6 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 |
10053 | Geometrical axioms imply the propositions, but the former may not be true [Russell] |
Full Idea: We must only assert of various geometries that the axioms imply the propositions, not that the axioms are true and therefore that the propositions are true. | |
From: Bertrand Russell (Foundations of Geometry [1897], Intro vii), quoted by Alan Musgrave - Logicism Revisited §4 | |
A reaction: Clearly the truth of the axioms can remain a separate issue from whether they actually imply the theorems. The truth of the axioms might be as much a metaphysical as an empirical question. Musgrave sees this as the birth of if-thenism. |
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. |
10052 | Geometry is united by the intuitive axioms of projective geometry [Russell, by Musgrave] |
Full Idea: Russell sought what was common to Euclidean and non-Euclidean systems, found it in the axioms of projective geometry, and took a Kantian view of them. | |
From: report of Bertrand Russell (Foundations of Geometry [1897]) by Alan Musgrave - Logicism Revisited §4 | |
A reaction: Russell's work just preceded Hilbert's famous book. Tarski later produced some logical axioms for geometry. |
7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
Full Idea: The six perfections are of giving, morality, patience, vigour, meditation, and wisdom. | |
From: Nagarjuna (Mahaprajnaparamitashastra [c.120], 88) | |
A reaction: What is 'morality', if giving is not part of it? I like patience and vigour being two of the virtues, which immediately implies an Aristotelian mean (which is always what is 'appropriate'). |