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3 ideas
| 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. |
| 16150 | One is, so numbers exist, so endless numbers exist, and each one must partake of being [Plato] |
| Full Idea: If one is, there must also necessarily be number - Necessarily - But if there is number, there would be many, and an unlimited multitude of beings. ..So if all partakes of being, each part of number would also partake of it. | |
| From: Plato (Parmenides [c.364 BCE], 144a) | |
| A reaction: This seems to commit to numbers having being, then to too many numbers, and hence to too much being - but without backing down and wondering whether numbers had being after all. Aristotle disagreed. |
| 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). |