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3 ideas
| 7555 | Zeno achieved the statement of the problems of infinitesimals, infinity and continuity [Russell on Zeno of Citium] |
| Full Idea: Zeno was concerned with three increasingly abstract problems of motion: the infinitesimal, the infinite, and continuity; to state the problems is perhaps the hardest part of the philosophical task, and this was done by Zeno. | |
| From: comment on Zeno (Citium) (fragments/reports [c.294 BCE]) by Bertrand Russell - Mathematics and the Metaphysicians p.81 | |
| A reaction: A very nice tribute, and a beautiful clarification of what Zeno was concerned with. |
| 9553 | Analytic geometry gave space a mathematical structure, which could then have axioms [Chihara] |
| Full Idea: With the invention of analytic geometry (by Fermat and then Descartes) physical space could be represented as having a mathematical structure, which could eventually lead to its axiomatization (by Hilbert). | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 02.3) | |
| A reaction: The idea that space might have axioms seems to be pythagoreanism run riot. I wonder if there is some flaw at the heart of Einstein's General Theory because of this? |
| 10192 | We can replace existence of sets with possibility of constructing token sentences [Chihara, by MacBride] |
| Full Idea: Chihara's 'constructability theory' is nominalist - mathematics is reducible to a simple theory of types. Instead of talk of sets {x:x is F}, we talk of open sentences Fx defining them. Existence claims become constructability of sentence tokens. | |
| From: report of Charles Chihara (A Structural Account of Mathematics [2004]) by Fraser MacBride - Review of Chihara's 'Structural Acc of Maths' p.81 | |
| A reaction: This seems to be approaching the problem in a Fregean way, by giving an account of the semantics. Chihara is trying to evade the Quinean idea that assertion is ontological commitment. But has Chihara retreated too far? How does he assert existence? |