A rational analysis of the approximate number system

Abstract

It is well-known in numerical cognition that higher numbers are represented with less absolute fidelity than lower numbers, often formalized as a logarithmic mapping. Previous derivations of this psychological law have worked by assuming that relative change in physical magnitude is the key psychologically-relevant measure (Fechner, 1860 ; Sun, Wang, Goyal, & Varshney, 2012 ; Portugal & Svaiter, 2011). Ideally, however, this property of psychological scales would be derived from more general, independent principles. This paper shows that a logarithmic number line is the one which minimizes the error between input and representation, relative to the probability that subjects would need to represent each number. This need probability is measured here through natural language and matches the form of need probabilities found in other literatures. The derivation does not presuppose anything like Weber's law and makes minimal assumptions about both the nature of internal representations and the form of the mapping. More generally, the results prove in a general setting that the optimal psychological scale will change with the square root of the probability of each input. For stimuli that follow a power-law need distribution this approach recovers either a logarithmic or power-law psychophysical mapping (Stevens, 1957, 1961, 1975).

Reflections of the social environment in chimpanzee memory: applying rational analysis beyond humans

“…Along with related findings for other cognitive processes (e.g. reasoning, resource allocation, numerical representation [6–10]), the rational analyses of human memory and human environments support the notion that information-processing systems adaptively exploit environmental structures to solve information-processing problems.…”

Zipf’s word frequency law in natural language: A critical review and future directions

“…Another domain where the meaning dependence of frequency is apparent is that of number words (Dehaene & Mehler 1992; Piantadosi, 2012). Figure 3 shows number word frequencies (e.g., “one,” “two,” “three,” etc.…”

The use of number words in natural language obeys Weber’s law.

“…Previous works on spontaneous verbal production have explored the mental organization of number concepts through the analysis of lexical frequencies (Dehaene & Mehler, 1992;Piantadosi, 2014Piantadosi, , 2016). Yet, this descriptive measure cannot satisfactorily replicate the main signatures of the ANS.…”

“…While this evidence on lexical frequency provides some indirect evidence for an ANS regulating verbal behavior about quantity, it has also some evident theoretical limitations, as it does not allow to replicate the core signatures of such system (i.e., ratio effect or Weber's law). Indeed, the frequency distribution of number words in language may simply reflect environmental pressures for representing small To appear in Journal of Experimental Psychology: General 9 numerical values, rather than being similar to the internal organization of the ANS (Piantadosi, 2016;Piantadosi & Cantlon, 2017).…”