13 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. |
| 17505 | Using proper names properly doesn't involve necessary and sufficient conditions [Putnam] |
| Full Idea: The important thing about proper names is that it would be ridiculous to think that having linguistic competence can be equated in their case with knowledge of a necessary and sufficient condition. | |
| From: Hilary Putnam (Explanation and Reference [1973], II B) |
| 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. |
| 11908 | Putnam bases essences on 'same kind', but same kinds may not share properties [Mackie,P on Putnam] |
| Full Idea: The only place for essentialism to come from in Putnam's semantic account is out of the 'same kind' relation. But if the same kind relation can be cashed out in terms that do not involve sharing properties (apart from 'being water') there is a gap. | |
| From: comment on Hilary Putnam (Explanation and Reference [1973]) by Penelope Mackie - How Things Might Have Been 10.4 | |
| A reaction: [This is the criticism of Salmon and Mellor] See Mackie's discussion for details. I would always have thought that relations result from essences, so could never be used to define them. |
| 17508 | Science aims at truth, not at 'simplicity' [Putnam] |
| Full Idea: Scientists are not trying to maximise some formal property of 'simplicity'; they are trying to maximise truth. | |
| From: Hilary Putnam (Explanation and Reference [1973], III B) | |
| A reaction: This seems to be aimed at the Mill-Ramsey-Lewis account of laws of nature, as the simplest axioms of experience. I'm with Putnam (as he was at this date). |
| 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! |
| 17506 | I now think reference by the tests of experts is a special case of being causally connected [Putnam] |
| Full Idea: In previous papers I suggested that the reference is fixed by a test known to experts; it now seems to me that this is just a special case of my use being causally connected to an introducing event. | |
| From: Hilary Putnam (Explanation and Reference [1973], II C) | |
| A reaction: I think he was probably right the first time, and has now wandered off course. |
| 7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
| Full Idea: The six perfections are of giving, morality, patience, vigour, meditation, and wisdom. | |
| From: Nagarjuna (Mahaprajnaparamitashastra [c.120], 88) | |
| A reaction: What is 'morality', if giving is not part of it? I like patience and vigour being two of the virtues, which immediately implies an Aristotelian mean (which is always what is 'appropriate'). |
| 17507 | Natural kind stereotypes are 'strong' (obvious, like tiger) or 'weak' (obscure, like molybdenum) [Putnam] |
| Full Idea: Natural kinds can be associated with 'strong' stereotypes (giving a strong picture of a typical member, like a tiger), or with 'weak' stereotypes (with no idea of a sufficient condition, such as molybdenum or elm). | |
| From: Hilary Putnam (Explanation and Reference [1973], II C) |
| 11904 | Express natural kinds as a posteriori predicate connections, not as singular terms [Putnam, by Mackie,P] |
| Full Idea: Putnam implies dispensing with the designation of natural kinds by singular terms in favour of the postulation of necessary but a posteriori connections between predicates. ...We might call this 'predicate essentialism', but not 'de re essentialism'. | |
| From: report of Hilary Putnam (Explanation and Reference [1973]) by Penelope Mackie - How Things Might Have Been 10.1 | |
| A reaction: It is characteristic of modern discussion that the logical form of natural kind statements is held to be crucial, rather than an account of nature in any old ways that do the job. So do I prefer singular terms, or predicate-connections. Hm. |