17 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! |
| 20653 | Six reduction levels: groups, lives, cells, molecules, atoms, particles [Putnam/Oppenheim, by Watson] |
| Full Idea: There are six 'reductive levels' in science: social groups, (multicellular) living things, cells, molecules, atoms, and elementary particles. | |
| From: report of H.Putnam/P.Oppenheim (Unity of Science as a Working Hypothesis [1958]) by Peter Watson - Convergence 10 'Intro' | |
| A reaction: I have the impression that fields are seen as more fundamental that elementary particles. What is the status of the 'laws' that are supposed to govern these things? What is the status of space and time within this picture? |
| 20086 | Nowadays sovereignty (once the basis of a state) has become relative [Reybrouck] |
| Full Idea: In the twenty-first century, sovereignty, once the basis of the nation state, has become a relative concept. ...Powerlessness is the key word of our time. | |
| From: David van Reybrouck (Against Elections [2013], 1 'Crisis') | |
| A reaction: The point is that nation states now have limited power, in the face of larger unions, multinational companies, and global problems. |
| 20090 | Today it seems almost impossible to learn the will of the people [Reybrouck] |
| Full Idea: Imagine having to develop a system today that would express the will of the people. | |
| From: David van Reybrouck (Against Elections [2013], 2 'electoral') | |
| A reaction: Our recent Brexit referendum didn't do the job, because it was confined to a single question. Van Reybrouck laughs at the idea of expressing it through a polling both. How about a council of 500, drawn by lots? Meet for three months. |
| 20087 | There are no united monolothic 'peoples', and no 'national gut feelings' [Reybrouck] |
| Full Idea: There is no such thing as one monolithic 'people' (every society has its diversity), nor is there anything that could be described as a 'national gut feeling'. | |
| From: David van Reybrouck (Against Elections [2013], 2 'populism') | |
| A reaction: Rousseau yearned for a republic no bigger than Geneva. I don't see why we should give up on the general will in huge modern societies. It is likely, though, to be an anodyne lowest common denominator. No bad thing, perhaps. |
| 20089 | Technocrats may be efficient, but they lose legitimacy as soon as they do unpopular things [Reybrouck] |
| Full Idea: Efficiency does not automatically generate legitimacy, and faith in the technocrat melts away as soon as spending cuts are implemented. | |
| From: David van Reybrouck (Against Elections [2013], 2 'democracy') | |
| A reaction: They can hang on to legitimacy if they can come up with some technical mumbo-jumbo like 'monetarism' which the people will swallow. |
| 20088 | Technocrats are expert managers, who replace politicians, and can be long-term and unpopular [Reybrouck] |
| Full Idea: Technocracy is a system where experts are charged with looking after the public interest. ...Technocrats are managers who replace politicians, so they can concentrate on long-term solutions and announce unpopular measures. | |
| From: David van Reybrouck (Against Elections [2013], 2 'technocracy') | |
| A reaction: I like technocrats. They just need to be accountable. In the UK we have far more respect for the governor of the Bank of England than for any politician. |
| 20085 | Democracy is the best compromise between legitimacy and efficiency [Reybrouck] |
| Full Idea: Democracy is the least bad form of all governments precisely because it attempts to find a healthy balance between legitimacy and efficiency. | |
| From: David van Reybrouck (Against Elections [2013], 1 'Crisis') | |
| A reaction: There seems to be a widespread feeling that democracy is declining in efficiency, and that may be because our remoteness from government decreases legitimacy, so we have less commitment to getting things done. |
| 20095 | A referendum result arises largely from ignorance [Reybrouck] |
| Full Idea: In a referendum you ask everyone to vote on a subject that usually only a few know anything about. | |
| From: David van Reybrouck (Against Elections [2013], 4 'remedies') | |
| A reaction: Tell me about it! I was forced to vote in the 2016 Brexit referendum, and felt thoroughly out of my depth on such a complex economic question. |
| 20094 | You don't really govern people if you don't involve them [Reybrouck] |
| Full Idea: Even with the best of intentions, those who govern the people without involving them, govern them only in a limited sense. | |
| From: David van Reybrouck (Against Elections [2013], 4 intro) | |
| A reaction: But if they are highly involved, who is governing who? Do we want the people to become happier about being governed, or do we want them more involved in doing the governing? |
| 20093 | In the 18th century democratic lots lost out to elections, that gave us a non-hereditary aristocracy [Reybrouck] |
| Full Idea: The drawing of lots, the most democratic of all political instruments, lost out in the eighteenth century to elections, a procedure that was not invented as a democratic instrument, but as a means of bringing a new non-hereditary aristocracy to power. | |
| From: David van Reybrouck (Against Elections [2013], 3 'democratisation') | |
| A reaction: This is the basic thesis of Van Reybrouck's book. He argues for the extensive use of lots ('sortition') for getting people involved in modern democracies. I love the idea that in a good democracy you get an occasional chance to rule. |
| 20091 | Representative elections were developed in order to avoid democracy [Reybrouck] |
| Full Idea: Bernard Manin (1995) revealed how, immediately after the American and French revolutions, the electoral-representative system was chosen with the intention of keeping at bay the tumult of democracy. | |
| From: David van Reybrouck (Against Elections [2013], 3 'procedure') | |
| A reaction: At the time America and France were two of the largest countries in the world, and communication and transport were slow. That has changed. |