The conceptual basis of numerical abilities: One-to-one correspondence versus the successor relation

Philosophical Psychology 21 (4):459 – 473 (2008)
In recent years, neologicists have demonstrated that Hume's principle, based on the one-to-one correspondence relation, suffices to construct the natural numbers. This formal work is shown to be relevant for empirical research on mathematical cognition. I give a hypothetical account of how nonnumerate societies may acquire arithmetical knowledge on the basis of the one-to-one correspondence relation only, whereby the acquisition of number concepts need not rely on enumeration (the stable-order principle). The existing empirical data on the role of the one-to-one correspondence relation for numerical abilities is assessed and additional empirical tests are proposed. In the final part, it is argued that the fact that the successor relation and the one-to-one correspondence relation can play independent roles in number concept acquisition may be a complication for testing the Whorfian hypothesis.
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DOI 10.1080/09515080802285255
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Gottlob Frege (1964). The Basic Laws of Arithmetic. Berkeley, University of California Press.

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