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.

Contents of the approximate number system

2021 ◽  
Vol 44 ◽  
Author(s):  
Jack C. Lyons
Keyword(s):  
Number System ◽  
Rational Numbers ◽  
Approximate Number System ◽  
Natural Numbers

Abstract Clarke and Beck argue that the approximate number system (ANS) represents rational numbers, like 1/3 or 3.5. I think this claim is not supported by the evidence. Rather, I argue, ANS should be interpreted as representing natural numbers and ratios among them; and we should view the contents of these representations are genuinely approximate.

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.

Classical Arithmetization

2019 ◽  
pp. 26-50
Author(s):  
John Stillwell
Keyword(s):  
Number System ◽  
Rational Numbers ◽  
Continuous Functions ◽  
Complex Numbers ◽  
Natural Numbers ◽  
Infinite Sums ◽  
Why Questions

This chapter describes how one proceeds from natural to rational numbers, then to real and complex numbers, and to continuous functions—thus arithmetizing the foundations of analysis and geometry. The definitions of integers and rational numbers show why questions about them can, in principle, be reduced to questions about natural numbers and their addition and multiplication. This is what it means to say that the natural numbers are a foundation for the integer and rational numbers. But the next steps in the arithmetization project go beyond algebra. By admitting sets of rational numbers, one can enlarge the number system to one that admits certain infinite operations, such as forming infinite sums. This is crucial to building a foundation for analysis. As such, the chapter turns to the foundations of the natural numbers themselves, the “Peano axioms,” which gives a first glimpse of the logic underlying the arithmetization project.

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.

A pair of rational double sequences

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Serik Altynbek ◽  
Heinrich Begehr
Keyword(s):  
Natural Number ◽  
Complex Plane ◽  
Rational Numbers ◽  
The Other ◽  
Number Sequence ◽  
Double Sequences ◽  
Natural Numbers ◽  
Reflection Process ◽  
Natural Way

Abstract Double sequences appear in a natural way in cases of iteratively given sequences if the iteration allows to determine besides the successors from the predecessors also the predecessors from their followers. A particular pair of double sequences is considered which appears in a parqueting-reflection process of the complex plane. While one end of each sequence is a natural number sequence, the other consists of rational numbers. The natural numbers sequences are not yet listed in OEIS Wiki. Complex versions from the double sequences are provided.

1. Numbers and algebra

2015 ◽  
pp. 1-10
Author(s):  
Peter M. Higgins
Keyword(s):  
Fundamental Theorem ◽  
Common Factor ◽  
Number System ◽  
Rational Numbers ◽  
Euclidean Algorithm ◽  
Greatest Common Divisor ◽  
Natural Numbers ◽  
Division Algorithm ◽  
Fundamental Theorem Of Arithmetic ◽  
And Algebra

‘Numbers and algebra’ introduces the number system and explains several terms used in algebra, including natural numbers, positive and negative integers, rational numbers, number factorization, the Fundamental Theorem of Arithmetic, Euclid’s Lemma, the Division Algorithm, and the Euclidean Algorithm. It proves that any common factor c of a and b is also a factor of any number of the form ax + by, and since the greatest common divisor (gcd) of a and b has this form, which may be found by reversing the steps of the Euclidean Algorithm, it follows that any common factor c of a and b divides their gcd d.

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.

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