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. |
| 20660 | At one level maths and nature are very similar, suggesting some deeper origin [Wolfram] |
| Full Idea: At some rather abstract level one can immediately recognise one basic similarity between nature and mathematics ...this suggests that the overall similarity between mathematics and nature must have a deeper origin. | |
| From: Stephen Wolfram (A New Kind of Science [2002], p.772), quoted by Peter Watson - Convergence 17 'Philosophy' | |
| A reaction: Personally I think mathematics has been derived by abstracting from the patterns in nature, and then further extrapolating from those abstractions. So the puzzle in nature is not the correspondence with mathematics, but the patterns. |
| 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! |
| 17371 | Some kinds are very explanatory, but others less so, and some not at all [Devitt] |
| Full Idea: Explanatory significance, hence naturalness, comes in degrees: positing some kinds may be very explanatory, positing others, only a little bit explanatory, positing others still, not explanatory at all. | |
| From: Michael Devitt (Natural Kinds and Biological Realism [2009], 4) | |
| A reaction: He mentions 'cousin' as a natural kind that is not very explanatory of anything. It interests us as humans, but not at all in other animals, it seems. ...Nice thought, though, that two squirrels might be cousins... |
| 20659 | Space and its contents seem to be one stuff - so space is the only existing thing [Wolfram] |
| Full Idea: It seems plausible that both space and its contents should somehow be made of the same stuff - so that in a sense space becomes the only thing in the universe. | |
| From: Stephen Wolfram (A New Kind of Science [2002], p.474), quoted by Peter Watson - Convergence 17 'Philosophy' | |
| A reaction: I presume the concept of a 'field' is what makes this idea possible. |
| 17372 | The higher categories are not natural kinds, so the Linnaean hierarchy should be given up [Devitt] |
| Full Idea: The signs are that the higher categories are not natural kinds and so the Linnaean hierarchy must be abandoned. ...This is not abandoning a hierarchy altogether, it is not abandoning a tree of life. | |
| From: Michael Devitt (Natural Kinds and Biological Realism [2009], 6) | |
| A reaction: Devitt's underlying point is that the higher and more general kinds do not have an essence (a specific nature), which is the qualification to be a natural kind. They explain nothing. Essence is the hallmark of natural kinds. Hmmm. |
| 17373 | Species pluralism says there are several good accounts of what a species is [Devitt] |
| Full Idea: Species pluralism is the view that there are several equally good accounts of what it is to be a species. | |
| From: Michael Devitt (Natural Kinds and Biological Realism [2009], 7) | |
| A reaction: Devitt votes for it, and cites Dupré, among many other. Given the existence of rival accounts, all making good points, it is hard to resist this view. |