6 ideas
| 9616 | A set is a collection into a whole of distinct objects of our intuition or thought [Cantor] |
| Full Idea: A set is any collection into a whole M of definite, distinct objects m ... of our intuition or thought. | |
| From: George Cantor (The Theory of Transfinite Numbers [1897], p.85), quoted by James Robert Brown - Philosophy of Mathematics Ch.2 | |
| A reaction: This is the original conception of a set, which hit trouble with Russell's Paradox. Cantor's original definition immediately invites thoughts about the status of vague objects. |
| 15896 | Cantor needed Power Set for the reals, but then couldn't count the new collections [Cantor, by Lavine] |
| Full Idea: Cantor grafted the Power Set axiom onto his theory when he needed it to incorporate the real numbers, ...but his theory was supposed to be theory of collections that can be counted, but he didn't know how to count the new collections. | |
| From: report of George Cantor (The Theory of Transfinite Numbers [1897]) by Shaughan Lavine - Understanding the Infinite I | |
| A reaction: I take this to refer to the countability of the sets, rather than the members of the sets. Lavine notes that counting was Cantor's key principle, but he now had to abandon it. Zermelo came to the rescue. |
| 19393 | What is not active is nothing [Leibniz] |
| Full Idea: We can now show from the inner truths of metaphysics that what is not active is nothing. | |
| From: Gottfried Leibniz (True Method in Philosophy and Theology [1686], p.64) | |
| A reaction: This is Leibniz's rebellion against the Cartesian idea that all that matters for natural existence is spatial extension. I agree (tentatively) with Leibniz's vision of nature here. Modern physics reveals a seething turmoil beneath the placid exterior. |
| 14979 | Being alone doesn't guarantee intrinsic properties; 'being alone' is itself extrinsic [Lewis, by Sider] |
| Full Idea: The property of 'being alone in the world' is an extrinsic property, even though it has had by an object that is alone in the world. | |
| From: report of David Lewis (Extrinsic Properties [1983]) by Theodore Sider - Writing the Book of the World 01.2 | |
| A reaction: I always choke on my cornflakes whenever anyone cites a true predicate as if it were a genuine property. This is a counterexample to Idea 14978. Sider offers another more elaborate example from Lewis. |
| 15454 | Extrinsic properties come in degrees, with 'brother' less extrinsic than 'sibling' [Lewis] |
| Full Idea: Properties may be more or less intrinsic; being a brother has more of an admixture of intrinsic structure than being a sibling does, yet both are extrinsic. | |
| From: David Lewis (Extrinsic Properties [1983], I) | |
| A reaction: I suppose the point is that a brother is intrinsically male - but then a sibling is intrinsically human. A totally extrinsic relation would be one between entities which shared virtually no categories of existence. |
| 15455 | Total intrinsic properties give us what a thing is [Lewis] |
| Full Idea: The way something is is given by the totality of its intrinsic properties. | |
| From: David Lewis (Extrinsic Properties [1983], I) | |
| A reaction: No. Some properties are intrinsic but trivial. The 'important' ones fix the identity (if the identity is indeed 'fixed'). |