### Ideas of R Kaplan / E Kaplan, by Theme

#### [American, fl. 2003, Husband and wife who have both taught mathematics at Harvard University.]

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###### 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 Full Idea: The problem with the Axiom of Choice is that it allows an initiate (by an ingenious train of reasoning) to cut a golf ball into a finite number of pieces and put them together again to make a globe as big as the sun. From: R Kaplan / E Kaplan (The Art of the Infinite , 9) A reaction: I'm not sure how this works (and I think it was proposed by the young Tarski), but it sounds like a real problem to me, for all the modern assumptions that Choice is fine.
###### 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 Full Idea: You have 1 and 0, something and nothing. Adding gives us the naturals. Subtracting brings the negatives into light; dividing, the rationals; only with a new operation, taking of roots, do the irrationals show themselves. From: R Kaplan / E Kaplan (The Art of the Infinite , 1 'Mind') A reaction: The suggestion is constructivist, I suppose - that it is only operations that produce numbers. They go on to show that complex numbers don't quite fit the pattern.
###### 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
 15711 The rationals are everywhere - the irrationals are everywhere else Full Idea: The rationals are everywhere - the irrationals are everywhere else. From: R Kaplan / E Kaplan (The Art of the Infinite , 1 'Nameless') A reaction: Nice. That is, the rationals may be dense (you can always find another one in any gap), but the irrationals are continuous (no gaps).
###### 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 Full Idea: The 'commutative' laws say the order in which you add or multiply two numbers makes no difference; ...the 'associative' laws declare that regrouping couldn't change a sum or product (e.g. a+(b+c)=(a+b)+c ). From: R Kaplan / E Kaplan (The Art of the Infinite , 2 'Tablets') A reaction: This seem utterly self-evident, but in more complex systems they can break down, so it is worth being conscious of them.
 15715 'Distributive' laws say if you add then multiply, or multiply then add, you get the same result Full Idea: The 'distributive' law says you will get the same result if you first add two numbers, and then multiply them by a third, or first multiply each by the third and then add the results (i.e. a · (b+c) = a · b + a · c ). From: R Kaplan / E Kaplan (The Art of the Infinite , 2 'Tablets') A reaction: Obviously this will depend on getting the brackets right, to ensure you are indeed doing the same operations both ways.
###### 14. Science / C. Induction / 3. Limits of Induction
 15713 The first million numbers confirm that no number is greater than a million Full Idea: The claim that no number is greater than a million is confirmed by the first million test cases. From: R Kaplan / E Kaplan (The Art of the Infinite , 2 'Intro') A reaction: Extrapolate from this, and you can have as large a number of cases as you could possibly think of failing to do the inductive job. Love it! Induction isn't about accumulations of cases. It is about explanation, which is about essence. Yes!