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Ideas for 'fragments/reports', 'Word and Object' and 'The Varieties of Necessity'

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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers

17905	Any progression will do nicely for numbers; they can all then be used to measure multiplicity [Quine]
	Full Idea: The condition on an explication of number can be put succinctly: any progression will do nicely. Russell once held that one must also be able to measure multiplicity, but this was a mistake; any progression can be fitted to that further condition.
	From: Willard Quine (Word and Object [1960], §54)
	A reaction: [compressed] This is the strongest possible statement that the numbers are the ordinals, and the Peano Axioms will define them. The Fregean view that cardinality comes first is redundant.

6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics

9556	Nearly all of mathematics has to quantify over abstract objects [Quine]
	Full Idea: Mathematics, except for very trivial portions such as very elementary arithmetic, is irredeemably committed to quantification over abstract objects.
	From: Willard Quine (Word and Object [1960], §55)
	A reaction: Personally I would say that we are no more committed to such things than actors in 'The Tempest' are committed to the existence of Prospero and Caliban (which is quite a strong commitment, actually).