Numbers, numerosities, and new directions

Abstract

In our target article, we argued that the number sense represents natural and rational numbers. Here, we respond to the 26 commentaries we received, highlighting new directions for empirical and theoretical research. We discuss two background assumptions, arguments against the number sense, whether the approximate number system (ANS) represents numbers or numerosities, and why the ANS represents rational (but not irrational) numbers.

Compositionality and constituent structure in the analogue mind

“…Alternatively, they could involve subjects representing some numerical range (e.g., 5-7 dots -Ball 2017) or (on my preferred view) a numerical range with some probability distribution or confidence level attached (Halberda 2016). All these possibilities can accommodate the abovementioned results provided that (e.g.,) noise in the extraction of number suffices to explain the ANS's characteristic imprecision and conformity to Weber's Law (Clarke & Beck 2021b). It is simply for ease of explication, that I will proceed by talking as though the ANS represents precise natural numbers (e.g., 6, 7, 8. .…”

“…11A further complication concerns whether the abovementioned results are better understood to involve a representation of cardinality as opposed to number , per se (Opfer et al. 2021; but see Clarke & Beck 2021b). This is an interesting possibility, deserving of discussion.…”

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., 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.…”

“…Somewhat distinctly, Peacocke (2021) objects that genuine rational number representation by the ANS requires that values be placed on a single "rational number line". Clarke and Beck (2021b) have responded to these concerns on broadly philosophical grounds. But while the current study was not designed to address these issues directly, we conjecture that there are empirically detectable differences between representations of ratios and fractions in the abovementioned experiments.…”

Rational Number Representation by the Approximate Number System

“…Somewhat distinctly, Peacocke (2021) objects that genuine rational number representation by the ANS requires that values be placed on a single "rational number line". Clarke and Beck (2021b) have responded to these concerns on broadly philosophical grounds. But while the current study was not designed to address these concerns directly, we conjecture that there are empirically tractable differences between representations of ratios and fractions.…”