The proposal by Clarke and Beck (C&B) that the approximate number system (ANS) represents rational numbers is specific and intriguing. In this comment, I shall speculate on representational mechanisms through which rationals are deployed by the mind as a numerical representation of magnitude continuum.

Representation of magnitude of sensations was long understood, through the ideas of “just-noticeable difference” (Weber fraction) and Fechner's logarithmic scaling of sensation in relation to the intensity of eliciting stimuli. With a series of psychophysical experiments using a variety of direct-scaling paradigms (e.g., magnitude estimation, magnitude production, and cross-modality matching), Stevens (Stevens, Reference Stevens1936, Reference Stevens1956; Stevens & Galanter, Reference Stevens and Galanter1957) unequivocally established this psychophysical scale as an ordered continuum admitting concatenation operation, and proposed the power-law to map physical intensities to a psychological scale (Stevens, Reference Stevens1957). Fechner's logarithmic scaling and Stevens' power scaling can be reconciled as the former can be taken as the extreme case of the latter, when the exponent β approaches zero:

$${1\over \beta} ( I^\beta -1) \to {\rm log}\,I.$$

When it comes to numerical representation of the magnitude scale, things become subtler because of co-existing, symbolic and non-symbolic aspects of numerals (Feigenson, Carey, & Spelke, Reference Feigenson, Carey and Spelke2002; Feigenson, Dehaene, & Spelke, Reference Feigenson, Dehaene and Spelke2004; Lemer, Dehaene, Spelke, & Cohen, Reference Lemer, Dehaene, Spelke and Cohen2003; Xu, Spelke, & Goddard, Reference Xu, Spelke and Goddard2005). Numerosity as the pre-linguistic faculty to represent numerical information, possibly congruent with the psychophysical scale mentioned above, is clearly different from counting (Carey, Reference Carey2001, Reference Carey2009; Le Corre & Carey, Reference Le Corre and Carey2007; Sarnecka & Carey, Reference Sarnecka and Carey2008), which involves the mastery of an array of cognitive routines such as a “successor” function, one-to-one correspondence, and “cardinality” and “equality” of sets for establishing the concept of integer (Carey & Barner, Reference Carey and Barner2019). Whether or not subitization (the process of immediate and accurate recognition of small numbers) plays any role for enabling the counting routine, the resulting system of integers acquires an approximate numerosity scale, and maps linearly or logarithmically to the psychophysical scale (Dehaene, Izard, Spelke, & Pica, Reference Dehaene, Izard, Spelke and Pica2008; Núñez, Cooperrider, & Wassmann, Reference Núñez, Cooperrider and Wassmann2012). This is possible because both counting (which is anchored on “successor function”) and numerosity estimation (which is subject to Weber's Law) are predicated on individuation of objects.

The authors referenced the work of He, Zhang, Zhou, and Chen (Reference He, Zhang, Zhou and Chen2009). In that paper, participants were briefly presented with visual displays of dots in random positions and were asked to judge their numerosities; the brevity of stimulus presentations precluded any counting strategies. In some displays, pairs of adjacent dots were connected by line segments whereas in others, line segments were freely hanging without touching the dots (see the reproduced Fig. 1d of C&B). The line segments were introduced to manipulate object individuation aspect of numerosity estimation. Results clearly showed that connecting the pairs of dots by line segments led to an underestimation of dot numbers in those patterns because of decreased individuation. Thus, we suggested “two different stages underlying numerosity perception: first, the individuation of items in a visual display, followed by magnitude estimation based on the distinct number of items” (p. 517).

Numerosity estimation mechanism (approximate integer scale) mentioned above is, of course, congruent with the psychophysical scale revealed in Steven's direct-scaling experiments. In the language of axiomatic foundation of measurement (FoM) (Krantz, Luce, Suppes, & Tversky, Reference Krantz, Luce, Suppes and Tversky1971), the common underlying measurement structure is that of ordered concatenation group. Their difference is that numerosity scale is countable, and hence the group of integers (Z, +), whereas the psychophysical scale is uncountable, and hence the group of reals (R, +) which is isomorphic to the group of positive reals (R₊, ×). Note that in any ordered concatenation structure (M, ${\oplus} $), there is only one binary operator, ${\oplus} $, on the set M (which can be R, R₊, or Z) of elements whose magnitude are in total “order.” This binary operator ${\oplus} $, takes in two elements of the set and outputs another element larger than any of its inputs. That (R, +) and (R₊, ×) are mathematically isomorphic can be easily seen by the identity: log(a × b) = log a + log b. Cohen and Narens (Cohen & Narens, Reference Cohen and Narens1979; Narens, Reference Narens1981) developed the elaborative theory of ratio-scalability for extensive concatenation systems and the associated scale types.

The magnitude system conceptualized as above does not automatically come with a multiplication or ⊗ operation that accompanies (and distributes across) the ${\oplus} $ operation. That is, the magnitude system is not an algebra with two interwoven operations. To remedy this in FoM, Narens (Luce & Narens, Reference Luce and Narens1987; Narens & Luce, Reference Narens and Luce1986) proposed to use the automorphism group on the representational space, either (R, +) or (R₊, ×), as a surrogate of ⊗. An automorphism σ of a measurement structure is a bijective transformation from the structure to itself (i.e., mapping one element to another) that preserves the order relationship among its elements:

$$\sigma ( a\oplus b) = \sigma ( a) \oplus \sigma ( b) , \;\quad a > b\;{\rm iff}\;\sigma ( a) > \sigma ( b) .$$

All automorphisms form (generally non-commutative) group, with group multiplication operation ⊗ implemented as successive application of two automorphisms: σ ₂(σ ₁(a)) = (σ ₂ ⊗ σ ₁)(a). Distributivity of ⊗ over ${\oplus} $ always holds, but ⊗ is, in general, a non-commutative multiplication.

In FoM (Krantz et al., Reference Krantz, Luce, Suppes and Tversky1971), commutative multiplication ⊗ of two elements is through the construction of a conjoint measurement structure. This is the structure involved in tradeoff of Utility and Risk in Value = (Utility, Risk), and Length and Width in Area = (Length, Width). In the current case, we have Q = (D, N) where D is the denominator and N the numerator of a rational number Q. Here, both (D, +) and (N, +) are integer or numerosity structure (Z, +).

Additive conjoint structure is axiomatized by various cancellation conditions across its two underlying “components.” Essential to an additive conjoint measurement is the independence assumptions about its components and the existence of tradeoff, or the “indifference curve” of equal value.

When the rational field Q = (D, N) is constructed this way, addition and multiplication of two rationals Q₁ and Q₂ can be quite convoluted. Therefore, achieving “dense” representation of magnitude scale using rationals, as C&B suggested, has a heavy price toll when computing the fraction addition and fraction multiplication using only the routines for (D, +) and (N, +). Interference effects should readily be expected.

To summarize, I endorse the proposal of C&B to construct ANS by rationals, and amend it with a suggestion that such representation is achieved through conjoint measurement and tradeoff between the approximate integer (numerosity) representations of the denominator and the numerator.