The approximate number system represents magnitude and precision

2021 ◽  
Vol 44 ◽  
Author(s):  
Charles R. Gallistel
Keyword(s):  
Fixed Point ◽  
Number System ◽  
Rational Numbers ◽  
Data Type ◽  
Approximate Number System ◽  
Point Data ◽  
Continuous Quantities

Abstract Numbers are symbols manipulated in accord with the axioms of arithmetic. They sometimes represent discrete and continuous quantities (e.g., numerosities, durations, rates, distances, directions, and probabilities), but they are often simply names. Brains, including insect brains, represent the rational numbers with a fixed-point data type, consisting of a significand and an exponent, thereby conveying both magnitude and precision.

Fixed Point Data Type Modeling for High Level Synthesis

2010 ◽  
Vol E93-C (3) ◽  
pp. 361-368
Author(s):  
Benjamin CARRION SCHAFER ◽  
Yusuke IGUCHI ◽  
Wataru TAKAHASHI ◽  
Shingo NAGATANI ◽  
Kazutoshi WAKABAYASHI
Keyword(s):  
Fixed Point ◽  
Data Type ◽  
High Level Synthesis ◽  
Point Data ◽  
High Level

The number sense does not represent numbers, but cardinality comparisons

2021 ◽  
Vol 44 ◽  
Author(s):  
José Luis Bermúdez
Keyword(s):  
Experimental Evidence ◽  
Number Sense ◽  
Number System ◽  
Rational Numbers ◽  
Approximate Number System

Abstract Against Clarke and Beck's proposal that the approximate number system (ANS) represents natural and rational numbers, I suggest that the experimental evidence is better accommodated by the (much weaker) thesis that the ANS represents cardinality comparisons. Cardinality comparisons do not stand in arithmetical relations and being able to apply them does not involve basic arithmetical concepts and operations.

A rational explanation for links between the ANS and math

2021 ◽  
Vol 44 ◽  
Author(s):  
Melissa E. Libertus ◽  
Shirley Duong ◽  
Danielle Fox ◽  
Leanne Elliott ◽  
Rebecca McGregor ◽  
...  
Keyword(s):  
Error Detection ◽  
Number System ◽  
Rational Numbers ◽  
Approximate Number System ◽  
Math Skills ◽  
Detection Mechanism ◽  
Affective Factors ◽  
Rational Explanation

Abstract The proposal by Clarke and Beck offers a new explanation for the association between the approximate number system (ANS) and math. Previous explanations have largely relied on developmental arguments, an underspecified notion of the ANS as an “error detection mechanism,” or affective factors. The proposal that the ANS represents rational numbers suggests that it may directly support a broader range of math skills.

The perception of quantity ain't number: Missing the primacy of symbolic reference

2021 ◽  
Vol 44 ◽  
Author(s):  
Rafael E. Núñez ◽  
Francesco d'Errico ◽  
Russell D. Gray ◽  
Andrea Bender
Keyword(s):  
Essential Role ◽  
Number System ◽  
Rational Numbers ◽  
Theoretical Construct ◽  
Approximate Number System

Abstract Clarke and Beck's defense of the theoretical construct “approximate number system” (ANS) is flawed in serious ways – from biological misconceptions to mathematical naïveté. The authors misunderstand behavioral/psychological technical concepts, such as numerosity and quantical cognition, which they disdain as “exotic.” Additionally, their characterization of rational numbers is blind to the essential role of symbolic reference in the emergence of number.

Ratio-based perceptual foundations for rational numbers, and perhaps whole numbers, too?

2021 ◽  
Vol 44 ◽  
Author(s):  
Edward M. Hubbard ◽  
Percival G. Matthews
Keyword(s):  
Numerical Cognition ◽  
Processing System ◽  
Number System ◽  
Rational Numbers ◽  
Approximate Number System ◽  
Access Points ◽  
Whole Numbers ◽  
Two Systems ◽  
Systems View

Abstract Clarke and Beck suggest that the ratio processing system (RPS) may be a component of the approximate number system (ANS), which they suggest represents rational numbers. We argue that available evidence is inconsistent with their account and advocate for a two-systems view. This implies that there may be many access points for numerical cognition – and that privileging the ANS may be a mistake.

Sizes, ratios, approximations: On what and how the ANS represents

2021 ◽  
Vol 44 ◽  
Author(s):  
Brian Ball
Keyword(s):  
Number System ◽  
Rational Numbers ◽  
Approximate Number System ◽  
Positive Integers

Abstract Clarke and Beck propose that the approximate number system (ANS) represents rational numbers. The evidence cited supports only the view that it represents ratios (and positive integers). Rational numbers are extensive magnitudes (i.e., sizes), whereas ratios are intensities. It is also argued that WHAT a system represents and HOW it does so are not as independent of one another as the authors assume.

Not so rational: A more natural way to understand the ANS

2021 ◽  
Vol 44 ◽  
Author(s):  
Eli Hecht ◽  
Tracey Mills ◽  
Steven Shin ◽  
Jonathan Phillips
Keyword(s):  
Number System ◽  
Rational Numbers ◽  
Approximate Number System ◽  
Natural Numbers ◽  
Wide Range ◽  
Natural Way

Abstract In contrast to Clarke and Beck's claim that that the approximate number system (ANS) represents rational numbers, we argue for a more modest alternative: The ANS represents natural numbers, and there are separate, non-numeric processes that can be used to represent ratios across a wide range of domains, including natural numbers.

The approximate number system represents rational numbers: The special case of an empty set

2021 ◽  
Vol 44 ◽  
Author(s):  
Michal Pinhas ◽  
Rut Zaks-Ohayon ◽  
Joseph Tzelgov
Keyword(s):  
Number System ◽  
Rational Numbers ◽  
Approximate Number System ◽  
Numerical Information ◽  
Special Case

Abstract We agree with Clarke and Beck that the approximate number system represents rational numbers, and we demonstrate our support by highlighting the case of the empty set – the non-symbolic manifestation of zero. It is particularly interesting because of its perceptual and semantic uniqueness, and its exploration reveals fundamental new insights about how numerical information is represented.

scholarly journals Perceived number is not abstract

2021 ◽  
Vol 44 ◽  
Author(s):  
Lauren S. Aulet ◽  
Stella F. Lourenco
Keyword(s):  
Number System ◽  
Rational Numbers ◽  
Second Order ◽  
Visual Illusions ◽  
Approximate Number System ◽  
Number Perception

Abstract To support the claim that the approximate number system (ANS) represents rational numbers, Clarke and Beck (C&B) argue that number perception is abstract and characterized by a second-order character. However, converging evidence from visual illusions and psychophysics suggests that perceived number is not abstract, but rather, is perceptually interdependent with other magnitudes. Moreover, number, as a concept, is second-order, but number, as a percept, is not.

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