On the linear representation of numbers: evidence from a new two-numbers-to-two positions task

Abstract

In the number-to-position methodology, a number is presented on each trial and the observer places it on a straight line in a position that corresponds to its felt subjective magnitude. In the novel modification introduced in this study, the two-numbers-to-two-positions method, a pair of numbers rather than a single number is presented on each trial and the observer places them in appropriate positions on the same line. Responses in this method indicate not only the subjective magnitude of each single number but, simultaneously, provide a direct estimation of their subjective numerical distance. The results of four experiments provide strong evidence for a linear representation of numbers and, commensurately, for the linear representation of numerical distances. We attribute earlier results that indicate a logarithmic representation to the ordered nature of numbers and to the task used and not to a truly non-linear underlying representation.

Can the Stroop effect serve as the gold standard of conflict monitoring and control? A conceptual critique

“…All too often, popular conjectures in psychology are taken for facts rather than for the theoretical notions they are. Two ready examples are the "mental number line" in numerical cognition (Dehaene, 1997; but see Bar et al, 2019) and the "attentional spotlight" (Posner et al, 1980; but see Shalev & Algom, 2000). It is important to keep in mind this caveat because, for many a student, the fact of control seems to be an article of faith.…”

Processing symbolic magnitude information conveyed by number words and by scalar adjectives

“…For example, when extracting a quantity from a visual scene, we are able to successfully distinguish a set of 15 dots from a set of 30 dots, but not 28 dots from 30 dots. Performance with nonsymbolic quantities in terms of accuracy and reaction times is dependent on the ratio between the two quantities to be compared such that larger ratios (i.e., larger relative difference in magnitude) lead to faster and more accurate responses than smaller ratios (i.e., smaller relative difference in magnitude; e.g., Buckley & Gillman, 1974;Feigenson et al, 2004;Halberda & Feigenson, 2008;Pica et al, 2004 this is consistent with Weber's law, see e.g., Bar et al, 2019 for a recent discussion). Such performance has been suggested to reflect the operation of the so-called Approximate Number System (ANS) which is thought to be an evolutionary old system shared with other animals (Barth et al, 2003;Dehaene, 1997;Feigenson et al, 2004;Gallistel & Gelman, 2000;Halberda & Feigenson, 2008).…”

Processing symbolic magnitude information conveyed by number words and by scalar adjectives

“…For example, when extracting a quantity from a visual scene, we are able to successfully distinguish a set of 15 dots from a set of 30 dots, but not 28 dots from 30 dots. Performance with nonsymbolic quantities in terms of accuracy and reaction times is dependent on the ratio between the two quantities to be compared such that larger ratios (i.e., larger relative difference in magnitude) lead to faster and more accurate responses than smaller ratios (i.e., smaller relative difference in magnitude; e.g., Buckley & Gillman, 1974;Feigenson, Dehaene, & Spelke, 2004;Halberda & Feigenson, 2008;Pica, Lemer, Izard, & Dehaene, 2004; this is consistent with Weber's law, see e.g., Bar, Fischer, & Algom, 2019 for a recent discussion). Such performance has been suggested to reflect the operation of the so-called Approximate Number System (ANS) which is thought to be an evolutionary old system shared with other animals (Barth, Kanwisher, & Spelke, 2003;Dehaene, 1997;Feigenson et al, 2004;Gallistel & Gelman, 2000;Halberda & Feigenson, 2008).…”

“…Numerical magnitude representations in ANS are thought to be continuous (or analog) distributions around a point (similar to a Gaussian distribution) which overlap with neighboring distributions. Two alternative accounts have been proposed 2 -either the spread of the distributions around points increases with increasing quantities or the spread of distributions is same for different quantities but the quantities are logarithmically compressed (e.g., Bar et al, 2019;Dehaene, Izard, Spelke, & Pica, 2008;Feigenson et al, 2004;Gallistel & Gelman, 1992;Merten & Nieder, 2008;Nieder, 2016; we leave this discussion aside as it does not matter for the purpose of the present study).…”