14 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. |
| 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! |
| 6968 | Some people think there are ethical facts, but of a 'queer' sort [Ayer] |
| Full Idea: If someone wishes to say that ethical statements are statements of fact, only it is a queer sort of fact, he is welcome to do so. | |
| From: A.J. Ayer (On the analysis of moral judgements [1949], p.233) | |
| A reaction: The word 'queer' was picked up by Mackie and developed into his error theory, that moral facts are a misunderstanding. Personally I think that moral facts might be teleological facts, but that is rather hard to demonstrate. |
| 6972 | A right attitude is just an attitude one is prepared to stand by [Ayer] |
| Full Idea: Asking whether the attitude that one has adopted is the right attitude comes down to asking whether one is prepared to stand by it. | |
| From: A.J. Ayer (On the analysis of moral judgements [1949], p.244) | |
| A reaction: I would have thought that someone who persisted in being ruthlessly selfish might nevertheless distinguish their behaviour from the grudging concession that the 'right' thing to do might be quite different. |
| 6973 | Moral theories are all meta-ethical, and are neutral as regards actual conduct [Ayer] |
| Full Idea: All moral theories, intuitionist, naturalistic, objectivist, emotive, and the rest, in so far as they are philosophical theories, are neutral as regards actual conduct; they belong to the field of meta-ethics, not ethics proper. | |
| From: A.J. Ayer (On the analysis of moral judgements [1949]) | |
| A reaction: Interestingly, Ayer doesn't seem willing to accept 'ethics proper' as being 'philosophical'. Given the modern rise of applied ethics, it seems suprising to say that even normative ethics is not philosophical. Utilitarianism seems not to be philosophical. |
| 6974 | Moral judgements cannot be the logical consequence of a moral philosophy [Ayer] |
| Full Idea: A moral philosopher will have his moral standards and will sometimes make moral judgements, but these moral judgements cannot be a logical consequence of his philosophy. | |
| From: A.J. Ayer (On the analysis of moral judgements [1949], p.247) | |
| A reaction: I take this to be an assertion of the is-ought distinction. Personally this strikes me as totally false. Ayer needs to think more deeply about moral philosophy! |
| 6971 | I would describe intuitions of good as feelings of approval [Ayer] |
| Full Idea: I suspect that the experiences which some philosophers want to describe as intuitions, or a quasi-sensory apprehensions, of good are not significantly different from those that I want to describe as feelings of approval. | |
| From: A.J. Ayer (On the analysis of moral judgements [1949], p.239) | |
| A reaction: This is the standard ground for rejecting intuitionism, along with the point that even if intuitions are not just feelings of approval, it seems impossible to tell the difference. |
| 6969 | Approval of historical or fictional murders gives us leave to imitate them [Ayer] |
| Full Idea: In saying that Brutus or Raskolnikov acted rightly, I am giving myself and others leave to imitate them should similar circumstances arise. | |
| From: A.J. Ayer (On the analysis of moral judgements [1949], p.237) | |
| A reaction: This seems to be a reply to the Frege-Geach Problem, of why we have emotional attitudes to crimes that mean nothing to us. Such crimes, however, involve our virtues, and don't depend on awaiting 'similar circumstances'. |
| 6970 | Moral judgements are not expressions, but are elements in a behaviour pattern [Ayer] |
| Full Idea: To say, as I once did, that moral judgements are merely expressive of certain feelings is an oversimplification; ..moral attitudes consist in certain patterns of behaviour, and the expression of a judgement is an element in the pattern. | |
| From: A.J. Ayer (On the analysis of moral judgements [1949], p.238) | |
| A reaction: This seems to switch from emotivism to what Frank Jackson calls 'moral functionalism', where morality is what gets us from certain emotional responses to willed actions. This strikes me, like most functional explanations, as wrong. |
| 5655 | Happiness is not satisfaction of desires, but fulfilment of values [Bradley, by Scruton] |
| Full Idea: For Bradley, the happiness of the individual is not to be understood in terms of his desires and needs, but rather in terms of his values - which is to say, in terms of those of his desires which he incorporates into his self. | |
| From: report of F.H. Bradley (Ethical Studies [1876]) by Roger Scruton - Short History of Modern Philosophy Ch.16 | |
| A reaction: Good. Bentham will reduce the values to a further set of desires, so that a value is a complex (second-level?) desire. I prefer to think of values as judgements, but I like Scruton's phrase of 'incorporating into his self'. Kant take note (Idea 1452). |