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! |
| 20327 | Modern attention has moved from the intrinsic properties of art to its relational properties [Lamarque/Olson] |
| Full Idea: In modern discussions, rather than look for intrinsic properties of objects, including aesthetic or formal properties, attention has turned to extrinsic or relational properties, notably of a social, historical, or 'institutional' nature. | |
| From: Lamargue,P/Olson,SH (Introductions to 'Aesthetics and the Phil of Art' [2004], Pt 1) | |
| A reaction: Lots of modern branches of philosophy have made this move, which seems to me like a defeat. We want to know why things have the relations they do. Just mapping the relations is superficial Humeanism. |
| 20326 | Early 20th cent attempts at defining art focused on significant form, intuition, expression, unity [Lamarque/Olson] |
| Full Idea: In the early twentieth century there were numerous attempts at defining the essence art. Significant form, intuition, the expression of emotion, organic unity, and other notions, were offered to this end. | |
| From: Lamargue,P/Olson,SH (Introductions to 'Aesthetics and the Phil of Art' [2004], Pt 1) | |
| A reaction: As far as I can see the whole of aesthetics was demolished in one blow by Marcel Duchamp's urinal. Artists announce: we will tell you what art is; you should just sit and listen. Compare the invention of an anarchic sport. |
| 20330 | The dualistic view says works of art are either abstract objects (types), or physical objects [Lamarque/Olson] |
| Full Idea: The dualistic view of the arts holds that works of art come in two fundamentally different kinds: those that are abstract entities, i.e. types, and those that are physical objects (tokens). | |
| From: Lamargue,P/Olson,SH (Introductions to 'Aesthetics and the Phil of Art' [2004], Pt 2) | |
| A reaction: Paintings are the main reason for retaining physical objects. Strawson 1974 argues that paintings are only physical because we cannot yet perfectly reproduce them. I agree. Works of art are types, not tokens. |
| 1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
| Full Idea: Archelaus was the first person to say that the universe is boundless. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.3 |
| 5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
| Full Idea: Archelaus wrote that life on Earth began in a primeval slime. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Malcolm Schofield - Archelaus | |
| A reaction: This sounds like a fairly clearcut assertion of the production of life by evolution. Darwin's contribution was to propose the mechanism for achieving it. We should honour the name of Archelaus for this idea. |