| 1958 | Hilbert's Programme |
| 05 | p.212 | 17809 | Gödel showed that the syntactic approach to the infinite is of limited value |
| 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) |
| 06 | p.213 | 17810 | The study of mathematical foundations needs new non-mathematical concepts |
| 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. |
| 13 | p.220 | 17811 | The natural conception of points ducks the problem of naming or constructing each point |
| 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. |