Cardinals, Ordinals, and the Prospects for a Fregean Foundation
Snyder (Eric), Shapiro (Stewart) & Samuels (Richard)
Source: Royal Institute of Philosophy Supplements, Volume 82 (Metaphysics) - July 2018, pp. 77-107
Paper - Abstract

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Authors’ Abstract

  1. There are multiple formal characterizations of the natural numbers available. Despite being inter-derivable, they plausibly codify different possible applications of the naturals – doing basic arithmetic, counting, and ordering – as well as different philosophical conceptions of those numbers: structuralist, cardinal, and ordinal.
  2. Some influential philosophers of mathematics have argued for a non-egalitarian attitude according to which one of those characterizations is ‘more basic’ or ‘more fundamental’ than the others.
  3. This paper addresses two related issues.
    1. First, we review some of these non-egalitarian arguments, lay out a laundry list of different, legitimate, notions of relative priority, and suggest that these arguments plausibly employ different such notions.
    2. Secondly, we argue that given a metaphysical-cum-epistemological gloss suggested by Frege’s foundationalist epistemology, the ordinals are plausibly more basic than the cardinals.
  4. This is just one orientation to relative priority one could take, however. Ultimately, we subscribe to an egalitarian attitude towards these formal characterizations: they are, in some sense, equally ‘legitimate’.

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