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. |
| 3144 | Everything is what it is, and not another thing [Butler] |
| Full Idea: Everything is what it is, and not another thing. | |
| From: Joseph Butler (works [1732]), quoted by Georges Rey - Contemporary Philosophy of Mind 2.4 |
| 21315 | A tree remains the same in the popular sense, but not in the strict philosophical sense [Butler] |
| Full Idea: When a man swears to the same tree having stood for fifty years in the same place, he means ...not that the tree has been all that time the same in the strict philosophical sense of the word. ...In a loose and popular sense they are said to be the same. | |
| From: Joseph Butler (Analogy of Religion [1736], App.1) | |
| A reaction: A helpful distinction which we should hang on. Of course, by the standards of modern physics, nothing is strictly the same from one Planck time to the next. All is flux. So we either drop the word 'same' (for objects) or relax a bit. |
| 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! |
| 21317 | Despite consciousness fluctuating, we are aware that it belongs to one person [Butler] |
| Full Idea: Though the successive consciousnesses which we have of our own existence are not the same, yet they are consciousnesses of one and the same thing or object; of the same person, self, or living agent. | |
| From: Joseph Butler (Analogy of Religion [1736], App.1) | |
| A reaction: Butler's arguments seems to be that he appears to be the same person, so he is the same person. He is explicitly disagreeing with Locke. |
| 21313 | If consciousness of events makes our identity, then if we have forgotten them we didn't exist then [Butler] |
| Full Idea: Though consciousness of what is past does ascertain our personal identity to ourselves, yet to say that it makes personal identity, or is necessary to our being the same persons is to say a person has not existed a single moment but what he can remember. | |
| From: Joseph Butler (Analogy of Religion [1736], App.1) | |
| A reaction: An over-cautious scepticism has crept in about the reliability of bodily identity. Now we can have photographs and CCTV to prove that we experienced events we have forgotten. Butler is right. |
| 21314 | Consciousness presupposes personal identity, so it cannot constitute it [Butler] |
| Full Idea: One would think it really self-evident that consciousness of personal identity presupposes, and therefore cannot constitute, personal identity, any more than knowledge can presuppose truth, which it presupposes. | |
| From: Joseph Butler (Analogy of Religion [1736], App.1) | |
| A reaction: It rather begs the question to dogmatically assert that mere consciousness presupposes a self, especially after Hume's criticisms. That consciousness implies a subject to experience needs arguing for. Is it the best explanation? |
| 21318 | If the self changes, we have no responsibilities, and no interest in past or future [Butler] |
| Full Idea: If personality is a transient thing ...then it follows that it is a fallacy to charge ourselves with any thing we did, or to imagine our present selves interested in any thing which befell us yesterday, or what will befall us tomorrow. | |
| From: Joseph Butler (Analogy of Religion [1736], App.1) | |
| A reaction: We seem to care about the past and future of our children, without actually being our children. Can't my future self be my descendant, a close one, instead of me? |
| 8066 | Butler exalts conscience, but it may be horribly misleading [Anscombe on Butler] |
| Full Idea: Butler exalts conscience, but appears ignorant that a man's conscience may tell him to do the vilest things. | |
| From: comment on Joseph Butler (Fifteen Sermons [1726]) by G.E.M. Anscombe - Modern Moral Philosophy p.176 | |
| A reaction: That would appear to be the end of conscience. To make conscience work, it must have a huge authority to back it, and also a fairly infallible means of knowing what it truly says, and that an impostor hasn't replaced it (e.g. via a bad upbringing). |
| 6005 | Animals are dangerous and nourishing, and can't form contracts of justice [Hermarchus, by Sedley] |
| Full Idea: Hermarchus said that animal killing is justified by considerations of human safety and nourishment and by animals' inability to form contractual relations of justice with us. | |
| From: report of Hermarchus (fragments/reports [c.270 BCE]) by David A. Sedley - Hermarchus | |
| A reaction: Could the last argument be used to justify torturing animals? Or could we eat a human who was too brain-damaged to form contracts? |