Rips et al/From Numerical Concepts to Concepts of Number
It’s always exciting to speculate what the authors will say when you see a fascinating title like this. I haven’t read it, and the following is NOT a response to the paper, but reactions to the abstract.
TITLE: From Numerical Concepts to Concepts of Number
AUTHORS: Lance J. Rips Amber Bloomfield, and Jennifer Asmuth
ABSTRACT: Many experiments with infants suggest that they possess quantitative abilities, and many experimentalists believe these abilities set the stage for later mathematics: the natural numbers and arithmetic. But the connection between these early and later skills is far from obvious. We evaluate two possible routes to mathematics and argue that neither is sufficient: (a) We first sketch what we think is the most likely model for infant abilities in this domain, and we examine proposals for extrapolating the natural number concept from these beginnings. Proposals for arriving at natural number by (empirical) induction presuppose the mathematical concepts they seek to explain. Moreover, standard experimental tests for children’s understanding of number terms do not necessarily tap these concepts. (b) True concepts of number do appear, however, when children are able to understand generalizations over all numbers, for example, the principle of additive commutativity (a + b = b + a). Theories of how children learn such principles usually rely on a process of mapping from physical object groupings. But both experimental results and theoretical considerations imply that direct mapping is insufficient for acquiring these principles. We suggest instead that children may arrive at natural numbers and arithmetic by constructing mathematical schemas on a base of innate abilities and math principles.
KEYWORDS: mathematical concepts; acquisition of natural numbers; representations of mathematics; theories of mathematical cognition
FULL TEXT: http://www.bbsonline.org/Preprints/Rips-08242006/Referees/
Innate Mathematical Principles … hmmm. Are mathematical principles out in the world, in our brain cells, or only restricted to the symbolic system we call math?
I believe the material world is lawful (give or take), but whatever governs it, it is not math, at least not the symbolic math that we know of.
If the mathematical principles are in our individual brains (or genes, guts, whatever), then do I have different numbers of them from my friend Sujai, who — I hate to admit — has far superior amount of the math stuff than I do? If we allow individual differences in math principles — as with many inherent traits — does it imply that we each have a different kind of math? Or, maybe we all have the same principles in our brains, but something else determines our math abilities. In that case why bother with that assumption in the first place? There is of course the obvious question — if it’s in our brains, why did it take millions of years to develop the math as we know it? And if the principles are fixed, how does one explain the coloful history of math, where so many ideas turned out to be un-principled?
Of course I have no proof, but whatever is out there, it is not a set of math principles. It is a matrix that generates principles in individual domains — be it syntax, math, or causality. And that matrix most likely does not reside in the material brains of individuals, but in the symbolic world created by a cultural process.
When humans are extinct, will there be math principles?
Yes, so long as the symbolic system we call math is preserved, carved in stones or in CDROMs. A symbolic intelligence will be able to derive the princples from the work — not from our pickled brains or the world homosapiens used to live in.