10 ideas
| 13489 | Von Neumann treated cardinals as a special sort of ordinal [Neumann, by Hart,WD] |
| Full Idea: Von Neumann's decision was to start with the ordinals and to treat cardinals as a special sort of ordinal. | |
| From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by William D. Hart - The Evolution of Logic 3 | |
| A reaction: [see Hart 73-74 for an explication of this] |
| 12336 | A von Neumann ordinal is a transitive set with transitive elements [Neumann, by Badiou] |
| Full Idea: In Von Neumann's definition an ordinal is a transitive set in which all of the elements are transitive. | |
| From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Alain Badiou - Briefings on Existence 11 |
| 18180 | Von Neumann numbers are preferred, because they continue into the transfinite [Maddy on Neumann] |
| Full Idea: Von Neumann's version of the natural numbers is in fact preferred because it carries over directly to the transfinite ordinals. | |
| From: comment on John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Penelope Maddy - Naturalism in Mathematics I.2 n9 |
| 18179 | For Von Neumann the successor of n is n U {n} (rather than {n}) [Neumann, by Maddy] |
| Full Idea: For Von Neumann the successor of n is n U {n} (rather than Zermelo's successor, which is {n}). | |
| From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Penelope Maddy - Naturalism in Mathematics I.2 n8 |
| 15925 | Each Von Neumann ordinal number is the set of its predecessors [Neumann, by Lavine] |
| Full Idea: Each Von Neumann ordinal number is the set of its predecessors. ...He had shown how to introduce ordinal numbers as sets, making it possible to use them without leaving the domain of sets. | |
| From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Shaughan Lavine - Understanding the Infinite V.3 |
| 19216 | Propositions (such as 'that dog is barking') only exist if their items exist [Williamson] |
| Full Idea: A proposition about an item exists only if that item exists... how could something be the proposition that that dog is barking in circumstances in which that dog does not exist? | |
| From: Timothy Williamson (Necessary Existents [2002], p.240), quoted by Trenton Merricks - Propositions | |
| A reaction: This is a view of propositions I can't make sense of. If I'm under an illusion that there is a dog barking nearby, when there isn't one, can I not say 'that dog is barking'? If I haven't expressed a proposition, what have I done? |
| 23896 | We see our character as a restricting limit, but also as an unshakable support [Weil] |
| Full Idea: Our character appears to us as a limit by which we do not want to be imprisoned, …but also as a support that we want to believe is unshakable. | |
| From: Simone Weil (On the Concept of Character [1941], p.100) | |
| A reaction: A nice perception. It is fairly easy to criticise, or even laugh at, one's own actions, but extremely hard to criticise our own character. Maybe we all wish we were more determined in our projects, but not much else. |
| 23893 | We don't see character in a single moment, but only over a period of time [Weil] |
| Full Idea: Character is constant over a period of time; the way a person is at a single moment does not at all reflect the character of this person. We do, however, concede that character changes. | |
| From: Simone Weil (On the Concept of Character [1941], p.98) | |
| A reaction: I do think, though, that there are moments in behaviour which are hugely revealing of character, even in a single remark. But I agree that most single moments do not show much. |
| 23894 | The concept of character is at the centre of morality [Weil] |
| Full Idea: We cannot pose a moral problem without putting the concept of character at its centre. | |
| From: Simone Weil (On the Concept of Character [1941], p.98) | |
| A reaction: The question for Aristotle (which I derive from Philippa Foot) is whether moral goodness simply is good character, or whether it is the actions (or even the consequences). Weil is close to modern virtue theory here. |
| 23895 | We modify our character by placing ourselves in situations, or by attending to what seems trivial [Weil] |
| Full Idea: We can modify our character, by putting ourselves in circumstances that will act on us from the outside, …or by the orientation of our attention in the moments that appear most insignificant or indifferent in our lives. | |
| From: Simone Weil (On the Concept of Character [1941], p.99) | |
| A reaction: I've never seen anyone address this question (apart from Aristotle's emphasis on training habits). Choosing your source for current affairs information strikes me as very important. What you read, what you watch, who you spend time with… |