17 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 |
| 7293 | It is legitimate to do harm if it is the unintended side-effect of an effort to achieve a good [Grayling] |
| Full Idea: The doctrine of double effect says that it is legitimate to do harm if the harm is the unintended side-effect of an effort to achieve a legitimate goal. | |
| From: A.C. Grayling (Among the Dead Cities [2006], Ch.6) | |
| A reaction: I think a key principle of morality is our duty to think about possible unnoticed consequences of our actions. To neglect concern for side-effects is wicked. Beyond that, the issue must concern the particulars of the situation. |
| 2705 | How can intuitionists distinguish universal convictions from local cultural ones? [Hare] |
| Full Idea: There are convictions which are common to most societies; but there are others which are not, and no way is given by intuitionists of telling which are the authoritative data. | |
| From: Richard M. Hare (Universal Prescriptivism [1991], p.454) | |
| A reaction: It seems unfair on intuitionists to say they haven't given a way to evaluate such things, given that they have offered intuition. The issue is what exactly they mean by 'intuition'. |
| 2712 | You can't use intuitions to decide which intuitions you should cultivate [Hare] |
| Full Idea: If it comes to deciding what intuitions and dispositions to cultivate, we cannot rely on the intuitions themselves, as intuitionists do. | |
| From: Richard M. Hare (Universal Prescriptivism [1991], p.461) | |
| A reaction: Makes intuitionists sound a bit dim. Surely Hume identifies dispositions (such as benevolence) which should be cultivated, because they self-evidently improve social life? |
| 2706 | Emotivists mistakenly think all disagreements are about facts, and so there are no moral reasons [Hare] |
| Full Idea: Emotivists concluded too hastily that because naturalism and intuitionism are false, you cannot reason about moral questions, because they assumed that the only questions you can reason about are factual ones. | |
| From: Richard M. Hare (Universal Prescriptivism [1991], p.455) | |
| A reaction: Personally I have a naturalistic view of ethics (based on successful functioning, as indicated by Aristotle), so not my prob. Why can't we reason about expressive emotions? We reason about art. |
| 2708 | An 'ought' statement implies universal application [Hare] |
| Full Idea: In any 'ought' statement there is implicit a principle which says that the statement applies to all precisely similar situations. | |
| From: Richard M. Hare (Universal Prescriptivism [1991], p.456) | |
| A reaction: No two situations can ever be 'precisely' similar. Indeed, 'precisely similar' may be an oxymoron (at least for situations). Kantians presumably like this idea. |
| 2704 | If morality is just a natural or intuitive description, that leads to relativism [Hare] |
| Full Idea: Non-descriptivists (e.g. prescriptivists) reject descriptivism in its naturalist or intuitionist form, because they are both destined to collapse into relativism. | |
| From: Richard M. Hare (Universal Prescriptivism [1991], p.453) | |
| A reaction: I'm not clear from this why prescriptism would not also turn out to be relativist, if it includes evaluations along with facts. |
| 2703 | Descriptivism say ethical meaning is just truth-conditions; prescriptivism adds an evaluation [Hare] |
| Full Idea: Ethical descriptivism is the view that ethical sentence-meaning is wholly determined by truth-conditions. …Prescriptivists think there is a further element of meaning, which expresses prescriptions or evaluations or attitudes which we assent to. | |
| From: Richard M. Hare (Universal Prescriptivism [1991], p.452) | |
| A reaction: Not sure I understand either of these. If all meaning consists of truth-conditions, that will apply to ethics. If meaning includes evaluations, that will apply to non-ethics. |
| 2707 | If there can be contradictory prescriptions, then reasoning must be involved [Hare] |
| Full Idea: Prescriptivists claim that there are rules of reasoning which govern non-descriptive as well as descriptive speech acts. The standard example is possible logical inconsistency between contradictory prescriptions. | |
| From: Richard M. Hare (Universal Prescriptivism [1991], p.455) | |
| A reaction: The example doesn't seem very good. Inconsistency can appear in any area of thought, but that isn't enough to infer full 'rules of reasoning'. I could desire two incompatible crazy things. |
| 2711 | Prescriptivism implies a commitment, but descriptivism doesn't [Hare] |
| Full Idea: Prescriptivists hold that moral judgements commit the speaker to motivations and actions, but non-moral facts by themselves do not do this. | |
| From: Richard M. Hare (Universal Prescriptivism [1991], p.459) | |
| A reaction: Surely hunger motivates to action? I suppose the key word is 'commit'. But lazy people are allowed to make moral judgements. |
| 2709 | Prescriptivism sees 'ought' statements as imperatives which are universalisable [Hare] |
| Full Idea: Universal prescriptivists hold that 'ought'-judgements are prescriptive like plain imperatives, but differ from them in being universalisable. | |
| From: Richard M. Hare (Universal Prescriptivism [1991], p.457) | |
| A reaction: Sounds a bit tautological. Which comes first, the normativity or the universalisability? |
| 2710 | Moral judgements must invoke some sort of principle [Hare] |
| Full Idea: To make moral judgements is implicitly to invoke some principle, however specific. | |
| From: Richard M. Hare (Universal Prescriptivism [1991], p.458) |
| 7292 | War must also have a good chance of success, and be waged with moderation [Grayling] |
| Full Idea: To Aquinas's three conditions for war (Idea 7291) modern theorists have added two others: that to be just a war must have a reasonable chance of success, and that the means used to conduct it must be proportional to the ends sought. | |
| From: A.C. Grayling (Among the Dead Cities [2006], Ch.6) | |
| A reaction: These two principles strike me as being much more civilized and humane than Aquinas's original contribution, suggesting that in our theoretical thinking we might be making some progress. |