11 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 |
| 7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
| Full Idea: The six perfections are of giving, morality, patience, vigour, meditation, and wisdom. | |
| From: Nagarjuna (Mahaprajnaparamitashastra [c.120], 88) | |
| A reaction: What is 'morality', if giving is not part of it? I like patience and vigour being two of the virtues, which immediately implies an Aristotelian mean (which is always what is 'appropriate'). |
| 8430 | Causal statements are used to explain, to predict, to control, to attribute responsibility, and in theories [Kim] |
| Full Idea: The function of causal statements is 1) to explain events, 2) for predictive usefulness, 3) to help control events, 4) with agents, to attribute moral responsibility, 5) in physical theory. We should judge causal theories by how they account for these. | |
| From: Jaegwon Kim (Causes and Counterfactuals [1973], p.207) | |
| A reaction: He suggests that Lewis's counterfactual theory won't do well on this test. I think the first one is what matters. Philosophy aims to understand, and that is achieved through explanation. Regularity and counterfactual theories explain very little. |
| 8396 | Many counterfactuals have nothing to do with causation [Kim, by Tooley] |
| Full Idea: Kim has pointed out that there are a number of counterfactuals that have nothing to do with causation. If John marries Mary, then if John had not existed he would not have married Mary, but that is not the cause of their union. | |
| From: report of Jaegwon Kim (Causes and Counterfactuals [1973], 5.2) by Michael Tooley - Causation and Supervenience | |
| A reaction: One might not think that this mattered, but it leaves the problem of distinguishing between the causal counterfactuals and the rest (and you mustn't mention causation when you are doing it!). |
| 8429 | Counterfactuals can express four other relations between events, apart from causation [Kim] |
| Full Idea: Counterfactuals can express 'analytical' dependency, or the fact that one event is part of another, or an action done by doing another, or (most interestingly) an event can determine another without causally determining it. | |
| From: Jaegwon Kim (Causes and Counterfactuals [1973], p.205) | |
| A reaction: [Kim gives example of each case] Counterfactuals can even express a relation that involves no dependency. Or they might just involve redescription, as in 'If Scott were still alive, then the author of "Waverley" would be too'. |
| 8428 | Causation is not the only dependency relation expressed by counterfactuals [Kim] |
| Full Idea: The sort of dependency expressed by counterfactual relations is considerably broader than strictly causal dependency, and causal dependency is only one among the heterogeneous group of dependency relationships counterfactuals can express. | |
| From: Jaegwon Kim (Causes and Counterfactuals [1973], p.205) | |
| A reaction: In 'If pigs could fly, one and one still wouldn't make three' there isn't even a dependency. Kim has opened up lines of criticism which make the counterfactual analysis of causation look very implausible to me. |
| 4781 | Many counterfactual truths do not imply causation ('if yesterday wasn't Monday, it isn't Tuesday') [Kim, by Psillos] |
| Full Idea: Kim gives a range of examples of counterfactual dependence without causation, as: 'if yesterday wasn't Monday, today wouldn't be Tuesday', and 'if my sister had not given birth, I would not be an uncle'. | |
| From: report of Jaegwon Kim (Causes and Counterfactuals [1973]) by Stathis Psillos - Causation and Explanation §3.3 | |
| A reaction: This is aimed at David Lewis. The objection seems like commonsense. "If you blink, the cat gets it". Causal claims involve counterfactuals, but they are not definitive of what causation is. |