having 252 and the second as 57, even approximately. By analogy, you or I can represent the sound of a lawn-mower as louder than the sound of distant thunder without representing the first as 90 decibels and the second as 62.
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