6 ideas
| 12427 | All of mathematics is properties of the whole numbers [Kronecker] |
| Full Idea: All the results of significant mathematical research must ultimately be expressible in the simple forms of properties of whole numbers. | |
| From: Leopold Kronecker (works [1885], Vol 3/274), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 09.5 | |
| A reaction: I've always liked Kronecker's line, but I'm beginning to realise that his use of the word 'number' is simply out-of-date. Natural numbers have a special status, but not sufficient to support this claim. |
| 10091 | God made the integers, all the rest is the work of man [Kronecker] |
| Full Idea: God made the integers, all the rest is the work of man. | |
| From: Leopold Kronecker (works [1885]), quoted by A.George / D.J.Velleman - Philosophies of Mathematics Intro | |
| A reaction: This famous remark was first quoted in Kronecker's obituary. A response to Dedekind, it seems. See Idea 10090. Did he really mean that negative numbers were the work of God? We took a long time to spot them. |
| 10645 | We reach concepts by clarification, or by definition, or by habitual experience [Price,HH] |
| Full Idea: We have three different ways in which we arrive at concepts or universals: there is a clarification, where we have a ready-made concept and define it; we have a combination (where a definition creates a concept); and an experience can lead to a habit. | |
| From: H.H. Price (Review of Aron 'Our Knowledge of Universals' [1946], p.190) | |
| A reaction: [very compressed] He cites Russell as calling the third one a 'condensed induction'. There seems to an intellectualist and non-intellectualist strand in the abstractionist tradition. |
| 10644 | A 'felt familiarity' with universals is more primitive than abstraction [Price,HH] |
| Full Idea: A 'felt familiarity' with universals seems to be more primitive than explicit abstraction. | |
| From: H.H. Price (Review of Aron 'Our Knowledge of Universals' [1946], p.188) | |
| A reaction: This I take to be part of the 'given' of the abstractionist view, which is quite well described in the first instance by Aristotle. Price says that it is 'pre-verbal'. |
| 10646 | Our understanding of 'dog' or 'house' arises from a repeated experience of concomitances [Price,HH] |
| Full Idea: Whether you call it inductive or not, our understanding of such a word as 'dog' or 'house' does arise from a repeated experience of concomitances. | |
| From: H.H. Price (Review of Aron 'Our Knowledge of Universals' [1946], p.191) | |
| A reaction: Philosophers don't use phrases like that last one any more. How else could we form the concept of 'dog' - if we are actually allowed to discuss the question of concept-formation, instead of just the logic of concepts. |
| 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'). |