13 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. |
| 472 | No things would be clear to us as entity or relationships unless there existed Number and its essence [Philolaus] |
| Full Idea: No existing things would be clear to anyone, either in themselves or in their relationship to one another, unless there existed Number and its essence. | |
| From: Philolaus (On the Cosmos (lost) [c.435 BCE], B11), quoted by John Stobaeus - Anthology 1.03.8 |
| 1757 | The Electra: she knows this man, but not that he is her brother [Eucleides, by Diog. Laertius] |
| Full Idea: The 'Electra': Electra knows that Orestes is her brother, but not that this man is Orestes, so she knows and does not know her brother simultaneously. | |
| From: report of Eucleides (fragments/reports [c.410 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Eu.4 | |
| A reaction: Hence we distinguish 'know of', 'know that' and 'know how'. Hence Russell makes 'knowledge by acquaintance' fundamental, and descriptions come later. |
| 22511 | Some reasonings are stronger than we are [Philolaus] |
| Full Idea: Some reasonings are stronger than we are. | |
| From: Philolaus (fragments/reports [c.425 BCE]), quoted by Aristotle - Eudemian Ethics 1225a33 | |
| A reaction: This endorses the Aristotle view of akrasia (as opposed to the Socratic view). This isolated remark seems to imply that we are more clearly embodiments of will than of reason. |
| 3028 | The chief good is unity, sometimes seen as prudence, or God, or intellect [Eucleides] |
| Full Idea: The chief good is unity, which is known by several names, for at one time people call it prudence, at another time God, at another intellect, and so on. | |
| From: Eucleides (fragments/reports [c.410 BCE]), quoted by Diogenes Laertius - Lives of Eminent Philosophers 02.9.2 | |
| A reaction: So the chief good is what unites and focuses our moral actions. Kant calls that 'the will'. |
| 1518 | Everything must involve numbers, or it couldn't be thought about or known [Philolaus] |
| Full Idea: Everything which is known has number, because otherwise it is impossible for anything to be the object of thought or knowledge. | |
| From: Philolaus (On the Cosmos (lost) [c.435 BCE], B04), quoted by John Stobaeus - Anthology 1.21.7b |
| 1519 | Harmony must pre-exist the cosmos, to bring the dissimilar sources together [Philolaus] |
| Full Idea: It would have been impossible for the dissimilar and incompatible sources to have been made into an orderly universe unless harmony had been present in some form or other. | |
| From: Philolaus (On the Cosmos (lost) [c.435 BCE], B06), quoted by John Stobaeus - Anthology 1.21.7d |
| 473 | There is no falsehood in harmony and number, only in irrational things [Philolaus] |
| Full Idea: The nature of number and harmony admits of no falsehood; for this is unrelated to them. Falsehood and envy belong to the nature of the Unlimited and the Unintelligent and the Irrational. | |
| From: Philolaus (On the Cosmos (lost) [c.435 BCE], B11), quoted by (who?) - where? |
| 469 | Existing things, and hence the Cosmos, are a mixture of the Limited and the Unlimited [Philolaus] |
| Full Idea: Since it is plain that existing things are neither wholly from the Limiting, nor wholly from the Unlimited, clearly the cosmos and its contents were fitted together from both the Limiting and the Unlimited. | |
| From: Philolaus (On the Cosmos (lost) [c.435 BCE], B02), quoted by John Stobaeus - Anthology 1.21.7a |
| 476 | Self-created numbers make the universe stable [Philolaus] |
| Full Idea: Number is the ruling and self-created bond which maintains the everlasting stability of the contents of the universe. | |
| From: Philolaus (On the Cosmos (lost) [c.435 BCE], B23), quoted by (who?) - where? |
| 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. |
| 1787 | Philolaus was the first person to say the earth moves in a circle [Philolaus, by Diog. Laertius] |
| Full Idea: Philolaus was the first person to affirm that the earth moves in a circle. | |
| From: report of Philolaus (On the Cosmos (lost) [c.435 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 08.Ph.3 |