7 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. |
| 3912 | I must exist in order to be mistaken, so that even if I am mistaken, I can't be wrong about my own existence [Augustine] |
| Full Idea: Since therefore I must exist in order to be mistaken, then even if I am mistaken, there can be no doubt that I am not mistaken in my knowledge that I exist…. I know that I exist, and I also know that I know. | |
| From: Augustine (City of God [c.427], Ch.XI.26) | |
| A reaction: Fine, but the main problem is his over-confidence about a stable personal identity that does the thinking. |
| 6683 | The contact of spirit and body is utterly amazing, and incomprehensible [Augustine] |
| Full Idea: The manner of contact of spirit with body, which produces a living being, is utterly amazing and beyond our powers of comprehension | |
| From: Augustine (City of God [c.427], XXI.10) | |
| A reaction: This leads to a rather clear objection against a theory which needs a miracle to explain a common natural phenomenon. At least Augustine was beginning to recognise that interaction is a bit of a problem. |
| 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 |
| 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. |
| 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. |