Clarke and Beck (C&B) argue that the approximate number system (ANS) represents rational numbers. This claim is based on a growing body of evidence that suggests humans are capable of comparing ratios between natural numbers. They argue that the most straightforward explanation for this ability is that the ANS represents rational numbers, which can capture ratios between numbers and can themselves be compared. We find C&B's careful argument that the ANS represents numbers, rather than non-numerical confounds, persuasive. However, we find their argument that the ANS represents rational numbers to be less careful, and less persuasive. We argue for a more modest alternative: The ANS represents natural numbers, and there are separate, non-numeric processes that can be used to represent ratios across a wide range of domains, including natural numbers.
The primary argument that the ANS represents rational numbers, as far as we can tell, is that people can represent ratios between natural numbers, and that to do this, they must be using rational number representations. This ratio processing system (RPS), they argue, is best understood as a component of the ANS, and thus the ANS must be representing rational numbers. Surprisingly, the central evidence for this connection between the RPS and ANS is that both systems seem to be governed by Weber's Law, and this similarity in performance is taken to be “suggestive of a shared system.” But, as C&B themselves point out, a great number of representational systems, including those for distance, duration, and weight, also seem to be governed by Weber's Law. Yet they specifically need to resist the claim that systems for representing duration, weight, or distance are part of a shared system that also includes number representation. They can't have it both ways, and so even C&B should not think that simply conforming to Weber's Law is sufficient evidence for being part of a shared system.
However, you might wonder whether C&B could provide some other kind of evidence that RPS and ANS form part of a shared system, perhaps evidence that they share some shared neural substrate. But C&B make it explicit that their conjecture concerns only a computational level of analysis, so even if some implementation-level link were discovered, it would not help their argument. They need to rely on functional properties of these systems, but the only one they have to offer is conforming to Weber's Law. Yet that property is shared so widely that it is not much evidence one way or another.
Finally, one might wonder whether C&B may actually be appealing to some more general principle in their argument that the RPS represents rational numbers. Perhaps, they are appealing to the argument that because the RPS clearly represents relationships between numbers, RPS representations themselves must be number representations. But this is not a good form of inference. It is not generally true that a representation of the relationship between two entities is of the same kind as the representation of those two entities. One might have an intuitive sense that a chair and couch are more similar to each other than a chair and a lamp. But clearly, the representation of the similarity between two items of furniture is not itself a furniture representation. Analogously, a representation of the relationship between numbers (such as a ratio) need not itself be a numerical representation, and a fortiori, need not be a rational number representation.
Given that this is not a generally valid form of inference, C&B must provide some other form of evidence that ratio representations are numeric. This conclusion is not obvious; for example, we are able to represent the ratios between lengths of lines according to Weber's Law, but these representations don't strike one as necessarily numeric.
The remaining question, then, is whether there is a better way to explain the representation of ratios between natural numbers. Here is one such alternative: The ability to represent ratios between natural numbers in accordance with Weber's law arises from the same general non-numeric ability that allows us to represent relationships between all kinds of things, whether it be a matter of length, weight, duration, color, or whatever else. When we discriminate between different numeric ratios, we may simply be applying this quite general ability to the genuinely numeric representations of the ANS.
And so, C&B have not proven that representations of ratios between natural numbers are necessarily rational number representations, nor have they provided strong evidence that the RPS is a component system of the ANS. A more natural way to understand the ANS is that it simply represents natural, not rational, numbers, and that ratio representations rely on a separate, domain-general process.
Clarke and Beck (C&B) argue that the approximate number system (ANS) represents rational numbers. This claim is based on a growing body of evidence that suggests humans are capable of comparing ratios between natural numbers. They argue that the most straightforward explanation for this ability is that the ANS represents rational numbers, which can capture ratios between numbers and can themselves be compared. We find C&B's careful argument that the ANS represents numbers, rather than non-numerical confounds, persuasive. However, we find their argument that the ANS represents rational numbers to be less careful, and less persuasive. We argue for a more modest alternative: The ANS represents natural numbers, and there are separate, non-numeric processes that can be used to represent ratios across a wide range of domains, including natural numbers.
The primary argument that the ANS represents rational numbers, as far as we can tell, is that people can represent ratios between natural numbers, and that to do this, they must be using rational number representations. This ratio processing system (RPS), they argue, is best understood as a component of the ANS, and thus the ANS must be representing rational numbers. Surprisingly, the central evidence for this connection between the RPS and ANS is that both systems seem to be governed by Weber's Law, and this similarity in performance is taken to be “suggestive of a shared system.” But, as C&B themselves point out, a great number of representational systems, including those for distance, duration, and weight, also seem to be governed by Weber's Law. Yet they specifically need to resist the claim that systems for representing duration, weight, or distance are part of a shared system that also includes number representation. They can't have it both ways, and so even C&B should not think that simply conforming to Weber's Law is sufficient evidence for being part of a shared system.
However, you might wonder whether C&B could provide some other kind of evidence that RPS and ANS form part of a shared system, perhaps evidence that they share some shared neural substrate. But C&B make it explicit that their conjecture concerns only a computational level of analysis, so even if some implementation-level link were discovered, it would not help their argument. They need to rely on functional properties of these systems, but the only one they have to offer is conforming to Weber's Law. Yet that property is shared so widely that it is not much evidence one way or another.
Finally, one might wonder whether C&B may actually be appealing to some more general principle in their argument that the RPS represents rational numbers. Perhaps, they are appealing to the argument that because the RPS clearly represents relationships between numbers, RPS representations themselves must be number representations. But this is not a good form of inference. It is not generally true that a representation of the relationship between two entities is of the same kind as the representation of those two entities. One might have an intuitive sense that a chair and couch are more similar to each other than a chair and a lamp. But clearly, the representation of the similarity between two items of furniture is not itself a furniture representation. Analogously, a representation of the relationship between numbers (such as a ratio) need not itself be a numerical representation, and a fortiori, need not be a rational number representation.
Given that this is not a generally valid form of inference, C&B must provide some other form of evidence that ratio representations are numeric. This conclusion is not obvious; for example, we are able to represent the ratios between lengths of lines according to Weber's Law, but these representations don't strike one as necessarily numeric.
The remaining question, then, is whether there is a better way to explain the representation of ratios between natural numbers. Here is one such alternative: The ability to represent ratios between natural numbers in accordance with Weber's law arises from the same general non-numeric ability that allows us to represent relationships between all kinds of things, whether it be a matter of length, weight, duration, color, or whatever else. When we discriminate between different numeric ratios, we may simply be applying this quite general ability to the genuinely numeric representations of the ANS.
And so, C&B have not proven that representations of ratios between natural numbers are necessarily rational number representations, nor have they provided strong evidence that the RPS is a component system of the ANS. A more natural way to understand the ANS is that it simply represents natural, not rational, numbers, and that ratio representations rely on a separate, domain-general process.
Conflict of interest
No conflicts of interest to declare.