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 |
| 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 |
| 18969 | How do you distinguish three beliefs from four beliefs or two beliefs? [Quine] |
| Full Idea: Suppose I say that I have given up precisely three beliefs since lunch. An over-coarse individuation could reduce the number to two, and an over-fine one could raise it to four. | |
| From: Willard Quine (Propositional Objects [1965], p.144) | |
| A reaction: Obviously if you ask how many beliefs I hold, it would be crazy to give a precise answer. But if I search for my cat, I give up my belief that it is in the kitchen, in the lounge and in the bathroom. That's precise enough to be three beliefs, I think. |
| 19569 | We have a basic epistemic duty to believe truth and avoid error [Chisholm, by Kvanvig] |
| Full Idea: Chisholm says our fundamental epistemic duties arise from the fundamental duty to (do one's best to) believe the truth and avoid error. | |
| From: report of Roderick Chisholm (Theory of Knowledge (2nd ed 1977) [1966]) by Jonathan Kvanvig - Truth is not the Primary Epistemic Goal 'Epistemic' | |
| A reaction: Since it strikes me as impossible to perceive something as being true, and yet still not believe it (except in moments of shock), I don't see why we need to introduce dubious claims about 'duty' here. Stupidity isn't a failure of duty. |
| 18967 | A 'proposition' is said to be the timeless cognitive part of the meaning of a sentence [Quine] |
| Full Idea: A 'proposition' is the meaning of a sentence. More precisely, since propositions are supposed to be true or false once and for all, it is the meaning of an eternal sentence. More precisely still, it is the 'cognitive' meaning, involving truth, not poetry. | |
| From: Willard Quine (Propositional Objects [1965], p.139) | |
| A reaction: Quine defines this in order to attack it. I equate a proposition with a thought, and take a sentence to be an attempt to express a proposition. I have no idea why they are supposed to be 'timeless'. Philosophers have some very odd ideas. |
| 18968 | The problem with propositions is their individuation. When do two sentences express one proposition? [Quine] |
| Full Idea: The trouble with propositions, as cognitive meanings of eternal sentences, is individuation. Given two eternal sentences, themselves visibly different linguistically, it is not sufficiently clear under when to say that they mean the same proposition. | |
| From: Willard Quine (Propositional Objects [1965], p.140) | |
| A reaction: If a group of people agree that two sentences mean the same thing, which happens all the time, I don't see what gives Quine the right to have a philosophical moan about some dubious activity called 'individuation'. |
| 18970 | The concept of a 'point' makes no sense without the idea of absolute position [Quine] |
| Full Idea: Unless we are prepared to believe that absolute position makes sense, the very idea of a point as an entity in its own right must be rejected as not merely mysterious but absurd. | |
| From: Willard Quine (Propositional Objects [1965], p.149) | |
| A reaction: The fact that without absolute position we can only think of 'points' as relative to a conceptual grid doesn't stop the grid from picking out actual locations in space, as shown by latitude and longitude. |