16 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| 15941 | For intuitionists excluded middle is an outdated historical convention [Brouwer] |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. |
| 8628 | I hold that algebra and number are developments of logic [Jevons] |
| Full Idea: I hold that algebra is a highly developed logic, and number but logical discrimination. | |
| From: William S. Jevons (The Principles of Science [1879], p.156), quoted by Gottlob Frege - Grundlagen der Arithmetik (Foundations) §15 | |
| A reaction: Thus Frege shows that logicism was an idea that was in the air before he started writing. Riemann's geometry and Boole's logic presumably had some influence here. |
| 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! |
| 22179 | Explanatory facts also predict, and predictive facts also explain [Hempel, by Okasha] |
| Full Idea: Hempel said every scientific explanation is potentially a prediction - it would have predicted the phenomenon in question, had it not already been known. But also the information used to make a prediction is potentially an explanation. | |
| From: report of Carl Hempel (Aspects of Scientific Explanation [1965]) by Samir Okasha - Philosophy of Science: Very Short Intro (2nd ed) 3 | |
| A reaction: Sounds too neatly glib to be quite true. If you explain a single event there is nothing to predict. You might predict accurately from a repetitive pattern, with no understanding at all of the pattern. |
| 21507 | Scientific explanation aims at a unifying account of underlying structures and processes [Hempel] |
| Full Idea: What theoretical scientific explanation aims at is an objective kind of insight that is achieved by a systematic unification, by exhibiting the phenomena as manifestations of common underlying structures and processes that conform to testable principles. | |
| From: Carl Hempel (Philosophy of Natural Science [1967], p.83), quoted by Laurence Bonjour - The Structure of Empirical Knowledge 5.3 | |
| A reaction: This is a pretty good statement of scientific essentialism, and structures and processes are what I take Aristotle to have had in mind when he sought 'what it is to be that thing'. Structures and processes give stability and powers. |
| 6755 | For Hempel, explanations are deductive-nomological or probabilistic-statistical [Hempel, by Bird] |
| Full Idea: Hempel proposes that explanations involve covering laws and antecedent conditions; this view (the 'covering law' view) has two versions, the deductive-nomological model and the probabilistic-statistical model of explanation. | |
| From: report of Carl Hempel (Aspects of Scientific Explanation [1965]) by Alexander Bird - Philosophy of Science Ch.2 | |
| A reaction: The obvious problem with this approach, it seem to me, is that the laws themselves need explanation, and I don't see how a law can be foundational unless there is a divine law-giver. Are the laws arbitrary and axiomatic? |
| 17083 | The covering-law model is for scientific explanation; historical explanation is quite different [Hempel] |
| Full Idea: To put forward the covering-law models of scientific explanation is not to deny that there are other contexts in which we speak of explanation. ….That it does not fit explaining the rules of Hanoverian succession is to miss the intent of our model. | |
| From: Carl Hempel (Aspects of Scientific Explanation [1965], p. 412-3), quoted by David-Hillel Ruben - Explaining Explanation Ch 1 | |
| A reaction: Important to get that clear. It then requires a clear demarcation between science and the rest, and it had better not rule out biology because it is having a love affair with physics. |
| 13052 | Hempel rejects causation as part of explanation [Hempel, by Salmon] |
| Full Idea: Hempel explicitly rejects the idea that causality plays any essential explanatory role. | |
| From: report of Carl Hempel (Aspects of Scientific Explanation [1965], p.352) by Wesley Salmon - Four Decades of Scientific Explanation 1.1 | |
| A reaction: Hempel champions the 'covering-law' model of explanation. It strikes me that Hempel is so utterly wrong about this that his views aren't even a candidate for correctness, but then for a long time his views were orthodoxy. |
| 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. |