Real models: The limits of behavioural evidence for understanding the ANS

Abstract

Clarke and Beck use behavioural evidence to argue that (1) approximate ratio computations are sufficient for claiming that the approximate number system (ANS) represents the rationals, and (2) the ANS does not represent the reals. We argue that pure behaviour is a poor litmus test for this problem, and that we should trust the psychophysical models that place ANS representations within the reals.

Rational number representation by the approximate number system

“…But while these findings are consistent with the suggestion that rational numbers are then extracted by a secondary stage of ANS processing, which extracts and represents ratios/fractions based on that system's prior extraction of natural number, it is an open question whether ratio comparisons of this sort are driven by the ANS itself (pace Clarke & Beck, 2021a). Perhaps most obviously, it could instead be that ratio comparisons result from participants' domain general reflection on the relationship between the natural numbers of items that their ANS has extractedi.e., the number of blue items, red items, or red and blue items in each rationot least, because the ANS's characteristic imprecision in representing natural numbers could potentially explain why subsequent ratio comparisons conform to Weber's Law in the way we observed (Dramkin & Odic, 2021).…”

“…The vast majority of ANS research focusses exclusively on the system's representation and discrimination of natural numbers; for instance, the system's capacity to discriminate 8 items from 16 items, or to add the number of whole items in two collections together. While some have proposed that the ANS goes beyond representing natural number by representing real numbers quite generally (e.g., Dramkin & Odic, 2021;Gallistel & Gelman, 2000), arguments in favour of this conjecture have been roundly criticized (Laurence & Margolis, 2005;Carey, 2009;Clarke & Beck, 2021a;Clarke & Beck, 2021b;Samuels & Snyder, 2024; see also Gallistel, 2021 who clarifies that he no longer endorses his original argument). Consequently, it is tempting to suppose that the ANS is exclusively in the business of representing whole or natural number.…”

“…For instance, having represented that there are 8 or 8ish blue items in one subset or collection and 10 or 10ish red items in another subset or collection, participants might appreciate that these numbers stand in an 8 to 10 ratio to one another, by employing post-perceptual cognitive resources involved in domain general reasoning. Indeed, Dramkin and Odic (2021) note that, if this were so, the resulting ratio discriminations (which conform to Weber's Law) might inherit their characteristic imprecision from the imprecision of the ANS's representations of natural number which serve as input to this central cognitive process of ratio discrimination. To adjudicate this concern, we conducted a second experiment in which participants engaged in a secondary verbal shadowing task, while performing the ratio comparison task tested in Experiment 1.…”

Rational Number Representation by the Approximate Number System

“…The vast majority of ANS research focusses exclusively on the system's representation and discrimination of natural numbers; for instance, the system's capacity to discriminate 8 items from 16 items, or to add the number of whole items in two collections together. While some have proposed that the ANS goes beyond representing natural number by representing real numbers quite generally (e.g., Gallistel & Gelman, 2000;Dramkin & Odic, 2021), arguments in favour of this conjecture have been roundly criticized ( Gallistel 2021 who clarifies that he no longer endorses his original argument). Consequently, it is tempting to suppose that the ANS is exclusively in the business of representing whole or natural number.…”

Rational Number Representation by the Approximate Number System

“…The vast majority of ANS research focusses exclusively on the system's representation and discrimination of natural numbers; for instance, the system's capacity to discriminate 8 items from 16 items, or to add the number of whole items in two collections together. While some have proposed that the ANS goes beyond representing natural numbers by representing real numbers quite generally (e.g., Gallistel & Gelman 2000;Dramkin & Odic 2021) arguments in favour of this conjecture have been roundly criticized (Laurence & Margolis 2005). Consequently, it is tempting to suppose that the ANS is exclusively in the business of representing whole or natural numbers.…”