12 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 |
| 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 |
| 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 |
| 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 |
| 22142 | In future, only logical limits can be placed on divine omnipotence [Anon (Par), by Boulter] |
| Full Idea: The Condemnation stipulated that all portions of the ancient intellectual heritage that placed non-logical limits on divine omnipotence were no longer to be tolerated. ...Philosophers now had to entertain the wildest ideas with all seriousness. | |
| From: report of Anon (Par) (The Condemnation of 1277 [1277]) by Stephen Boulter - Why Medieval Philosophy Matters 3 | |
| A reaction: Boulter identifies this as 'the ultimate source of Hume's philosophical delirium'. Presumably the angels-on-a-pinhead stuff originated with this. It is crazy to think that the only limit on possible existence is logic. Can God make a planet of uranium? |
| 16716 | It is heresy to require self-evident foundational principles in order to be certain [Anon (Par)] |
| Full Idea: Heresy 151: 'To have certainty regarding any conclusion, it must be founded on self-evident principles'. | |
| From: Anon (Par) (The Condemnation of 1277 [1277], 151), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 20.3 | |
| A reaction: The correct view is obviously to found certainty on faith and authority. It is one thing to be told that foundationalism is a poor theory, but another to be told it is a heresy, and thus a potential capital crime! |
| 2799 | Bayes' theorem explains why very surprising predictions have a higher value as evidence [Horwich] |
| Full Idea: Bayesianism can explain the fact that in science surprising predictions have greater evidential value, as the equation produces a higher degree of confirmation. | |
| From: Paul Horwich (Bayesianism [1992], p.42) |
| 2798 | Probability of H, given evidence E, is prob(H) x prob(E given H) / prob(E) [Horwich] |
| Full Idea: Bayesianism says ideally rational people should have degrees of belief (not all-or-nothing beliefs), corresponding with probability theory. Probability of H, given evidence E, is prob(H) X prob(E given H) / prob(E). | |
| From: Paul Horwich (Bayesianism [1992], p.41) |
| 1866 | It is heresy to teach that history repeats every 36,000 years [Anon (Par)] |
| Full Idea: It is heresy to teach that with all the heavenly bodies coming back to the same point after a period of thirty-six thousand years, the same effects as now exist will reappear. | |
| From: Anon (Par) (The Condemnation of 1277 [1277], §92) |
| 1865 | It is heresy to teach that natural impossibilities cannot even be achieved by God [Anon (Par)] |
| Full Idea: It is heresy to teach that what is absolutely impossible according to nature cannot be brought about by God or another agent. | |
| From: Anon (Par) (The Condemnation of 1277 [1277], §17) |
| 1864 | It is heresy to teach that we can know God by his essence in this mortal life [Anon (Par)] |
| Full Idea: It is heresy to teach that we can know God by his essence in this mortal life. | |
| From: Anon (Par) (The Condemnation of 1277 [1277], §9) |