A natural view is that the approximate number system (ANS) represents… well, approximate number. That is, (i) its outputs are abstract, sortal-dependent, second-order representations, serving as answers to the question “how many?” (rather than, say, “how much?”); but (ii) these representations are semantically approximate: their contents are something like 13ish, and not so precise as 13, certainly not 13.7. This would be an especially good view for Clarke and Beck (C&B) to embrace. If they're right to reject the sensitivity principle then (i) and (ii) are consistent; there need be nothing oxymoronic about “approximate number.”
Instead, however, they claim that ANS represents rational numbers, like 1/3, 3.5, and 2.75, a claim not easily squared with this natural view.
C&B's argument for thinking that ANS represents rational numbers is based on studies indicating a sensitivity not just to numerosities but also to ratios among these numerosities (e.g., Denison & Xu, Reference Denison and Xu2014; McCrink & Wynn, Reference McCrink and Wynn2007). These studies, however, don't support C&B's claim. It is one thing to say that ANS represents (approximate and natural) numbers and ratios among them, and quite another to say that it represents the rational numbers. The former is much more plausible than the latter. Among the few examples C&B give of non-natural rational numbers are 3.5 (sect. 7.3, para. 1) and 2.75 (sect. 7.3, para. 10), but they present no evidence that ANS represents such numbers, indeed no evidence that ANS represents any non-natural numbers greater than 1. No evidence suggests that ANS represents some aggregate as containing 3.5 items, or as containing 2.75 times as many yellow items as red ones.
C&B should have said that ANS employs contents like 1/3ish in addition to 13ish. By claiming that ANS represents rational numbers, like 2.75, rather than approximate ratios, C&B seem to attribute to ANS greater precision than had it merely represented natural numbers, when what it needed was less. They address this tension by insisting that, although ANS represents precise quantities, it represents them imprecisely; it’s the representation that’s imprecise, not what it represents.
There are two ways to read this claim.
On the first interpretation, C&B are saying that although an ANS representation has precise truth conditions, it is, perhaps because of the physical or syntactic properties of the representational vehicle, easily confused for some other representation, with similar but distinct truth conditions. This is a kind of “vehicular imprecision” rather than “semantic imprecision.” A vehicular-imprecision-with-semantic-precision view makes the same behavioral predictions as a semantic imprecision view, but the former implies that, because the ANS truth-conditions are precise (e.g., 13, rather than 13ish), ANS representations will be in error very often, perhaps much more often than not. (Even if we discriminate 13 from 14 at levels somewhat better than chance, and even if that shows that we're applying 13 contents more than half of those times, we would need to discriminate 13 from the disjunction 11-or-12-or-14-or-15 [etc.] at better than chance in order to be applying 13 correctly more often than not.) A semantic imprecision view will ascribe much less error, as a 13ish verdict is presumably true of a 14-item array. Everything else equal, a theory that ascribes less representational error is to be preferred over one that posits more, and C&B, on this interpretation, seem to be ascribing error quite gratuitously by insisting on precise contents.
A second interpretation sees C&B espousing semantic imprecision after all, allowing them to embrace the natural view I started with. Here, the precision lies not in the representation, but in the representational “target” (Cummins, Reference Cummins1996; C&B call it “referent,” [sect. 2.2; note 1; sect. 6, para. 10]), that is, the thing to which the representation is applied, and of which that content is predicated. Although Jones has a precise weight, we might represent that weight imprecisely (sect. 5.1, para. 2). This avoids the problems for the first interpretation, but it's no longer the claim that ANS represents rational numbers, in the only sense that could be relevant. Suppose I misrepresent a dog as a cat. I thereby apply a cat representation to what is, in fact, a dog; that dog is the target of this cat. If we were to say – misleadingly, with C&B – that the dog was the “referent” of cat, then we would be tempted to claim that cat represents (/means, /refers to) dogs, but this is clearly false, at least on any standard construal. Yet it's exactly this reasoning that C&B use to argue that ANS represents rational numbers.
If C&B hold merely that rational numbers are targets of ANS representations, then it's unclear where they disagree with Carey (Reference Carey2009) and Núñez (Reference Núñez2017), both of whom are surely aware that aggregates typically contain precise numbers of items and thus agree that ANS represents a precise thing imprecisely in this sense.
But anyway, C&B can't – or can't only – be saying that we apply a 16ish to instances of 16.29 in the world. They’re saying we apply 16.29s, if they’re saying anything. The claim that ANS represents rational numbers is supposed to be explanatory, but it can't be explanatory if it's only a claim about the targets of ANS representations and not the contents. It is completely unexplanatory to claim that ANS represents a 2:1 ratio of yellow to red items, if that claim is only a statement about the stimulus and leaves open all possibilities about how ANS represents that stimulus.
Because they reject the sensitivity principle, however, C&B didn't need any of this trouble. If sensitivity is false, then precision never needed to figure into C&B’s account. They could have simply claimed that ANS represents approximate natural numbers and ratios among these.
A natural view is that the approximate number system (ANS) represents… well, approximate number. That is, (i) its outputs are abstract, sortal-dependent, second-order representations, serving as answers to the question “how many?” (rather than, say, “how much?”); but (ii) these representations are semantically approximate: their contents are something like 13ish, and not so precise as 13, certainly not 13.7. This would be an especially good view for Clarke and Beck (C&B) to embrace. If they're right to reject the sensitivity principle then (i) and (ii) are consistent; there need be nothing oxymoronic about “approximate number.”
Instead, however, they claim that ANS represents rational numbers, like 1/3, 3.5, and 2.75, a claim not easily squared with this natural view.
C&B's argument for thinking that ANS represents rational numbers is based on studies indicating a sensitivity not just to numerosities but also to ratios among these numerosities (e.g., Denison & Xu, Reference Denison and Xu2014; McCrink & Wynn, Reference McCrink and Wynn2007). These studies, however, don't support C&B's claim. It is one thing to say that ANS represents (approximate and natural) numbers and ratios among them, and quite another to say that it represents the rational numbers. The former is much more plausible than the latter. Among the few examples C&B give of non-natural rational numbers are 3.5 (sect. 7.3, para. 1) and 2.75 (sect. 7.3, para. 10), but they present no evidence that ANS represents such numbers, indeed no evidence that ANS represents any non-natural numbers greater than 1. No evidence suggests that ANS represents some aggregate as containing 3.5 items, or as containing 2.75 times as many yellow items as red ones.
C&B should have said that ANS employs contents like 1/3ish in addition to 13ish. By claiming that ANS represents rational numbers, like 2.75, rather than approximate ratios, C&B seem to attribute to ANS greater precision than had it merely represented natural numbers, when what it needed was less. They address this tension by insisting that, although ANS represents precise quantities, it represents them imprecisely; it’s the representation that’s imprecise, not what it represents.
There are two ways to read this claim.
On the first interpretation, C&B are saying that although an ANS representation has precise truth conditions, it is, perhaps because of the physical or syntactic properties of the representational vehicle, easily confused for some other representation, with similar but distinct truth conditions. This is a kind of “vehicular imprecision” rather than “semantic imprecision.” A vehicular-imprecision-with-semantic-precision view makes the same behavioral predictions as a semantic imprecision view, but the former implies that, because the ANS truth-conditions are precise (e.g., 13, rather than 13ish), ANS representations will be in error very often, perhaps much more often than not. (Even if we discriminate 13 from 14 at levels somewhat better than chance, and even if that shows that we're applying 13 contents more than half of those times, we would need to discriminate 13 from the disjunction 11-or-12-or-14-or-15 [etc.] at better than chance in order to be applying 13 correctly more often than not.) A semantic imprecision view will ascribe much less error, as a 13ish verdict is presumably true of a 14-item array. Everything else equal, a theory that ascribes less representational error is to be preferred over one that posits more, and C&B, on this interpretation, seem to be ascribing error quite gratuitously by insisting on precise contents.
A second interpretation sees C&B espousing semantic imprecision after all, allowing them to embrace the natural view I started with. Here, the precision lies not in the representation, but in the representational “target” (Cummins, Reference Cummins1996; C&B call it “referent,” [sect. 2.2; note 1; sect. 6, para. 10]), that is, the thing to which the representation is applied, and of which that content is predicated. Although Jones has a precise weight, we might represent that weight imprecisely (sect. 5.1, para. 2). This avoids the problems for the first interpretation, but it's no longer the claim that ANS represents rational numbers, in the only sense that could be relevant. Suppose I misrepresent a dog as a cat. I thereby apply a cat representation to what is, in fact, a dog; that dog is the target of this cat. If we were to say – misleadingly, with C&B – that the dog was the “referent” of cat, then we would be tempted to claim that cat represents (/means, /refers to) dogs, but this is clearly false, at least on any standard construal. Yet it's exactly this reasoning that C&B use to argue that ANS represents rational numbers.
If C&B hold merely that rational numbers are targets of ANS representations, then it's unclear where they disagree with Carey (Reference Carey2009) and Núñez (Reference Núñez2017), both of whom are surely aware that aggregates typically contain precise numbers of items and thus agree that ANS represents a precise thing imprecisely in this sense.
But anyway, C&B can't – or can't only – be saying that we apply a 16ish to instances of 16.29 in the world. They’re saying we apply 16.29s, if they’re saying anything. The claim that ANS represents rational numbers is supposed to be explanatory, but it can't be explanatory if it's only a claim about the targets of ANS representations and not the contents. It is completely unexplanatory to claim that ANS represents a 2:1 ratio of yellow to red items, if that claim is only a statement about the stimulus and leaves open all possibilities about how ANS represents that stimulus.
Because they reject the sensitivity principle, however, C&B didn't need any of this trouble. If sensitivity is false, then precision never needed to figure into C&B’s account. They could have simply claimed that ANS represents approximate natural numbers and ratios among these.
Acknowledgments
The author thanks Sam Clarke for comments.
Conflict of interest
None.