17 ideas
| 3340 | Von Neumann defines each number as the set of all smaller numbers [Neumann, by Blackburn] |
| Full Idea: Von Neumann defines each number as the set of all smaller numbers. | |
| From: report of John von Neumann (works [1935]) by Simon Blackburn - Oxford Dictionary of Philosophy p.280 |
| 15943 | Limitation of Size is not self-evident, and seems too strong [Lavine on Neumann] |
| Full Idea: Von Neumann's Limitation of Size axiom is not self-evident, and he himself admitted that it seemed too strong. | |
| From: comment on John von Neumann (An Axiomatization of Set Theory [1925]) by Shaughan Lavine - Understanding the Infinite VII.1 |
| 3355 | Von Neumann wanted mathematical functions to replace sets [Neumann, by Benardete,JA] |
| Full Idea: Von Neumann suggested that functions be pressed into service to replace sets. | |
| From: report of John von Neumann (works [1935]) by José A. Benardete - Metaphysics: the logical approach Ch.23 |
| 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] |
| 22716 | Von Neumann defined ordinals as the set of all smaller ordinals [Neumann, by Poundstone] |
| Full Idea: At age twenty, Von Neumann devised the formal definition of ordinal numbers that is used today: an ordinal number is the set of all smaller ordinal numbers. | |
| From: report of John von Neumann (works [1935]) by William Poundstone - Prisoner's Dilemma 02 'Sturm' | |
| A reaction: I take this to be an example of an impredicative definition (not predicating something new), because it uses 'ordinal number' in the definition of ordinal number. I'm guessing the null set gets us started. |
| 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 |
| 13672 | All the axioms for mathematics presuppose set theory [Neumann] |
| Full Idea: There is no axiom system for mathematics, geometry, and so forth that does not presuppose set theory. | |
| From: John von Neumann (An Axiomatization of Set Theory [1925]), quoted by Stewart Shapiro - Foundations without Foundationalism 8.2 | |
| A reaction: Von Neumann was doubting whether set theory could have axioms, and hence the whole project is doomed, and we face relativism about such things. His ally was Skolem in this. |
| 8845 | An experience's having propositional content doesn't make it a belief [Pryor] |
| Full Idea: To say that experiences have propositional content is not to say that experiences are beliefs. | |
| From: James Pryor (There is immediate Justification [2005], §4) | |
| A reaction: This is important for opponents of foundationalism, because they will not allow a raw experience to act as a justification on its own. Even if concepts, or even propositions, are offered by experience, the crucial evaluation must preceded knowledge. |
| 8842 | The best argument for immediate justification is not the Regress Argument, but considering examples [Pryor] |
| Full Idea: The best argument for immediate justification is not the Regress Argument, but from considering examples, such as I have a headache, I am raising my arm, I am imagining my grandmother, or seeing how dominoes could fill a chessboard. | |
| From: James Pryor (There is immediate Justification [2005], §3) | |
| A reaction: Most of his examples depend on the fact that they cannot be challenged by anyone else, because they are within his own mind. The dominoes require complex thought. The first two could be erroneous if he was dreaming. |
| 8843 | Impure coherentists accept that perceptions can justify, unlike pure coherentists [Pryor] |
| Full Idea: Pure coherentists claim that a belief can only be justified by its relations to other beliefs; impure coherentists are willing to give some non-beliefs, such as perceptual experiences, a justifying role. | |
| From: James Pryor (There is immediate Justification [2005], §4) | |
| A reaction: I think I would vote for the pure version. The distinction that is needed, I think, is between justification and evidence. You have to surmise causal links and explanations before you can see an experience as evidence, and then justification. |
| 8844 | Coherentism rests on the claim that justifications must be beliefs, with propositional content [Pryor] |
| Full Idea: The Master Argument for coherentism is the claim that a justifier requires asserted propositional content, and that only beliefs represent propositions assertively. | |
| From: James Pryor (There is immediate Justification [2005], §4) | |
| A reaction: I think this claim (which Pryor attacks) is correct. A key point is that almost any experience can be delusional, and in need of critical evaluation. We would even only accept an experience as being necessarily veridical after critical evaluation. |
| 8846 | Reasons for beliefs can be cited to others, unlike a raw headache experience [Pryor] |
| Full Idea: If you have reasons for your belief, they should be considerations you could in principle cite, or give, to someone who doubted or challenged the belief. You can't give some else a non-propositional state like a headache. | |
| From: James Pryor (There is immediate Justification [2005], §6) | |
| A reaction: On the whole I agree, but if someone asked you to justify your claim that there is a beautiful sunset over the harbour, you could just say 'Look!'. Headaches are too private. The person must still see that the sunset is red, and not the window. |
| 8847 | Beliefs are not chosen, but you can seek ways to influence your belief [Pryor] |
| Full Idea: Ordinarily we make no intentional choices about what to believe, but one can choose to believe something, and then seek ways to get oneself to believe it. | |
| From: James Pryor (There is immediate Justification [2005], §7) | |
| A reaction: Deliberately reading the articles of a philosopher that you seem to agree with would be an example. Presumably the belief that this is a good belief and should be given support is not itself voluntarily chosen. Ultimately we are helpless. See Idea 1854. |
| 1513 | The Egyptians were the first to say the soul is immortal and reincarnated [Herodotus] |
| Full Idea: The Egyptians were the first to claim that the soul of a human being is immortal, and that each time the body dies the soul enters another creature just as it is being born. | |
| From: Herodotus (The Histories [c.435 BCE], 2.123.2) |