5 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 |
| 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 |
| 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. |
| 19382 | Abstracta are abbreviated ways of talking; there are just substances, and truths about them [Leibniz] |
| Full Idea: I consider abstracta not as real things but as abbreviated ways of talking ...and to that extent I am a nominalist, at least provisionally ...It suffices to posit only substances as real things, and, to assert truths about these. | |
| From: Gottfried Leibniz (On the Reality of Accidents [1688]), quoted by Richard T.W. Arthur - Leibniz | |
| A reaction: I am a modern nominalist, in my hostility to a serious ontological commitment to abstracta. You get into trouble, though, if you say there are only objects or substances. Physics says reality may all be 'fields', or something.... 'Truths' is good. |
| 14409 | I am a presentist, and all language and common sense supports my view [Bigelow] |
| Full Idea: I am a presentist: nothing exists which is not present. Everyone believed this until the nineteenth century; it is writing into the grammar of natural languages; it is still assumed in everyday life, even by philosophers who deny it. | |
| From: John Bigelow (Presentism and Properties [1996], p.36), quoted by Trenton Merricks - Truth and Ontology | |
| A reaction: The most likely deniers of presentism seem to be physicists and cosmologists who have overdosed on Einstein. On the whole I vote for presentism, but what justifies truths about the past and future. Traces existing in the present? |