Combining Philosophers

Ideas for Empedocles, Isaac Newton and John Bigelow

unexpand these ideas     |    start again     |     choose another area for these philosophers

display all the ideas for this combination of philosophers

5 ideas

6. Mathematics / A. Nature of Mathematics / 2. Geometry
18079 Newton developed a kinematic approach to geometry [Newton, by Kitcher]
     Full Idea: The reduction of the problems of tangents, normals, curvature, maxima and minima were effected by Newton's kinematic approach to geometry.
     From: report of Isaac Newton (Principia Mathematica [1687]) by Philip Kitcher - The Nature of Mathematical Knowledge 10.1
     A reaction: This approach apparently contrasts with that of Leibniz.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
13152 We can talk of 'innumerable number', about the infinite points on a line [Newton]
     Full Idea: If any man shall take the words number and sum in a larger sense, to understand things which are numberless and sumless (such as the infinite points on a line), I could allow him the contradictious phrase 'innumerable number' without absurdity.
     From: Isaac Newton (Letters to Bentley [1692], 1693.02.25)
     A reaction: [compressed] I take the key point here to be the phrase of taking number 'in a larger sense'. Like the word 'atom' in physics, the word 'number' retains its traditional reference, but has considerably shifted its scope. Amateurs must live with this.
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
13151 Not all infinites are equal [Newton]
     Full Idea: It is an error that all infinites are equal.
     From: Isaac Newton (Letters to Bentley [1692], 1693.01.17)
     A reaction: There follows a discussion of the mathematicians' view of infinity. Cantor was not the first to notice that there is more than one sort of of infinity.
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / l. Limits
18082 Quantities and ratios which continually converge will eventually become equal [Newton]
     Full Idea: Quantities and the ratios of quantities, which in any finite time converge continually to equality, and, before the end of that time approach nearer to one another by any given difference become ultimately equal.
     From: Isaac Newton (Principia Mathematica [1687], Lemma 1), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 10.2
     A reaction: Kitcher observes that, although Newton relies on infinitesimals, this quotation expresses something close to the later idea of a 'limit'.
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.