The Innateness Hypothesis and Mathematical Concepts

In historical claims for nativism, mathematics is a paradigmatic example of innate knowledge. Claims by contemporary developmental psychologists of elementary mathematical skills in human infants are a legacy of this. However, the connection between these skills and more formal mathematical concepts and methods remains unclear. This paper assesses the current debates surrounding nativism and mathematical knowledge by teasing them apart into two distinct claims. First, in what way does the experimental evidence from infants, nonhuman animals and neuropsychology support the nativist hypothesis? Second, granting that infants have some elementary mathematical skills, does this mean that such skills play an important role in the development of mathematical knowledge?

[1]  W. Thurston.  On Proof and Progress in Mathematics , 1994, math/9404236.

[2]  H. D. Cruz,et al.  An enhanced argument for innate elementary geometric knowledge and its philosophical implications , 2008 .

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

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

[5]  M. Haith.  Who put the cog in infant cognition ? Is rich interpretation too costly ? , 1998 .

[6]  Justin Halberda,et al.  Individual differences in non-verbal number acuity correlate with maths achievement , 2008, Nature.

[7]  K. Wynn,et al.  Large-Number Addition and Subtraction by 9-Month-Old Infants , 2004, Psychological science.

[8]  René Descartes,et al.  Le discours de la méthode , 1900 .

[9]  Reuben Hersh,et al.  18 Unconventional Essays on the Nature of Mathematics , 2013 .

[10]  Geoffrey B. Saxe,et al.  Effects of schooling on arithmetical understandings: studies with oksapmin children in Papua New Guinea , 1985 .

[11]  Judith V. Grabiner,et al.  Is Mathematical Truth Time-Dependent? , 1974 .

[12]  J. Piaget,et al.  The Child's Conception of Number , 1953 .

[13]  S E Antell,et al.  Perception of numerical invariance in neonates. , 1983, Child development.

[14]  Elizabeth S. Spelke,et al.  Symbolic arithmetic knowledge without instruction , 2007, Nature.

[15]  Michael D. Resnik.  Mathematics as a Science of Patterns: Epistemology , 1982 .

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

[17]  Stanislas Dehaene,et al.  Numerical Transformations in Five-month-old Human Infants. , 1997 .

[18]  Karen Wynn,et al.  Addition and subtraction by human infants , 1992, Nature.

[19]  R. Jaeger,et al.  Salamanders (Plethodon cinereus) go for more: rudiments of number in an amphibian , 2003, Animal Cognition.

[20]  Brian Butterworth,et al.  Numerical thought with and without words: Evidence from indigenous Australian children , 2008, Proceedings of the National Academy of Sciences.

[21]  K. Hiraki,et al.  Baby arithmetic: one object plus one tone , 2004, Cognition.

[22]  Elizabeth M Brannon,et al.  The development of ordinal numerical knowledge in infancy , 2002, Cognition.

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

[24]  Patrick Bateson,et al.  Innateness and the Sciences , 2006 .

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

[26]  Pierre R. Dasen,et al.  Yupno Number System and Counting , 1994 .

[27]  Julie L. Booth,et al.  Development of numerical estimation in young children. , 2004, Child development.

[28]  L. Decock.  The Conceptual Basis of Numerical Abilities: One-to-One Correspondence Versus the Successor Relation , 2008 .

[29]  Wes Sharrock,et al.  Mathematical Relativism: logic, grammar, and arithmetic in cultural comparison , 2006 .

[30]  R. Samuels.  Innateness in cognitive science , 2004, Trends in Cognitive Sciences.

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

[32]  Thomas Tymoczko.  New Directions in the Philosophy of Mathematics , 1985 .

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

[34]  Andreas Nieder,et al.  Neuronal population coding of continuous and discrete quantity in the primate posterior parietal cortex , 2007, Proceedings of the National Academy of Sciences.

[35]  S. Shapiro.  Philosophy of mathematics : structure and ontology , 1997 .

[36]  Richard Samuels,et al.  Nativism in cognitive science , 2002 .

[37]  L. Cohen,et al.  How infants process addition and subtraction events , 2002 .

[38]  H. D. Cruz,et al.  An Extended Mind Perspective on Natural Number Representation , 2008 .

[39]  H. Putnam.  What is mathematical truth , 1975 .

[40]  Jonathan I. Flombaum,et al.  Rhesus monkeys (Macaca mulatta) spontaneously compute addition operations over large numbers , 2005, Cognition.

[41]  R. Church,et al.  A mode control model of counting and timing processes. , 1983, Journal of experimental psychology. Animal behavior processes.

[42]  D. Biro,et al.  Chimpanzee Numerical Competence: Cardinal and Ordinal Skills , 2008 .

[43]  Tetsuro Matsuzawa,et al.  Primate Origins of Human Cognition and Behavior , 2001, Springer Japan.

[44]  Andrea Berger,et al.  Infant brains detect arithmetic errors , 2006, Proceedings of the National Academy of Sciences.

[45]  Michael W. L. Chee,et al.  Neural correlates of symbolic and non-symbolic arithmetic , 2005, Neuropsychologia.

[46]  I. Kant,et al.  Critique of Pure Reason: Glossary , 1998 .

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

[48]  Eric Margolis,et al.  Number and natural language , 2005 .

[49]  Peter Carruthers,et al.  The Innate Mind, Volume 3 , 2005 .

[50]  Elizabeth M Brannon,et al.  The multisensory representation of number in infancy. , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[51]  Melissa E. Libertus,et al.  Comment on "Log or Linear? Distinct Intuitions of the Number Scale in Western and Amazonian Indigene Cultures" , 2009, Science.

[52]  Philip J. Davis,et al.  The Mathematical Experience , 1982 .

[53]  Katherine D. Kinzler,et al.  Core knowledge. , 2007, Developmental science.