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Ideas for 'fragments/reports', 'Universal Arithmetick' and 'On the Nature of Mathematical Reasoning'

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2 ideas

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / d. Actual infinite

15923	Poincaré rejected the actual infinite, claiming definitions gave apparent infinity to finite objects [Poincaré, by Lavine]
	Full Idea: Poincaré rejected the actual infinite. He viewed mathematics that is apparently concerned with the actual infinite as actually concerning the finite linguistic definitions the putatively describe actually infinite objects.
	From: report of Henri Poincaré (On the Nature of Mathematical Reasoning [1894]) by Shaughan Lavine - Understanding the Infinite

6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic

17783	A number is not a multitude, but a unified ratio between quantities [Newton]
	Full Idea: By a Number we understand not so much a Multitude of Unities, as the abstracted Ratio of any Quantity to another Quantity of the same Kind, which we take for unity.
	From: Isaac Newton (Universal Arithmetick [1669]), quoted by John Mayberry - What Required for Foundation for Maths? p.407-2
	A reaction: This needs a metaphysics of 'kinds' (since lines can't have ratios with solids). Presumably Newton wants the real numbers to be more basic than the natural numbers. This is the transition from Greek to modern.