11 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! |
| 7506 | God made man in his own image [Anon (Tor)] |
| Full Idea: And God said, let us make man in our image, after our likeness. | |
| From: Anon (Tor) (01: Book of Genesis [c.750 BCE], 1.26) | |
| A reaction: Since we are obviously not identical in every way with God, we can presumably choose in which respects we think of ourselves as being like Him. Reason, understanding, beauty, goodness, consciousness? A troublesome verse, challenged by Darwin. |
| 16782 | The names of all the types of creature were given forever by Adam [Anon (Tor)] |
| Full Idea: Whatsoever Adam called any living creature, the same is its name. And Adam called all the beasts by their names, and all the fowls of the air, and all the cattle of the field. | |
| From: Anon (Tor) (01: Book of Genesis [c.750 BCE], 02:20) |
| 9111 | God is not wise, but more-than-wise; God is not good, but more-than-good [William of Ockham] |
| Full Idea: God is not wise, but more-than-wise; God is not good, but more-than-good. | |
| From: William of Ockham (Reportatio [1330], III Q viii) | |
| A reaction: [He is quoting 'Damascene'] I quote this for interest, but I very much doubt whether Damascene or William knew what it meant, and I certainly don't. There seems to have been a politically correct desire to invent super-powers for God. |
| 4013 | And God saw the light, that it was good [Anon (Tor)] |
| Full Idea: And God saw the light, that it was good. | |
| From: Anon (Tor) (01: Book of Genesis [c.750 BCE], 01.04) | |
| A reaction: The text seems to suggest that God did not decide that it was good, but that it conformed to a standard of goodness. |
| 9112 | We could never form a concept of God's wisdom if we couldn't abstract it from creatures [William of Ockham] |
| Full Idea: What we abstract is said to belong to perfection in so far as it can be predicated of God and can stand for Him. For if such a concept could not be abstracted from a creature, then in this life we could not arrive at a cognition of God's wisdom. | |
| From: William of Ockham (Reportatio [1330], III Q viii) | |
| A reaction: This seems to be the germ of an important argument. Without the ability to abstract from what is experienced, we would not be able to apply general concepts to things which are beyond experience. It is a key idea for empiricism. |