19 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| 15941 | For intuitionists excluded middle is an outdated historical convention [Brouwer] |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. |
| 24162 | Planck introduced the idea that energy can be quantized [Baggott] |
| Full Idea: By deriving his radiation law, Planck had inadvertently introduced the idea that energy itself could be 'quantized'. | |
| From: Jim Baggott (The Quantum Story: 40 moments [2011], 01) | |
| A reaction: He earlier assumed energy is continuously variable. I presume this means that the older idea of energy is now subsumed into the concept of fields, which are quantized into particles. The powers of nature are found in the fields. |
| 20645 | Heat is a state of vibration, not a substance [Joule] |
| Full Idea: We consider heat not as a substance but as a state of vibration. | |
| From: James Joule (works [1870]), quoted by Peter Watson - Convergence 01 'Nature's' | |
| A reaction: The puzzle is that giving accurate accounts of vibrations, heat and movement require a quantitative substance, energy. But all we have here is movement, and the denial of a substance. Energy is 'nature's currency system'. |
| 20972 | Joule showed that energy converts to heat, and heat to energy [Joule, by Papineau] |
| Full Idea: James Joule established the equivalence of heat and mechanical energy, in the sense of showing that a specific amount of heat will always be produced by the expenditure of a given amount of energy, and vice versa. | |
| From: report of James Joule (works [1870]) by David Papineau - Thinking about Consciousness App 4.2 | |
| A reaction: This was a major step towards the law of conservation of energy. |
| 21731 | Fields can be 'scalar', or 'vector', or 'tensor', or 'spinor' [Baggott] |
| Full Idea: Fields can be 'scalar', with no particular direction (pointing, but not pushing or pulling); or 'vector', with a direction (like magnetism, or Newtonian gravity); or 'tensor' (needing further parameters); or 'spinor' (depending on spin orientation). | |
| From: Jim Baggott (Farewell to Reality: fairytale physics [2013], 2 'Quantum') | |
| A reaction: [compressed] So the question is, why do they differ? What is it in the nature of each field the result in a distinctive directional feature? |
| 21730 | A 'field' is a property with a magnitude, distributed across all of space and time [Baggott] |
| Full Idea: A 'field' is defined in terms of the magnitude of some physical property distributed over every point in time and space. | |
| From: Jim Baggott (Farewell to Reality: fairytale physics [2013], 2 'Quantum') | |
| A reaction: If it involves a 'property', normal usage entails that there is some entity which possesses the property. So what's the entity? Eh? Eh? You don't know! Disappointed... |
| 24163 | Free electrons have clouds of virtual particles, arising from field interaction [Baggott] |
| Full Idea: A free electron doesn't simply persist as a point particle travelling along a predetermined, classical path; it is surrounded by a swarm of virtual particles arsising from self-interactions with its own magnetic field. | |
| From: Jim Baggott (The Quantum Story: 40 moments [2011], 19) | |
| A reaction: It seems to me important for amateurs and mere philosophers to hang on to this idea of virtual particles, because they undermine any attempt to impose a macro picture on sub-atomic events. |
| 24161 | Thermodynamics sees nature as a continuous flow of energy, as radiation and as substance [Baggott] |
| Full Idea: Thermodynamics reinforced a vision of nature as one of harmonious flow. Energy, which could be neither created nor destroyed, flowed continuously between radiation and material substance, in themselves unbroken continua. | |
| From: Jim Baggott (The Quantum Story: 40 moments [2011], 01) | |
| A reaction: Interestingly, Einstein's Special Relativity e = mc2 seems to endorse this view, by equation energy and mass. I've always wanted to know what energy is, but no one seems to know. |
| 21732 | The current standard model requires 61 particles [Baggott] |
| Full Idea: The current model requires 61 particles: three generations of two leptons and two flavours of quark, in three different colours (making 24); the anti-particles of all of these (48); 12 force particles (photon, W1, Z0, 8 gluons), and a Higgs boson. | |
| From: Jim Baggott (Farewell to Reality: fairytale physics [2013], 6 n) |
| 24160 | Particle measurements don't seem to reflect their reality [Baggott] |
| Full Idea: It seems that we can no longer assume that the particle properties we measure necessarily reflect or represent the properties of the particles as they really are. | |
| From: Jim Baggott (The Quantum Story: 40 moments [2011], Pref) | |
| A reaction: [He cites a 2006 experiment] This gives an interesting response to the Copenhagen Interpretation - that observers appear to be creating the reality they observe, because they only have 'observations', with no reality to correspond to them. I like it. |