16 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! |
| 21740 | I doubt whether ethics is part of philosophy [Russell] |
| Full Idea: I hardly think myself that ethics ought to be included in the domain of philosophy. | |
| From: Bertrand Russell (An Outline of Philosophy [1927], Ch.22) | |
| A reaction: He declines to give his reasons. The implication of the chapter is that ethics is essentially a social and political matter, so that individual ethical guidelines are unimportant. Maybe the woolliness of ethics was also an impediment. |
| 21741 | 'You ought to do p' primarily has emotional content, expressing approval [Russell] |
| Full Idea: A sentence like 'You ought to do so-and-so' primarily has an emotional content. It means ' this is the act towards which I feel the emotion of approval'. | |
| From: Bertrand Russell (An Outline of Philosophy [1927], Ch 22) | |
| A reaction: I don't understand how I can say 'you ought to do p', and very clearly mean that the situation would be altogether better if p, only to be told by some philosopher that what I thought was a sensible judgement is actually an emotional outburst. |
| 21746 | Unlike hate, all desires can be satisfied by love [Russell] |
| Full Idea: If harmonious desires are what we should seek, love is better than hate, since, when two people love each other, both can be satisfied, whereas when they hate each other one at most can achieve the object of his desire. | |
| From: Bertrand Russell (An Outline of Philosophy [1927], Ch 22) | |
| A reaction: A wonderful example of cool philosophical objectivity! Of course it is not true, because the fact that two people love one another doesn't not prevent them from having some incompatible desires, as every couple knows. |
| 21747 | Goodness is a combination of love and knowledge [Russell] |
| Full Idea: The good life is one inspired by love and guided by knowledge. | |
| From: Bertrand Russell (An Outline of Philosophy [1927], Ch 22) | |
| A reaction: Forty years later, Russell's famous filmed message to posteriority said exactly this. In decision making, get the facts; in relationships, show love and tolerance. I find both parts inspiring. |
| 21743 | In wartime, happiness is hating the enemy, because it gives the war a purpose [Russell] |
| Full Idea: During and immediately after the war [14-18], those who hated the Germans were happier than those who still regarded them as human beings, because they could feel that what was being done served a good purpose. | |
| From: Bertrand Russell (An Outline of Philosophy [1927], Ch 22) | |
| A reaction: A striking remark. There are lots of situations where hatred seems to increase happiness. Russell is roughly defending consequentialism. |
| 21742 | Originally virtue was obedience, to gods, government, or custom [Russell] |
| Full Idea: Historically, virtue consisted at first of obedience to authority, whether that of the gods, the government, or custom. | |
| From: Bertrand Russell (An Outline of Philosophy [1927], Ch 22) | |
| A reaction: Russell proceeds to demolish such a theory, which he finds it fairly easy to do. In Nietzsche's terms, he is only describing slave virtue. Each role in the world has its own virtues (and functions). Which gods are the most virtuous? |
| 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'). |
| 21745 | Act so as to produce harmonious rather than discordant desires [Russell] |
| Full Idea: The supreme moral rule should be: Act so as to produce harmonious rather than discordant desires. | |
| From: Bertrand Russell (An Outline of Philosophy [1927], Ch 22) | |
| A reaction: Russell makes no reference to Kant, but this is obviously intended to rebut the more rationalist Kantian view of what is imperative. The use of 'harmonious' chimes in best with Plato's account of the soul in 'Republic'. |
| 21744 | Legally curbing people's desires is inferior to improving their desires [Russell] |
| Full Idea: To force a man to curb his desires, as we do by the criminal law, is not nearly so satisfactory as to cause him genuinely to feel the desires which promote socially harmonious conduct. | |
| From: Bertrand Russell (An Outline of Philosophy [1927], Ch 22) | |
| A reaction: It is hard to disagree, but improving the desires of selfish and even vicious people is a rather challenging task. |
| 22891 | We could be aware of time if senses briefly vibrated, extending their experience of movement [Russell, by Bardon] |
| Full Idea: Russell suggested, in defence of an empiricist theory of time-awareness, that a sense organ goes on vibrating, like a piano string, for while after the stimulation. Thus we can see the movement of a second hand, seen in several places at once. | |
| From: report of Bertrand Russell (An Outline of Philosophy [1927]) by Adrian Bardon - Brief History of the Philosophy of Time 2 'Realism' | |
| A reaction: Hm. If they were vibrating the last experience, they couldn't pick up the new one. When something fast happens the brain resonates fortissimo! If your eyes are moving it will be different neurons that get fired at each instant. |