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| 15941 | For intuitionists excluded middle is an outdated historical convention [Brouwer] |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! |
| 15148 | Powers give explanations, without being necessary for some class membership [Chakravartty] |
| Full Idea: Powers explain behaviours regardless of whether they are necessary for membership in a particular class of things. | |
| From: Anjan Chakravarrty (Inessential Aristotle: Powers without Essences [2012], 3) | |
| A reaction: This seems right, and is important for driving a wedge between powers and essences. If there are essences, they are not simply some bunch of powers. |
| 15145 | A kind essence is the necessary and sufficient properties for membership of a class [Chakravartty] |
| Full Idea: The modern concept of a kind essence is a set of intrinsic properties that are individually necessary and jointly sufficient for the membership of something in a class of things, or 'kind'. | |
| From: Anjan Chakravarrty (Inessential Aristotle: Powers without Essences [2012], 2) | |
| A reaction: I am always struck by the problem that the kind itself is constructed from the individuals, so circularity always seems to loom. |
| 15147 | Cluster kinds are explained simply by sharing some properties, not by an 'essence' [Chakravartty] |
| Full Idea: The fact that members of some cluster kinds are subjects of causal generalizations reflects the degree to which they share causally efficacious properties, not the fact that they may be composed of essence kinds per se. | |
| From: Anjan Chakravarrty (Inessential Aristotle: Powers without Essences [2012], 2) | |
| A reaction: I think this is right. I am a fan of individual essences, but not of kind essences. I take kinds, and kind explanations, to be straightforward inductive generalisations from individuals. Extreme stabilities give the illusion of a kind essence. |
| 15144 | Explanation of causal phenomena concerns essential kinds - but also lack of them [Chakravartty] |
| Full Idea: Scientific practices such as prediction and explanation regarding causal phenomena are concerned not merely with kinds having essences, but also with kinds lacking them. | |
| From: Anjan Chakravarrty (Inessential Aristotle: Powers without Essences [2012], 1) | |
| A reaction: Not quite clear what he has in mind, but explanation should certainly involve a coherent picture, and not just the citation of some underlying causal mechanism. |
| 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. |
| 15146 | Some kinds, such as electrons, have essences, but 'cluster kinds' do not [Chakravartty] |
| Full Idea: Many of the kinds we theorize about and experiment on today simply do not have essences. We can distinguish 'essence kinds', such as electrons, and 'cluster kinds', such as biological species. | |
| From: Anjan Chakravarrty (Inessential Aristotle: Powers without Essences [2012], 2) | |
| A reaction: This is an important point for essentialists. He offers a strict criterion, in Idea 15145, for mind membership, but we might allow species to have essences by just relaxing the criteria a bit, and acknowledging some vagueness, especially over time. |
| 15151 | Many causal laws do not refer to kinds, but only to properties [Chakravartty] |
| Full Idea: Causal laws often do not make reference to kinds of objects at all, but rather summarize relations between quantitative, causally efficacious properties of objects. | |
| From: Anjan Chakravarrty (Inessential Aristotle: Powers without Essences [2012], 3) | |
| A reaction: This would only be a serious challenge if it was not possible to translate talk of properties into talk of kinds, and vice versa. |
| 1513 | The Egyptians were the first to say the soul is immortal and reincarnated [Herodotus] |
| Full Idea: The Egyptians were the first to claim that the soul of a human being is immortal, and that each time the body dies the soul enters another creature just as it is being born. | |
| From: Herodotus (The Histories [c.435 BCE], 2.123.2) |