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It will be recognized as the most powerful presentation yet of a neo-Fregean program.doctrine that can be advanced with respect to any branch of mathematics.
The point is so simple that it needs a sophisticated intellect to overlook it; and it shows Frege to have been right, as against Dedekind, to have made the use of the natural numbers as finite cardinals intrinsic to their characterisation., Chapter 3): ‘Men do not think they know a thing unless they have grasped the ‘why’ of it’.
So proper foundational knowledge is of-a-piece with what Jaegwon Kim calls ‘explanatory knowledge’ – knowledge why – as opposed to mere descriptive knowledge – knowledge that ( to have an (or the) explanation for what one knows.
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.
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. 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.
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.
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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