Combining Texts

All the ideas for 'Mathematics and Indispensibility', 'The Art of the Infinite' and 'Introduction to a Secret Encyclopaedia'

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9 ideas

1. Philosophy / F. Analytic Philosophy / 2. Analysis by Division

13099

Analysing right down to primitive concepts seems beyond our powers [Leibniz]

3. Truth / A. Truth Problems / 8. Subjective Truth

5022	We hold a proposition true if we are ready to follow it, and can't see any objections [Leibniz]

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

15717

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

15712

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

15711

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

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

15714

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

15715

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

7. Existence / D. Theories of Reality / 11. Ontological Commitment / e. Ontological commitment problems

9558	All scientific tests will verify mathematics, so it is a background, not something being tested [Sober]

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

15713

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