10 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. |
| 8793 | If observation is knowledge, it is not just an experience; it is a justification in the space of reasons [Sellars] |
| Full Idea: In characterizing an observational episode or state as 'knowing', we are not giving an empirical description of it; we are placing it in the logical space of reasons, of justifying and being able to justify what one says. | |
| From: Wilfrid Sellars (Does Emp.Knowledge have Foundation? [1956], p.123) | |
| A reaction: McDowell has made the Kantian phrase 'the logical space of reasons' very popular. This is a very nice statement of the internalist view of justification, with which I sympathise more and more. It is a rationalist coherentist view. It needn't be mystical! |
| 8792 | Observations like 'this is green' presuppose truths about what is a reliable symptom of what [Sellars] |
| Full Idea: Observational knowledge of any particular fact, e.g. that this is green, presupposes that one knows general facts of the form 'X is a reliable symptom of Y'. | |
| From: Wilfrid Sellars (Does Emp.Knowledge have Foundation? [1956], p.123) | |
| A reaction: This is a nicely observed version of the regress problem with justification. I would guess that foundationalists would simply deny that this further knowledge is required; 'this is green' arises out of the experience, but it is not an inference. |
| 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! |
| 3643 | The concept of mind excludes body, and vice versa [Descartes] |
| Full Idea: The concept of body includes nothing at all which belongs to the mind, and the concept of mind includes nothing at all which belongs to the body. | |
| From: René Descartes (Reply to Fourth Objections [1641], 225) | |
| A reaction: A headache? Hunger? The mistake, I think, is to regard the mind as entirely conscious, thus creating a sharp boundary between two aspects of our lives. As shown by blindsight, I take many of my central mental operations to be pre- or non-conscious. |
| 8791 | The concept of 'green' involves a battery of other concepts [Sellars] |
| Full Idea: One can only have the concept of green by having a whole battery of concepts of which it is one element. | |
| From: Wilfrid Sellars (Does Emp.Knowledge have Foundation? [1956], p.120) | |
| A reaction: This points in the direction of holism about language and thought, but need not imply it. It might be that concepts have to be learned in small families. It is not clear, though, what is absolutely essential to 'green', except that it indicates colour. |