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Full Idea
It is far from obvious that knowing what 'just as many' means requires knowing what a one-one correspondence is. The notion of a one-one correspondence is very sophisticated, and it is far from clear that five-year-olds have any grasp of it.
Gist of Idea
Understanding 'just as many' needn't involve grasping one-one correspondence
Source
Richard G. Heck (Cardinality, Counting and Equinumerosity [2000], 4)
Book Ref
-: 'Notre Dame Journal of Formal Logic' [-], p.198
A Reaction
The point is that children decide 'just as many' by counting each group and arriving at the same numeral, not by matching up. He cites psychological research by Gelman and Galistel.
| 17448 | In counting, numerals are used, not mentioned (as objects that have to correlated) [Heck] |
| 17449 | We can understand cardinality without the idea of one-one correspondence [Heck] |
| 17450 | Understanding 'just as many' needn't involve grasping one-one correspondence [Heck] |
| 17451 | We can know 'just as many' without the concepts of equinumerosity or numbers [Heck] |
| 17454 | Children can use numbers, without a concept of them as countable objects [Heck] |
| 17455 | Is counting basically mindless, and independent of the cardinality involved? [Heck] |
| 17456 | Counting is the assignment of successively larger cardinal numbers to collections [Heck] |
| 17453 | The meaning of a number isn't just the numerals leading up to it [Heck] |
| 17457 | A basic grasp of cardinal numbers needs an understanding of equinumerosity [Heck] |
| 17458 | Equinumerosity is not the same concept as one-one correspondence [Heck] |
| 17459 | Frege's Theorem explains why the numbers satisfy the Peano axioms [Heck] |