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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.
Gist of Idea
A number is not a multitude, but a unified ratio between quantities
Source
Isaac Newton (Universal Arithmetick [1669]), quoted by John Mayberry - What Required for Foundation for Maths? p.407-2
Book Ref
'Philosophy of Mathematics: anthology', ed/tr. Jacquette,Dale [Blackwell 2002], p.407
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.
Related Idea
Idea 17781 Real numbers were invented, as objects, to simplify and generalise 'quantity' [Mayberry]
| 11041 | Some quantities are discrete, like number, and others continuous, like lines, time and space [Aristotle] |
| 17843 | The idea of 'one' is the foundation of number [Aristotle] |
| 17850 | Each many is just ones, and is measured by the one [Aristotle] |
| 17851 | Number is plurality measured by unity [Aristotle] |
| 1600 | Euclid's common notions or axioms are what we must have if we are to learn anything at all [Euclid, by Roochnik] |
| 17783 | A number is not a multitude, but a unified ratio between quantities [Newton] |
| 9800 | Arithmetic is based on definitions, and Sums of equals are equal, and Differences of equals are equal [Mill] |
| 14116 | Numbers were once defined on the basis of 1, but neglected infinities and + [Russell] |
| 10205 | Mathematics originally concerned the continuous (geometry) and the discrete (arithmetic) [Shapiro] |
| 19093 | Greek mathematics is wholly sensory, where ours is wholly inferential [Macbeth] |