Clarke, Beck, and I all agree that finite cardinal numbers “have a ‘second-order character’ that non-numerical quantities lack” (sect. 5.3, para. 4). This means that numbers are properties that apply to objects relative to a given sortal-kind whereas first-order properties apply to objects tout court. We disagree on the question of whether approximate number system (ANS) explanations of our quantitative judgments require positing a sensory relation to numbers (cf. Azzouni, Reference Azzouni2010). To me, this idea is very odd because second-order entities are no part of the sensible realm, and so have to be grasped by thinking rather than sensing (Marshall, Reference Marshall2017). Why do Clarke and Beck (C&B) find it necessary, or even remotely plausible, to say that numbers are both second-order and sensible?
They seem to think that plausibility is inherited from Tyler Burge's theory that the ANS operates over objects that have already been individuated by perceptual representations of kinds that Burge calls “perceptual attributives” (Reference Burge2010). These attributives are supposed to be the perceptual correspondents of concepts, elements of perceptual states that represent properties and kinds. The problem is that an examination of Burge's work does not reveal any evidence that the ANS operates in this way. Rather, Burge simply draws on his theory of perceptual attribution to assert without empirical argument that the ANS can carry out processes of individuation and enumeration that are analogous to transitive counting. Although not as circular as positing a homunculus who can count transitively, this does smuggle conditions required for counting into the description of the ANS. Such speculation may show that perception of numbers is possible; but it does not constitute evidence that the ANS actually operates in this way. (By the way, if there is no such evidence, this would weaken C&B's criticism of Burge's proposal that the ANS represents the magnitudes and ratios described by Eudoxus, a criticism that appeals to the fact that the ANS represents second-order entities.)
C&B offer some evidence that the ANS draws on perceptual attributives – namely, the dumbbell studies, on which they have to hang an awful lot. This is because as I show in my (Marshall, Reference Marshall2018), there are various essential properties of numbers that are not represented by the ANS. Furthermore, what C&B call the “weak sensitivity principle” is satisfied only if the ANS can represent some essential property of numbers; further, they argue that it does so in virtue of representing their essentially second-order nature. They also admit that if the principle is not satisfied, then it is unclear “what else could make it the case that [numbers are] being represented rather than some other entity.” Hence, they have to hang so much on the dumbbell studies.
C&B avoid committing themselves to Burge's model by distinguishing Marr's computational and algorithmic levels of explanation, allowing that the ANS could function at the former level to represent numbers qua second-order entities, while leaving open exactly how this is done at the algorithmic level. The problem is that what is represented places constraints on how it is represented and so one cannot offer a theory at the computational level in complete abstraction from the algorithmic level (cf. Burge, ibid, p. 93, fn. 43). As things stand, the situation seems to be that if we could sense numbers, then the ANS would have to operate as Burge says; but there is little evidence that it does. To defeat this argument by showing how else the ANS could operate, more needs to be said about how their alternative “indirect model” of the ANS could generate representations of second-order numbers from representations of “a mishmash of non-numerical magnitudes” (sect. 4, para. 11).
Developing this line of thought, C&B argue that positing a sensory relation to numbers would be needed to unify ANS explanations of behavior and to explain the common function that our sensitivity to the aforementioned mishmash would be serving. However, the same explanatory benefits are gained by positing a sensory relation to multitudes of sensible objects, where these objects taken collectively possess non-numerical but quantitative properties from which an approximate representation of their multitude might be extracted by the ANS.
They might object that I cannot substitute talk of multitudes for talk of numerosities, because only the latter are, like numbers, second-order: just as we speak of the number of F's, so we speak of the numerosity of F's. I respond that the relevant sortal-kind of the objects making up the multitude can be discerned from the context if necessary. When I sense that there are two oranges, I sense an orange, another, and no more. When I sense that there are just as many oranges as there are apples, I sense an orange, another, and no more, an apple and another and no more, and match them up so that none are left over. If I cut off half an apple, leaving the remainder, I sense that there is an apple and a half, which is to say one and a half apples worth of apple, as well as three and half fruits worth of fruit. Therefore, this proposal also encompasses what C&B (mistakenly) call our sense of rational numbers. In any case, the sortal-kind is the natural analog of a unit of measurement that is provided by and can be discerned from the context. The connection to numbers is as follows:
The sensible world contains multitudes from which cardinal numbers are abstracted. The latter are then applied to the world as it is organized by sortals. Numbers are not simply properties of the sensible world but properties of the world as it is organized (cf. Gaifman, Reference Gaifman2005). The sensible world also contains magnitudes (as well as natural analogs of units of measurement), which fall into ratios from which rational numbers are abstracted. The ANS does not represent numbers; but, in virtue of approximately representing multitudes and ratios of magnitudes it is correlated approximately with numbers, because numbers by their nature apply to multitudes relative to sortals as well as to magnitudes and their ratios.
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Clarke, Beck, and I all agree that finite cardinal numbers “have a ‘second-order character’ that non-numerical quantities lack” (sect. 5.3, para. 4). This means that numbers are properties that apply to objects relative to a given sortal-kind whereas first-order properties apply to objects tout court. We disagree on the question of whether approximate number system (ANS) explanations of our quantitative judgments require positing a sensory relation to numbers (cf. Azzouni, Reference Azzouni2010). To me, this idea is very odd because second-order entities are no part of the sensible realm, and so have to be grasped by thinking rather than sensing (Marshall, Reference Marshall2017). Why do Clarke and Beck (C&B) find it necessary, or even remotely plausible, to say that numbers are both second-order and sensible?
They seem to think that plausibility is inherited from Tyler Burge's theory that the ANS operates over objects that have already been individuated by perceptual representations of kinds that Burge calls “perceptual attributives” (Reference Burge2010). These attributives are supposed to be the perceptual correspondents of concepts, elements of perceptual states that represent properties and kinds. The problem is that an examination of Burge's work does not reveal any evidence that the ANS operates in this way. Rather, Burge simply draws on his theory of perceptual attribution to assert without empirical argument that the ANS can carry out processes of individuation and enumeration that are analogous to transitive counting. Although not as circular as positing a homunculus who can count transitively, this does smuggle conditions required for counting into the description of the ANS. Such speculation may show that perception of numbers is possible; but it does not constitute evidence that the ANS actually operates in this way. (By the way, if there is no such evidence, this would weaken C&B's criticism of Burge's proposal that the ANS represents the magnitudes and ratios described by Eudoxus, a criticism that appeals to the fact that the ANS represents second-order entities.)
C&B offer some evidence that the ANS draws on perceptual attributives – namely, the dumbbell studies, on which they have to hang an awful lot. This is because as I show in my (Marshall, Reference Marshall2018), there are various essential properties of numbers that are not represented by the ANS. Furthermore, what C&B call the “weak sensitivity principle” is satisfied only if the ANS can represent some essential property of numbers; further, they argue that it does so in virtue of representing their essentially second-order nature. They also admit that if the principle is not satisfied, then it is unclear “what else could make it the case that [numbers are] being represented rather than some other entity.” Hence, they have to hang so much on the dumbbell studies.
C&B avoid committing themselves to Burge's model by distinguishing Marr's computational and algorithmic levels of explanation, allowing that the ANS could function at the former level to represent numbers qua second-order entities, while leaving open exactly how this is done at the algorithmic level. The problem is that what is represented places constraints on how it is represented and so one cannot offer a theory at the computational level in complete abstraction from the algorithmic level (cf. Burge, ibid, p. 93, fn. 43). As things stand, the situation seems to be that if we could sense numbers, then the ANS would have to operate as Burge says; but there is little evidence that it does. To defeat this argument by showing how else the ANS could operate, more needs to be said about how their alternative “indirect model” of the ANS could generate representations of second-order numbers from representations of “a mishmash of non-numerical magnitudes” (sect. 4, para. 11).
Developing this line of thought, C&B argue that positing a sensory relation to numbers would be needed to unify ANS explanations of behavior and to explain the common function that our sensitivity to the aforementioned mishmash would be serving. However, the same explanatory benefits are gained by positing a sensory relation to multitudes of sensible objects, where these objects taken collectively possess non-numerical but quantitative properties from which an approximate representation of their multitude might be extracted by the ANS.
They might object that I cannot substitute talk of multitudes for talk of numerosities, because only the latter are, like numbers, second-order: just as we speak of the number of F's, so we speak of the numerosity of F's. I respond that the relevant sortal-kind of the objects making up the multitude can be discerned from the context if necessary. When I sense that there are two oranges, I sense an orange, another, and no more. When I sense that there are just as many oranges as there are apples, I sense an orange, another, and no more, an apple and another and no more, and match them up so that none are left over. If I cut off half an apple, leaving the remainder, I sense that there is an apple and a half, which is to say one and a half apples worth of apple, as well as three and half fruits worth of fruit. Therefore, this proposal also encompasses what C&B (mistakenly) call our sense of rational numbers. In any case, the sortal-kind is the natural analog of a unit of measurement that is provided by and can be discerned from the context. The connection to numbers is as follows:
The sensible world contains multitudes from which cardinal numbers are abstracted. The latter are then applied to the world as it is organized by sortals. Numbers are not simply properties of the sensible world but properties of the world as it is organized (cf. Gaifman, Reference Gaifman2005). The sensible world also contains magnitudes (as well as natural analogs of units of measurement), which fall into ratios from which rational numbers are abstracted. The ANS does not represent numbers; but, in virtue of approximately representing multitudes and ratios of magnitudes it is correlated approximately with numbers, because numbers by their nature apply to multitudes relative to sortals as well as to magnitudes and their ratios.
Financial support
This paper was written with the generous and exclusive support of a postdoctoral fellowship at Instituto de Investigaciones Filosóficas, Universidad Nacional Autónoma de México and without a grant from any funding agency, commercial, or not for profit sectors.
Conflict of interest
None.