11 ideas
| 21554 | Sets always exceed terms, so all the sets must exceed all the sets [Lackey] |
| Full Idea: Cantor proved that the number of sets in a collection of terms is larger than the number of terms. Hence Cantor's Paradox says the number of sets in the collection of all sets must be larger than the number of sets in the collection of all sets. | |
| From: Douglas Lackey (Intros to Russell's 'Essays in Analysis' [1973], p.127) | |
| A reaction: The sets must count as terms in the next iteration, but that is a normal application of the Power Set axiom. |
| 21553 | It seems that the ordinal number of all the ordinals must be bigger than itself [Lackey] |
| Full Idea: The ordinal series is well-ordered and thus has an ordinal number, and a series of ordinals to a given ordinal exceeds that ordinal by 1. So the series of all ordinals has an ordinal number that exceeds its own ordinal number by 1. | |
| From: Douglas Lackey (Intros to Russell's 'Essays in Analysis' [1973], p.127) | |
| A reaction: Formulated by Burali-Forti in 1897. |
| 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 |
| 6019 | If someone squashed a horse to make a dog, something new would now exist [Mnesarchus] |
| Full Idea: If, for the sake of argument, someone were to mould a horse, squash it, then make a dog, it would be reasonable for us on seeing this to say that this previously did not exist but now does exist. | |
| From: Mnesarchus (fragments/reports [c.120 BCE]), quoted by John Stobaeus - Anthology 179.11 | |
| A reaction: Locke would say it is new, because the substance is the same, but a new life now exists. A sword could cease to exist and become a new ploughshare, I would think. Apply this to the Ship of Theseus. Is form more important than substance? |
| 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. |
| 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? |
| 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 |