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| 15941 | For intuitionists excluded middle is an outdated historical convention [Brouwer] |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! |
| 16083 | Aristotelian matter seriously threatens the intrinsic unity and substantiality of its object [Gill,ML] |
| Full Idea: On the interpretation of Aristotelian matter that I shall propose, matter seriously threatens the intrinsic unity, and hence the substantiality, of the object to which it contributes. | |
| From: Mary Louise Gill (Aristotle on Substance [1989], Intro) | |
| A reaction: Presumably the thought is that if an object is form+matter (hylomorphism), then forms are essentially unified, but matter is essentially unified and sloppy. |
| 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. |
| 536 | We should follow the law in public, and nature in private [Antiphon] |
| Full Idea: A man can best conduct himself in harmony with justice, if when in company of witnesses he upholds the laws, and when alone without witnesses he upholds the edicts of nature. | |
| From: Antiphon (fragments/reports [c.439 BCE], B44), quoted by Anon (Oxy) - Oxyrhynchus Papyrus XI.1364 | |
| A reaction: I'm not sure how you identify the 'edicts of nature', without guidance from other people or the law. Natural behaviour can be pretty grim. |
| 1557 | To gain the greatest advantage only treat law as important when other people are present [Antiphon] |
| Full Idea: The way to get maximum advantage to yourself from justice is to treat the laws as important when other people are present, but when there is nobody else with you to value the demands of nature. | |
| From: Antiphon (fragments/reports [c.439 BCE], B44A), quoted by Anon (Oxy) - Oxyrhynchus Papyrus 1364A | |
| A reaction: This looks like a pretty good description of the majority of people active in politics. |
| 540 | The way you spend your time will form your character [Antiphon] |
| Full Idea: One's character must necessarily grow like that with which one spends the greater part of the day. | |
| From: Antiphon (fragments/reports [c.439 BCE], B62), quoted by John Stobaeus - Anthology 3.31.41 |
| 539 | Nothing is worse for mankind than anarchy [Antiphon] |
| Full Idea: Nothing is worse for mankind than anarchy. | |
| From: Antiphon (fragments/reports [c.439 BCE], B61), quoted by (who?) - where? |
| 17006 | Prime matter has no place in Aristotle's theories, and passages claiming it are misread [Gill,ML] |
| Full Idea: I argue that prime matter has no place in Aristotle's elemental theory. ..References to prime matter are found in Aristotle's work because his theory was thought to need the doctrine. If I am right, these passages will all admit of another interpretation. | |
| From: Mary Louise Gill (Aristotle on Substance [1989], App) | |
| A reaction: If correct, this strikes me as important for the history of ideas, because scholastics got themselves in a right tangle over prime matter. See Pasnau on it. It pushed the 17th century into corpuscularianism. |
| 16093 | Prime matter is actually nothing and potentially everything (or potentially an element) [Gill,ML] |
| Full Idea: Prime matter is supposed to be actually nothing and potentially everything or, at any rate, potentially the simplest bodies - earth, water, air and fire. | |
| From: Mary Louise Gill (Aristotle on Substance [1989], Ch.1) | |
| A reaction: The view that the four elements turn out to be prime matter is distinctive of Gill's approach. Prime matter sounds like quark soup in the early universe. |