Exact Equality and Successor Function: Two Key Concepts on the Path towards Understanding Exact Numbers

Humans possess two nonverbal systems capable of representing numbers, both limited in their representational power: the first one represents numbers in an approximate fashion, and the second one conveys information about small numbers only. Conception of exact large numbers has therefore been thought to arise from the manipulation of exact numerical symbols. Here, we focus on two fundamental properties of the exact numbers as prerequisites to the concept of EXACT NUMBERS: the fact that all numbers can be generated by a successor function and the fact that equality between numbers can be defined in an exact fashion. We discuss some recent findings assessing how speakers of Mundurucú (an Amazonian language), and young Western children (3–4 years old) understand these fundamental properties of numbers.

[1]  Brian Butterworth,et al.  Verbal Counting and Spatial Strategies in Numerical Tasks: Evidence from Indigenous Australia , 2008 .

[2]  E. Spelke,et al.  Large number discrimination in 6-month-old infants , 2000, Cognition.

[3]  C. Gallistel,et al.  Nonverbal Counting in Humans: The Psychophysics of Number Representation , 1999 .

[4]  Margaret C. Donaldson,et al.  Conservation accidents , 1975, Cognition.

[5]  E. Spelke,et al.  Language and Conceptual Development series Core systems of number , 2004 .

[6]  Gabriel Segal.  The Innate Mind: Foundations and the Future , 2007 .

[7]  Elizabeth S Spelke,et al.  The development of language and abstract concepts: the case of natural number. , 2008, Journal of experimental psychology. General.

[8]  S. Dehaene,et al.  Exact and Approximate Arithmetic in an Amazonian Indigene Group , 2004, Science.

[9]  C. Gallistel,et al.  Non-verbal numerical cognition: from reals to integers , 2000, Trends in Cognitive Sciences.

[10]  C. Gallistel,et al.  The Child's Understanding of Number , 1979 .

[11]  Susan Carey,et al.  One, two, three, four, nothing more: An investigation of the conceptual sources of the verbal counting principles , 2007, Cognition.

[12]  J. G. Wallace,et al.  Cognitive development: An information-processing view , 1976 .

[13]  S. Carey,et al.  How counting represents number: What children must learn and when they learn it , 2008, Cognition.

[14]  P. Gordon.  Numerical Cognition Without Words: Evidence from Amazonia , 2004, Science.

[15]  S. Gelman,et al.  Six does not just mean a lot: preschoolers see number words as specific , 2004, Cognition.

[16]  J. Cantlon,et al.  Shared System for Ordering Small and Large Numbers in Monkeys and Humans , 2006, Psychological science.

[17]  Elizabeth S. Spelke,et al.  Children’s understanding of the relationship between addition and subtraction , 2008, Cognition.

[18]  Denise Brandão de Oliveira e Britto,et al.  The faculty of language , 2007 .

[19]  Rochel Gelman,et al.  Where Integers Come From , 2008 .

[20]  Kelly S. Mix,et al.  Multiple cues for quantification in infancy: is number one of them? , 2002, Psychological bulletin.

[21]  Susan Carey,et al.  What representations might underlie infant numerical knowledge , 1999 .

[22]  James R. Hurford,et al.  Language and Number: The Emergence of a Cognitive System , 1987 .

[23]  K. Fuson.  Children's Counting and Concepts of Number , 1987 .

[24]  S. Carey,et al.  Re-visiting the competence/performance debate in the acquisition of the counting principles , 2006, Cognitive Psychology.

[25]  Justin Halberda,et al.  Infants chunk object arrays into sets of individuals , 2004, Cognition.

[26]  Michael C. Frank,et al.  Number as a cognitive technology: Evidence from Pirahã language and cognition , 2008, Cognition.

[27]  E. Miller,et al.  Coding of Cognitive Magnitude Compressed Scaling of Numerical Information in the Primate Prefrontal Cortex , 2003, Neuron.

[28]  Elizabeth S Spelke,et al.  Preschool children's mapping of number words to nonsymbolic numerosities. , 2005, Child development.

[29]  J. Piaget.  The Child's Conception of Number , 1953 .

[30]  Noam Chomsky,et al.  The faculty of language: what is it, who has it, and how did it evolve? , 2002, Science.

[31]  S. Dehaene.  Symbols and quantities in parietal cortex: elements of a mathematical theory of number representation and manipulation , 2007 .

[32]  Philippe Pinel,et al.  Tuning Curves for Approximate Numerosity in the Human Intraparietal Sulcus , 2004, Neuron.

[33]  T. Simon,et al.  The foundations of numerical thinking in a brain without numbers , 1999, Trends in Cognitive Sciences.

[34]  Karen Wynn,et al.  Children's understanding of counting , 1990, Cognition.

[35]  Stanislas Dehaene,et al.  Moving along the Number Line: Operational Momentum in Nonsymbolic Arithmetic , 2006 .

[36]  Rochel Gelman,et al.  Language and Conceptual Development series Number and language : how are they related ? , 2004 .

[37]  Pierre Pica,et al.  Core knowledge of geometry in an Amazonian indigene group. , 2006, Science.

[38]  Pierre Pica,et al.  Theoretical Implications of the Study of Numbers and Numerals in Mundurucu , 2008 .

[39]  Eli Maor,et al.  The universal history of numbers : from prehistory to the invention of the computer , 2001 .

[40]  S. Carey.  Bootstrapping & the origin of concepts , 2004, Daedalus.

[41]  J. Mehler,et al.  Cognitive Capacity of Very Young Children , 1967, Science.

[42]  Lisa Feigenson,et al.  The equality of quantity , 2007, Trends in Cognitive Sciences.