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Ideas for 'fragments/reports', 'Principia Mathematica' and 'The Question of Ontology'

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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 / 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 / c. Fregean numbers
12215 The existence of numbers is not a matter of identities, but of constituents of the world [Fine,K]
     Full Idea: On saying that a particular number exists, we are not saying that there is something identical to it, but saying something about its status as a genuine constituent of the world.
     From: Kit Fine (The Question of Ontology [2009], p.168)
     A reaction: This is aimed at Frege's criterion of identity, which is to be an element in an identity relation, such as x = y. Fine suggests that this only gives a 'trivial' notion of existence, when he is interested in a 'thick' sense of 'exists'.
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
12211 It is plausible that x^2 = -1 had no solutions before complex numbers were 'introduced' [Fine,K]
     Full Idea: It is not implausible that before the 'introduction' of complex numbers, it would have been incorrect for mathematicians to claim that there was a solution to the equation 'x^2 = -1' under a completely unrestricted understanding of 'there are'.
     From: Kit Fine (The Question of Ontology [2009])
     A reaction: I have adopted this as the crucial test question for anyone's attitude to platonism in mathematics. I take it as obvious that complex numbers were simply invented so that such equations could be dealt with. They weren't 'discovered'!
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
12209 The indispensability argument shows that nature is non-numerical, not the denial of numbers [Fine,K]
     Full Idea: Arguments such as the dispensability argument are attempting to show something about the essentially non-numerical character of physical reality, rather than something about the nature or non-existence of the numbers themselves.
     From: Kit Fine (The Question of Ontology [2009], p.160)
     A reaction: This is aimed at Hartry Field. If Quine was right, and we only believe in numbers because of our science, and then Field shows our science doesn't need it, then Fine would be wrong. Quine must be wrong, as well as Field.