17 ideas
| 12452 | Our dislike of contradiction in logic is a matter of psychology, not mathematics [Brouwer] |
| Full Idea: Not to the mathematician, but to the psychologist, belongs the task of explaining why ...we are averse to so-called contradictory systems in which the negative as well as the positive of certain propositions are valid. | |
| From: Luitzen E.J. Brouwer (Intuitionism and Formalism [1912], p.79) | |
| A reaction: Was the turning point of Graham Priest's life the day he read this sentence? I don't agree. I take the principle of non-contradiction to be a highly generalised observation of how the world works (and Russell agrees with me). |
| 15941 | For intuitionists excluded middle is an outdated historical convention [Brouwer] |
| Full Idea: From the intuitionist standpoint the dogma of the universal validity of the principle of excluded third in mathematics can only be considered as a phenomenon of history of civilization, like the rationality of pi or rotation of the sky about the earth. | |
| From: Luitzen E.J. Brouwer (works [1930]), quoted by Shaughan Lavine - Understanding the Infinite VI.2 | |
| A reaction: [Brouwer 1952:510-11] |
| 18119 | Mathematics is a mental activity which does not use language [Brouwer, by Bostock] |
| Full Idea: Brouwer made the rather extraordinary claim that mathematics is a mental activity which uses no language. | |
| From: report of Luitzen E.J. Brouwer (Mathematics, Science and Language [1928]) by David Bostock - Philosophy of Mathematics 7.1 | |
| A reaction: Since I take language to have far less of a role in thought than is commonly believed, I don't think this idea is absurd. I would say that we don't use language much when we are talking! |
| 18247 | Brouwer saw reals as potential, not actual, and produced by a rule, or a choice [Brouwer, by Shapiro] |
| Full Idea: In his early writing, Brouwer took a real number to be a Cauchy sequence determined by a rule. Later he augmented rule-governed sequences with free-choice sequences, but even then the attitude is that Cauchy sequences are potential, not actual infinities. | |
| From: report of Luitzen E.J. Brouwer (works [1930]) by Stewart Shapiro - Philosophy of Mathematics 6.6 | |
| A reaction: This is the 'constructivist' view of numbers, as espoused by intuitionists like Brouwer. |
| 12451 | Scientific laws largely rest on the results of counting and measuring [Brouwer] |
| Full Idea: A large part of the natural laws introduced by science treat only of the mutual relations between the results of counting and measuring. | |
| From: Luitzen E.J. Brouwer (Intuitionism and Formalism [1912], p.77) | |
| A reaction: His point, I take it, is that the higher reaches of numbers have lost touch with the original point of the system. I now see the whole issue as just depending on conventions about the agreed extension of the word 'number'. |
| 18118 | Brouwer regards the application of mathematics to the world as somehow 'wicked' [Brouwer, by Bostock] |
| Full Idea: Brouwer regards as somehow 'wicked' the idea that mathematics can be applied to a non-mental subject matter, the physical world, and that it might develop in response to the needs which that application reveals. | |
| From: report of Luitzen E.J. Brouwer (Mathematics, Science and Language [1928]) by David Bostock - Philosophy of Mathematics 7.1 | |
| A reaction: The idea is that mathematics only concerns creations of the human mind. It presumably has no more application than, say, noughts-and-crosses. |
| 12454 | Intuitionists only accept denumerable sets [Brouwer] |
| Full Idea: The intuitionist recognises only the existence of denumerable sets. | |
| From: Luitzen E.J. Brouwer (Intuitionism and Formalism [1912], p.80) | |
| A reaction: That takes you up to omega, but not beyond, presumably because it then loses sight of the original intuition of 'bare two-oneness' (Idea 12453). I sympathise, but the word 'number' has shifted its meaning a lot these days. |
| 12453 | Neo-intuitionism abstracts from the reuniting of moments, to intuit bare two-oneness [Brouwer] |
| Full Idea: Neo-intuitionism sees the falling apart of moments, reunited while remaining separated in time, as the fundamental phenomenon of human intellect, passing by abstracting to mathematical thinking, the intuition of bare two-oneness. | |
| From: Luitzen E.J. Brouwer (Intuitionism and Formalism [1912], p.80) | |
| A reaction: [compressed] A famous and somewhat obscure idea. He goes on to say that this creates one and two, and all the finite ordinals. |
| 8728 | Intuitionist mathematics deduces by introspective construction, and rejects unknown truths [Brouwer] |
| Full Idea: Mathematics rigorously treated from the point of view of deducing theorems exclusively by means of introspective construction, is called intuitionistic mathematics. It deviates from classical mathematics, which believes in unknown truths. | |
| From: Luitzen E.J. Brouwer (Consciousness, Philosophy and Mathematics [1948]), quoted by Stewart Shapiro - Thinking About Mathematics 1.2 | |
| A reaction: Clearly intuitionist mathematics is a close cousin of logical positivism and the verification principle. This view would be anathema to Frege, because it is psychological. Personally I believe in the existence of unknown truths, big time! |
| 10117 | Intuitonists in mathematics worried about unjustified assertion, as well as contradiction [Brouwer, by George/Velleman] |
| Full Idea: The concern of mathematical intuitionists was that the use of certain forms of inference generates, not contradiction, but unjustified assertions. | |
| From: report of Luitzen E.J. Brouwer (Intuitionism and Formalism [1912]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.6 | |
| A reaction: This seems to be the real origin of the verificationist idea in the theory of meaning. It is a hugely revolutionary idea - that ideas are not only ruled out of court by contradiction, but that there are other criteria which should also be met. |
| 23146 | Motives produce intentions, which lead to actions [Driver] |
| Full Idea: Motives will cause persons for form intentions; it is intentions which more directly guide actions. | |
| From: Julia Driver (The Virtues and Human Nature [1996], 3) | |
| A reaction: This is invites the question of whether there is a sharp distinction between the motive and the action. Detectives look for motives, but law courts look for intentions. |
| 23147 | Good intentions are not necessary for virtue [Driver] |
| Full Idea: I deny the claim that good intentions are necessary for virtue. | |
| From: Julia Driver (The Virtues and Human Nature [1996], 3) | |
| A reaction: Presumably one could continually do the right thing, because it was your duty or your job, without actually being well motivated for it. |
| 23144 | Virtue should be defined by consequences, not by states of mind [Driver] |
| Full Idea: The behavioural aspects of virtue are more important than its phenomenology, because virtue is best defined along consequentialist lines. | |
| From: Julia Driver (The Virtues and Human Nature [1996], Intro) | |
| A reaction: This is the thesis of her paper. Quite persuasive. Consequences are, of course, important in all moral theories (even Kant's). She doesn't rely on human nature. The social virtues vary according to the circumstances, such as gossiping in wartime. |
| 23148 | Virtues are character traits or dispositions which produce good consequences for others [Driver] |
| Full Idea: A moral virtue is a character trait (a disposition or cluster of dispositions) which, generally speaking, produces good consequences for others. | |
| From: Julia Driver (The Virtues and Human Nature [1996], 3) | |
| A reaction: There are self-directed virtues, such as keeping fit and healthy. There are virtues for ways to receive the kindness of others. That said, I like this idea. |
| 23150 | Control of pregnancy and knowledge of paternity have downgraded chastity [Driver] |
| Full Idea: Women now have more control over becoming pregnant. Men can now be more certain of paternity, without the constraint of female chastity. Hence chastity is no longer considered a moral virtue. | |
| From: Julia Driver (The Virtues and Human Nature [1996], 5) | |
| A reaction: A persuasive argument that virtues are defined by their consequences (to which I add my example of gossiping in wartime). Different social situations and crises promote or relegate the status of certain virtues (such as food hoarding). |
| 23149 | If generosity systematically turned recipients into parasites, it wouldn't be a virtue [Driver] |
| Full Idea: If generosity towards the needy in the long run produced [social] parasites, and if generosity did this systematically, then it would not be a moral virtue. | |
| From: Julia Driver (The Virtues and Human Nature [1996], 5) | |
| A reaction: A very persuasive example. Hume has similar views - that we encourage those emotions which have good social outcomes. |
| 20716 | False prophets will perform wonders to deceive even the elect [Mark] |
| Full Idea: For false messiahs and false prophets will appear and perform signs and wonders to deceive, if possible, even the elect. | |
| From: St Mark (02: Gospel of St Mark [c.66], 13:22), quoted by Brian Davies - Introduction to the Philosophy of Religion | |
| A reaction: This casts a rather different light on the miracles of Jesus, since they were performed in a context in which even Jesus believed that lots of people (and not just the son of God) could perform miracles. Undermines any Argument from Miracles. |