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Ideas for 'Parmenides', 'In Praise of Philosophy' and 'Philosophy of Mathematics'

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5 ideas

5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
9605 If a proposition is false, then its negation is true [Brown,JR]
     Full Idea: The law of excluded middle says if a proposition is false, then its negation is true
     From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 1)
     A reaction: Surely that is the best statement of the law? How do you write that down? ¬(P)→¬P? No, because it is a semantic claim, not a syntactic claim, so a truth table captures it. Semantic claims are bigger than syntactic claims.
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
9649 Axioms are either self-evident, or stipulations, or fallible attempts [Brown,JR]
     Full Idea: The three views one could adopt concerning axioms are that they are self-evident truths, or that they are arbitrary stipulations, or that they are fallible attempts to describe how things are.
     From: James Robert Brown (Philosophy of Mathematics [1999], Ch.10)
     A reaction: Presumably modern platonists like the third version, with others choosing the second, and hardly anyone now having the confidence to embrace the first.
5. Theory of Logic / L. Paradox / 3. Antinomies
13986 Plato found antinomies in ideas, Kant in space and time, and Bradley in relations [Plato, by Ryle]
     Full Idea: Plato (in 'Parmenides') shows that the theory that 'Eide' are substances, and Kant that space and time are substances, and Bradley that relations are substances, all lead to aninomies.
     From: report of Plato (Parmenides [c.364 BCE]) by Gilbert Ryle - Are there propositions? 'Objections'
14150 Plato's 'Parmenides' is perhaps the best collection of antinomies ever made [Russell on Plato]
     Full Idea: Plato's 'Parmenides' is perhaps the best collection of antinomies ever made.
     From: comment on Plato (Parmenides [c.364 BCE]) by Bertrand Russell - The Principles of Mathematics §337
5. Theory of Logic / L. Paradox / 4. Paradoxes in Logic / c. Berry's paradox
9638 Berry's Paradox finds a contradiction in the naming of huge numbers [Brown,JR]
     Full Idea: Berry's Paradox refers to 'the least integer not namable in fewer than nineteen syllables' - a paradox because it has just been named in eighteen syllables.
     From: James Robert Brown (Philosophy of Mathematics [1999], Ch. 5)
     A reaction: Apparently George Boolos used this quirky idea as a basis for a new and more streamlined proof of Gödel's Theorem. Don't tell me you don't find that impressive.