Clarke and Beck (C&B) reject Burge's (Reference Burge2010) hypothesis that the approximate number system (ANS) represents Eudoxian pure magnitudes. They maintain that ANS represents natural and rational numbers. But there are strong reasons – under-appreciated in the target article – to favor pure magnitudes as what ANS represents. C&B's considerations against the pure magnitudes hypothesis have no force.

Pure magnitudes can measure both continuous magnitudes (e.g., an object's weight) and the magnitude of an aggregate's membership. (An aggregate is a concrete analog of a set; Burge, Reference Burge1977.) Pure magnitudes always apply only relative to some property that delimits what is measured. In the continuous case, pure magnitudes apply to something (e.g., some water) relative to a dimension (like weight). In measuring an aggregate's membership, pure magnitudes are like natural numbers in having to apply relative to the sortal that determines an aggregate's members. (The same physical stuff can constitute 1 deck, 4 suits, or 52 cards; Frege, Reference Frege1884.) Crucially, pure magnitudes occur in ratios. Pure magnitudes are associative and commutative under analogs of addition and multiplication; and, for any magnitudes a and b, exactly one of these conditions holds: a = b + c, for some magnitude c; b = a + c, for some magnitude c; or a = b (Scott, Reference Scott1963). They do not, however, bear successor relations: there is no “next” pure magnitude.

The fundamental advantage of the pure magnitudes hypothesis is its comparative explanatory economy. First and most importantly, pure magnitudes have all the structure necessary to explain extant evidence relevant to ANS. Ratio-dependent discrimination behavior comprises the core data in ANS research. As mentioned, pure magnitudes occur in ratios. Number hypotheses concerning ANS representations predict capacities that the data do not support. For example, there is no evidence that ANS capacities include counting, one-to-one matching, or a successor operation. These are basic to a competence in representing natural numbers. They are no part of a competence for representing pure magnitudes. Second, on our view, pure magnitudes are already represented in perception: Continuous magnitudes are measured there by pure magnitudes relative to a dimension (weight and distance). Measuring continuous magnitudes by number would require both a dimension and a unit of measurement. Perception appears to be unit-free (Burge, Reference Burge2021; Peacocke, Reference Peacocke1986, Reference Peacocke2019). In sum, the pure magnitudes hypothesis is supported by the evidence, does not posit more than is necessary, and accords well with explanations of perceptual magnitude representation.

C&B, citing Burge (Reference Burge and Woodfield1982), correctly note that some representational competencies do not require sensitivity to every essential feature of what they represent. They also, citing Burge (Reference Burge2005), caution against confusing what is represented with how it is represented (the mode of presentation). Accordingly, one might think that ANS can represent natural numbers despite the absence of counting, one-to-one matching, or a successor operation, and thus that the numbers hypothesis need not postulate these capacities. We deny this. In typical cases, representational competence despite limited sensitivity to essential features is grounded in causal connections to the subject matter or in reliance on interlocutors. These factors are irrelevant here. The main evidence for competence in representing numbers for example, in developmental studies is evidence of capacities for counting, one-to-one matching, and a successor operation. These capacities constitute our main grip on whether numbers are represented.

Why, then, do C&B reject the pure magnitudes hypothesis? They offer two main considerations.

The first invokes the sortal-dependence of membership estimation stressed by Burge (Reference Burge2010). ANS relies on a sortal's distinguishing and grouping the members of an aggregate. Natural numbers must measure a magnitude relative to a sortal. A pure magnitude, by contrast, does not have to measure a magnitude relative to a sortal (as when it measures weight). C&B claim that pure magnitudes “are thus poorly suited to capturing the contents of ANS representations.” This argument has no weight. When pure magnitudes measure aggregate membership, they must hold relative to a sortal. That pure magnitudes can also measure continuous magnitudes without a sortal is irrelevant. ANS representations of pure magnitudes can thus be sensitive to the sortal-dependence of the magnitude of aggregates' membership, just as attributions of number would be.

C&B's second consideration is that we should favor the hypothesis that postulates representations of “entities we have independent reason to posit in our scientifically informed ontology.” This consideration is prima facie (agents can represent there to be entities that do not exist) and is overwhelmed by considerations of explanatory economy of the kind we advance. Furthermore, here the consideration does not favor the numbers hypothesis. C&B claim that scientific explanations refer to numbers, not to numerosities – and so, presumably, not to numerosities construed as pure magnitudes. But, in mathematical science, pure magnitudes are in as good-standing as numbers (Scott, Reference Scott1963). And, as noted, empirical science is already committed to attributions of pure magnitudes in its explanation of perception. C&B enlist Burge's (Reference Burge2010) use of ethology to settle whether frog vision represents flies or undetached fly parts. The case seems disanalogous. Ethology can break a tie between causal candidates. Different considerations are needed for mathematical entities, which apply to concrete particulars with causal powers only via further properties. Those considerations favor the pure magnitudes hypothesis.

C&B characterize pure magnitudes as “exotic” and numerosities more generally as “recherché” and “peculiar.” Quanticals are deemed “mysterious.” These formulations could suggest that positing “ersatz” numbers is problematic because they are unfamiliar and ill-understood. However, pure magnitudes have been theoretically well-understood since the ancient Greeks. Through their presentation in Euclid's Elements, they were central to mathematical and scientific practice up through the early modern period (Stein, Reference Stein1990; Sutherland, Reference Sutherland2006). They are indeed unfamiliar to, and not reflectively understood by, many possessors of ANS, which after all include organisms that may well lack supra-perceptual powers. But numbers are similarly unfamiliar to such creatures. Theorists need not be deterred.

The pure magnitudes hypothesis explains the behavioral data without invoking unevinced capacities (as the neologism “numerosity” cautioned against) and cites resources already deployed in perception. Numbers are more familiar to us. ANS represents pure magnitudes.