24 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| 15941 | For intuitionists excluded middle is an outdated historical convention [Brouwer] |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] |
| 4568 | If 'Queen of England' does not refer if there is no queen, its meaning can't refer if there is one [Cooper,DE] |
| Full Idea: If 'the Queen of England' is not a referring expression when there is no queen, nor can it be one when there is a queen - since the meaning of the expression is the same in either case. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §4.1) | |
| A reaction: I'm not convinced. Does this mean that since I can point with my finger at nothing, I therefore do not indicate anything when there is an object at which I am pointing. Sounds silly to me. |
| 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! |
| 4574 | If some peoples do not have categories like time or cause, they can't be essential features of rationality [Cooper,DE] |
| Full Idea: If our most basic concepts, like time, space, substance or causality, are not shared by some peoples, it puts paid to the cherished ideal of philosophers to discover a set of concepts or categories which any rational human must employ in his thinking. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §5.2) | |
| A reaction: This seems to be a place where a priori philosophy (Aristotle,Kant,Hegel) meets empirical research (Whorf). However, interpreting the research is so fraught with problems it drives you back to the a priori… |
| 4573 | If it is claimed that language correlates with culture, we must be able to identify the two independently [Cooper,DE] |
| Full Idea: If it is claimed that linguistic differences significantly correlate with cultural differences, it must therefore be possible to identify the linguistic differences independently from the cultural ones. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §5.1) | |
| A reaction: This is a basic objection to any extreme relativist version of the S-P hypothesis. They are part of the conspiracy to overemphasise language in philosophy, and they are wrong. |
| 4575 | A person's language doesn't prove their concepts, but how are concepts deduced apart from language? [Cooper,DE] |
| Full Idea: It would be absurd to say the Hopi lack the concept of time because they lack tensed verbs, ..but how do we find out what a man's concepts are except in terms of his language? | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §5.2) | |
| A reaction: Presumably we should look at animals, where concepts must be inferred in order to explain behaviour. I don't see why introspection (scientifically wicked) should not also be employed to detect our own non-verbal concepts. How are new words invented? |
| 20653 | Six reduction levels: groups, lives, cells, molecules, atoms, particles [Putnam/Oppenheim, by Watson] |
| Full Idea: There are six 'reductive levels' in science: social groups, (multicellular) living things, cells, molecules, atoms, and elementary particles. | |
| From: report of H.Putnam/P.Oppenheim (Unity of Science as a Working Hypothesis [1958]) by Peter Watson - Convergence 10 'Intro' | |
| A reaction: I have the impression that fields are seen as more fundamental that elementary particles. What is the status of the 'laws' that are supposed to govern these things? What is the status of space and time within this picture? |
| 4561 | Many sentences set up dispositions which are irrelevant to the meanings of the sentences [Cooper,DE] |
| Full Idea: Many sentences set up dispositions which are irrelevant to the meanings of the sentences. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §2.3) | |
| A reaction: Yet another telling objection to behaviourism. When I look at broccoli I may have a disposition to be sick, but that isn't part of the concept of broccoli. |
| 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. |
| 4564 | I can meaningfully speculate that humans may have experiences currently impossible for us [Cooper,DE] |
| Full Idea: It is not meaningless for me to postulate the potential for humans to sense in a manner which is at present unimaginable and indescribable. There is no reason to believe me, but I might be right. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §3.1) | |
| A reaction: The key counterexample to verificationist theories of meaning is wild speculations, which are clearly meaningful, though frequently far beyond any likely human experience. Logical positivists are allergic to imagination. |
| 4565 | The verification principle itself seems neither analytic nor verifiable [Cooper,DE] |
| Full Idea: It seems that the positivists must admit that there is at least one statement which is meaningful, but which is neither verifiable nor analytic - namely, the statement of the principle of verification itself. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §3.1) | |
| A reaction: Some people think this objection is decisive, but I think any theory must be permitted a few metatheoretic assertions or axioms which are beyond discussion. Ayer thought the VP might be treated as analytic. Everyone has to start somewhere. |
| 4563 | 'How now brown cow?' is used for elocution, but this says nothing about its meaning [Cooper,DE] |
| Full Idea: The sentence 'How now brown cow?' has its use in elocutions classes, yet this aspect of its use tells us nothing about its meaning. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §2.4) | |
| A reaction: Indeed, and also there are weird sentence of which we can assemble a meaning, but cannot think of any conceivable use ('rats swim in purple marmalade'). |
| 4562 | Most people know how to use the word "Amen", but they do not know what it means [Cooper,DE] |
| Full Idea: Most people know how to use the word "Amen", but they do not know what it means. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §2.4) | |
| A reaction: Personally I find examples like this decisive against the 'use' theory of meaning. Maybe the defence is that the theory works for sentences, and individual words (like passwords) are peripheral. |
| 4571 | Reference need not be a hit-or-miss affair [Cooper,DE] |
| Full Idea: Reference need not be a hit-or-miss affair. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §4.2) | |
| A reaction: Sounds right. If the basic scenario is picking someone out in a crowd, your listener may think they know which person you are talking about, with a high degree of probability. |
| 4566 | Any thesis about reference is also a thesis about what exists to be referred to [Cooper,DE] |
| Full Idea: Any thesis about reference is also going to be a thesis about what there is in existence to refer to. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §4) | |
| A reaction: I see the point, but we must not put the cart before the horse. I may have an intuition that something exists, but not know how to refer to it (because of my small vocabulary). |
| 4572 | If predicates name things, that reduces every sentence to a mere list of names [Cooper,DE] |
| Full Idea: If predicates are names of entities, then subject/predicate sentences are pairs of names, since subjects are names (or referring expressions). But a pair of names is not a sentence at all, it is a mere list. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §4.4) | |
| A reaction: If that is meant to demolish universals it is too quick. Concatenating names is not the same as listing them. A relationship is asserted. There is a (mysterious) Platonic 'partaking' between form and particular. Perhaps. |
| 4576 | An analytic truth is one which becomes a logical truth when some synonyms have been replaced [Cooper,DE] |
| Full Idea: The definition of analytic truth which has, I believe, the most chance of success is one in terms of synonymy; ..an analytic truth is one which can be transformed into a logical truth once synonyms are replaced by synonyms. | |
| From: David E. Cooper (Philosophy and the Nature of Language [1973], §7.1) | |
| A reaction: Sounds promising, though there is Quine's notorious problem of circularity in all these concepts. If synonymy is conventional, then so is analyticity. I personally feel that the circle can be broken. |