scholarly journals The emergence of children’s natural number concepts: Current theoretical challenges

2021 ◽  
Author(s):  
Francesco Sella ◽  
Emily Slusser ◽  
Darko Odic ◽  
Attila Krajcsi
Keyword(s):  
Natural Number ◽  
Number Concepts

Do mental magnitudes form part of the foundation for natural number concepts? Don't count them out yet

2008 ◽  
Vol 31 (6) ◽  
pp. 644-645 ◽  
Author(s):  
Hilary Barth
Keyword(s):  
Natural Number ◽  
Number Concepts ◽  
Form Part

AbstractThe current consensus among most researchers is that natural number is not built solely upon a foundation of mental magnitudes. On their way to the conclusion that magnitudes do not form any part of that foundation, Rips et al. pass rather quickly by theories suggesting that mental magnitudes might play some role. These theories deserve a closer look.

The origins of number: Getting developmental

2008 ◽  
Vol 31 (6) ◽  
pp. 662-662
Author(s):  
Kelly S. Mix
Keyword(s):  
Natural Number ◽  
Real Time ◽  
Explanatory Power ◽  
Future Research ◽  
Number Concepts ◽  
Developmental Issues

AbstractRips et al. raise important questions about the relation between infant quantification and achievement of natural number concepts. However, they may be oversimplifying the interactions that characterize actual development in real time. Though they propose a worthwhile agenda for future research, its explanatory power will be limited if it does not address developmental issues with greater sensitivity.

The generative basis of natural number concepts

2008 ◽  
Vol 12 (6) ◽  
pp. 213-218 ◽  
Author(s):  
Alan M. Leslie ◽  
Rochel Gelman ◽  
C.R. Gallistel
Keyword(s):  
Natural Number ◽  
Number Concepts

Music training, engagement with sequence, and the development of the natural number concept in young learners

2008 ◽  
Vol 31 (6) ◽  
pp. 652-653 ◽  
Author(s):  
Martin F. Gardiner
Keyword(s):  
Natural Number ◽  
Learning Support ◽  
Number Concepts ◽  
Music Training ◽  
Number Concept ◽  
Young Learners ◽  
Musical Pitch ◽  
Innate Ability

AbstractStudies by Gardiner and colleagues connecting musical pitch and arithmetic learning support Rips et al.'s proposal that natural number concepts are constructed on a base of innate abilities. Our evidence suggests that innate ability concerning sequence (“Basic Sequencing Capability” or BSC) is fundamental. Mathematical engagement relating number to BSC does not develop automatically, but, rather, should be encouraged through teaching.

Natural number concepts: No derivation without formalization

2008 ◽  
Vol 31 (6) ◽  
pp. 666-667
Author(s):  
Paul Pietroski ◽  
Jeffrey Lidz
Keyword(s):  
Natural Number ◽  
Building Blocks ◽  
Logical Space ◽  
Number Concepts ◽  
New Concepts

AbstractThe conceptual building blocks suggested by developmental psychologists may yet play a role in how the human learner arrives at an understanding of natural number. The proposal of Rips et al. faces a challenge, yet to be met, faced by all developmental proposals: to describe the logical space in which learners ever acquire new concepts.

Why cardinalities are the “natural” natural numbers

2008 ◽  
Vol 31 (6) ◽  
pp. 659-659
Author(s):  
Mathieu Le Corre
Keyword(s):  
Natural Number ◽  
Prior Knowledge ◽  
Numerical Cognition ◽  
Independent Sets ◽  
Natural Numbers ◽  
Number Concepts

AbstractAccording to Rips et al., numerical cognition develops out of two independent sets of cognitive primitives – one that supports enumeration, and one that supports arithmetic and the concepts of natural numbers. I argue against this proposal because it incorrectly predicts that natural number concepts could develop without prior knowledge of enumeration.

The Natural Numbers

2018 ◽  
Author(s):  
Øystein Linnebo
Keyword(s):  
Natural Number ◽  
Peano Arithmetic ◽  
Number Sequence ◽  
Natural Numbers ◽  
Ordinal Numbers ◽  
Cardinal Numbers

How are the natural numbers individuated? That is, what is our most basic way of singling out a natural number for reference in language or in thought? According to Frege and many of his followers, the natural numbers are cardinal numbers, individuated by the cardinalities of the collections that they number. Another answer regards the natural numbers as ordinal numbers, individuated by their positions in the natural number sequence. Some reasons to favor the second answer are presented. This answer is therefore developed in more detail, involving a form of abstraction on numerals. Based on this answer, a justification for the axioms of Dedekind–Peano arithmetic is developed.

scholarly journals The Limits of Computation

Axiomathes ◽  
2021 ◽  
Author(s):  
Andrew Powell
Keyword(s):  
Natural Number ◽  
Set Theory ◽  
Lambda Calculus ◽  
Second Order ◽  
Functional Interpretation ◽  
Arbitrary Type ◽  
Computable Functions

AbstractThis article provides a survey of key papers that characterise computable functions, but also provides some novel insights as follows. It is argued that the power of algorithms is at least as strong as functions that can be proved to be totally computable in type-theoretic translations of subsystems of second-order Zermelo Fraenkel set theory. Moreover, it is claimed that typed systems of the lambda calculus give rise naturally to a functional interpretation of rich systems of types and to a hierarchy of ordinal recursive functionals of arbitrary type that can be reduced by substitution to natural number functions.

scholarly journals ADDITIVE BASES AND NIVEN NUMBERS

2021 ◽  
pp. 1-8
Author(s):  
CARLO SANNA
Keyword(s):  
Positive Integer ◽  
Natural Number ◽  
Riemann Hypothesis ◽  
Additive Basis

Abstract Let $g \geq 2$ be an integer. A natural number is said to be a base-g Niven number if it is divisible by the sum of its base-g digits. Assuming Hooley’s Riemann hypothesis, we prove that the set of base-g Niven numbers is an additive basis, that is, there exists a positive integer $C_g$ such that every natural number is the sum of at most $C_g$ base-g Niven numbers.

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