10 ideas
| 15717 | Using Choice, you can cut up a small ball and make an enormous one from the pieces [Kaplan/Kaplan] |
| 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 [2003], 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. |
| 15712 | 1 and 0, then add for naturals, subtract for negatives, divide for rationals, take roots for irrationals [Kaplan/Kaplan] |
| 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 [2003], 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. |
| 15711 | The rationals are everywhere - the irrationals are everywhere else [Kaplan/Kaplan] |
| Full Idea: The rationals are everywhere - the irrationals are everywhere else. | |
| From: R Kaplan / E Kaplan (The Art of the Infinite [2003], 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). |
| 15714 | 'Commutative' laws say order makes no difference; 'associative' laws say groupings make no difference [Kaplan/Kaplan] |
| 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 [2003], 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 [Kaplan/Kaplan] |
| 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 [2003], 2 'Tablets') | |
| A reaction: Obviously this will depend on getting the brackets right, to ensure you are indeed doing the same operations both ways. |
| 19489 | For me, fictions are internally true, without a significant internal or external truth-value [Yablo] |
| Full Idea: A 'myth' or fiction for me is a true internal statement (a statement endorsed by the rules) whose external truth value is as may be, the point being that that truth value is from an internal standpoint quite irrelevant. | |
| From: Stephen Yablo (Does Ontology Rest on a Mistake? [1998], IX) | |
| A reaction: This contrasts with Carnap, for whom talk of 'ghosts' is false in an internal thing-framework. Yablo seems here to say a statement can be true while having no truth value. Presumably he is relaxing the internal rules. |
| 19490 | Make-believe can help us to reason about facts and scientific procedures [Yablo] |
| Full Idea: Make-believe games can make it easier to reason about facts, to systematize them, to visualize them, to spot connections with other facts, and to evaluate potential lines of research. | |
| From: Stephen Yablo (Does Ontology Rest on a Mistake? [1998], XI) | |
| A reaction: This is the key pragmatic defence of the fictionalist view of abstract objects. Fictions are devices to help us think better. I think a lot of ontology turns out that way. |
| 19491 | 'The clouds are angry' can only mean '...if one were attributing emotions to clouds' [Yablo] |
| Full Idea: It is an open question whether the clouds that we call 'angry' are literally F, for any F other than 'such that it would be natural and proper to regard them as angry if one were going to attribute emotions to clouds'. | |
| From: Stephen Yablo (Does Ontology Rest on a Mistake? [1998], XII) | |
| A reaction: His point is that it is TRUE, in those circumstances, that the clouds are angry. Thus fictions are a valid and useful part of ordinary sensible course, giving real information. I like it. |
| 15713 | The first million numbers confirm that no number is greater than a million [Kaplan/Kaplan] |
| 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 [2003], 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! |
| 468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
| Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice. | |
| From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where? |