Both versions of logicism—strong and weak—maintain that in expressing them; or, as Kant might have preferred it, by virtue of internal relations among the concepts involved.
A successful logicist reduction of any branch of mathematics will therefore show that its truths (strong version) or its theorems (weak version) are analytic.
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.
So there must be such an object in order to get an ordinal at all.
So the smallest ordinal, in that sense, is one (or ‘first’). But there is a zero ordinal, in the mathematical sense.
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.
Here, Bob Hale and Crispin Wright assemble the key writings that lead to their distinctive neo-Fregean approach to the philosophy of mathematics.
In addition to fourteen previously published papers, the volume features a new paper on the Julius Caesar problem; a substantial new introduction mapping out the program and the contributions made to it by the various papers; a section explaining which issues most require further attention; and bibliographies of references and further useful sources.