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Ideas for 'works', 'Intuitionism and Formalism' and 'Lectures 1930-32 (student notes)'

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5 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
18738 We don't get 'nearer' to something by adding decimals to 1.1412... (root-2) [Wittgenstein]
     Full Idea: We say we get nearer to root-2 by adding further figures after the decimal point: 1.1412.... This suggests there is something we can get nearer to, but the analogy is a false one.
     From: Ludwig Wittgenstein (Lectures 1930-32 (student notes) [1931], Notes)
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / g. Applying mathematics
12451 Scientific laws largely rest on the results of counting and measuring [Brouwer]
     Full Idea: A large part of the natural laws introduced by science treat only of the mutual relations between the results of counting and measuring.
     From: Luitzen E.J. Brouwer (Intuitionism and Formalism [1912], p.77)
     A reaction: His point, I take it, is that the higher reaches of numbers have lost touch with the original point of the system. I now see the whole issue as just depending on conventions about the agreed extension of the word 'number'.
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
18708 Infinity is not a number, so doesn't say how many; it is the property of a law [Wittgenstein]
     Full Idea: 'Infinite' is not an answer to the question 'How many?', since the infinite is not a number. ...Infinity is the property of a law, not of an extension.
     From: Ludwig Wittgenstein (Lectures 1930-32 (student notes) [1931], A VII.2)
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
12454 Intuitionists only accept denumerable sets [Brouwer]
     Full Idea: The intuitionist recognises only the existence of denumerable sets.
     From: Luitzen E.J. Brouwer (Intuitionism and Formalism [1912], p.80)
     A reaction: That takes you up to omega, but not beyond, presumably because it then loses sight of the original intuition of 'bare two-oneness' (Idea 12453). I sympathise, but the word 'number' has shifted its meaning a lot these days.
12453 Neo-intuitionism abstracts from the reuniting of moments, to intuit bare two-oneness [Brouwer]
     Full Idea: Neo-intuitionism sees the falling apart of moments, reunited while remaining separated in time, as the fundamental phenomenon of human intellect, passing by abstracting to mathematical thinking, the intuition of bare two-oneness.
     From: Luitzen E.J. Brouwer (Intuitionism and Formalism [1912], p.80)
     A reaction: [compressed] A famous and somewhat obscure idea. He goes on to say that this creates one and two, and all the finite ordinals.