Combining Philosophers

All the ideas for H.Putnam/P.Oppenheim, R Kaplan / E Kaplan and Paul J. Cohen

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8 ideas

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX

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Using Choice, you can cut up a small ball and make an enormous one from the pieces [Kaplan/Kaplan]

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number

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1 and 0, then add for naturals, subtract for negatives, divide for rationals, take roots for irrationals [Kaplan/Kaplan]

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers

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The rationals are everywhere - the irrationals are everywhere else [Kaplan/Kaplan]

6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / f. Arithmetic

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'Commutative' laws say order makes no difference; 'associative' laws say groupings make no difference [Kaplan/Kaplan]

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'Distributive' laws say if you add then multiply, or multiply then add, you get the same result [Kaplan/Kaplan]

6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism

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We could accept the integers as primitive, then use sets to construct the rest [Cohen]

14. Science / C. Induction / 3. Limits of Induction

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The first million numbers confirm that no number is greater than a million [Kaplan/Kaplan]

14. Science / D. Explanation / 2. Types of Explanation / j. Explanations by reduction

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Six reduction levels: groups, lives, cells, molecules, atoms, particles [Putnam/Oppenheim, by Watson]