5 ideas
| 17809 | Gödel showed that the syntactic approach to the infinite is of limited value [Kreisel] |
| Full Idea: Usually Gödel's incompleteness theorems are taken as showing a limitation on the syntactic approach to an understanding of the concept of infinity. | |
| From: Georg Kreisel (Hilbert's Programme [1958], 05) |
| 17810 | The study of mathematical foundations needs new non-mathematical concepts [Kreisel] |
| Full Idea: It is necessary to use non-mathematical concepts, i.e. concepts lacking the precision which permit mathematical manipulation, for a significant approach to foundations. We currently have no concepts of this kind which we can take seriously. | |
| From: Georg Kreisel (Hilbert's Programme [1958], 06) | |
| A reaction: Music to the ears of any philosopher of mathematics, because it means they are not yet out of a job. |
| 3032 | I can form no notion of what the good is [Amphis] |
| Full Idea: What the good is I no more can form a notion of, than of the good of Plato. | |
| From: Amphis (comedies (frags) [c.350 BCE]), quoted by Diogenes Laertius - Lives of Eminent Philosophers 03.1.22 | |
| A reaction: It was evidently a running joke in the ancient world that no one could define Plato's Form of the Good. He was said to have written a book on it, now lost. |
| 21097 | Modern monarchies are (like republics) rule by law, rather than by men [Hume] |
| Full Idea: In modern times monarchical government seems to have made the greatest advances towards perfection. It may now be affirmed of civilized monarchies, what was formerly said in praise of republics alone, that they are a government of laws, not of men. | |
| From: David Hume (Of Civil Liberty [1750], p.54) | |
| A reaction: Dreams of simple 'government by law' disappeared with the rise of modern media, which can be controlled by wealth. |
| 17811 | The natural conception of points ducks the problem of naming or constructing each point [Kreisel] |
| Full Idea: In analysis, the most natural conception of a point ignores the matter of naming the point, i.e. how the real number is represented or by what constructions the point is reached from given points. | |
| From: Georg Kreisel (Hilbert's Programme [1958], 13) | |
| A reaction: This problem has bothered me. There are formal ways of constructing real numbers, but they don't seem to result in a name for each one. |