26 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| 15941 | For intuitionists excluded middle is an outdated historical convention [Brouwer] |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! |
| 16078 | Clay is intrinsically and atomically the same as statue (and that lacks 'modal properties') [Rudder Baker] |
| Full Idea: Arguments for statue being the clay are: that the clay is intrinsically like the statue, that the clay has the same atoms as the statue', that objects don't have modal properties such as being necessarily F, and the reference of 'property' changes. | |
| From: Lynne Rudder Baker (Why Constitution is not Identity [1997], II) | |
| A reaction: [my summary of the arguments she identifies - see text for details] Rudder Baker attempts to refute all four of these arguments, in defence of constitution as different from identity. |
| 16077 | The clay is not a statue - it borrows that property from the statue it constitutes [Rudder Baker] |
| Full Idea: I argue that a lump of clay borrows the property of being a statue from the statue. The lump is a statue because, and only because, there is something that the lump constitutes that is a statue. | |
| From: Lynne Rudder Baker (Why Constitution is not Identity [1997], n9) | |
| A reaction: It is skating on very thin metaphysical ice to introduce the concept of 'borrowing' a property. I've spent the last ten minutes trying to 'borrow' some properties, but without luck. |
| 16080 | Is it possible for two things that are identical to become two separate things? [Rudder Baker] |
| Full Idea: A strong intuition shared by many philosophers is that some things that are in fact identical might not have been identical. | |
| From: Lynne Rudder Baker (Why Constitution is not Identity [1997], IV) | |
| A reaction: This flies in the face of the Kripkean view that if Hesperus=Phosphorus then the identity is necessary. I don't think I have an intuition that some given thing might have been two things - indeed the thought seems totally weird. Amoeba? Statue/clay? |
| 16076 | Constitution is not identity, as consideration of essential predicates shows [Rudder Baker] |
| Full Idea: I want to resuscitate an essentialist argument against the view that constitution is identity, of the form 'x is essentially F, y is not essentially F, so x is not y'. | |
| From: Lynne Rudder Baker (Why Constitution is not Identity [1997], Intro) | |
| A reaction: The point is that x might be essentially F and y only accidentally F. Thus a statue is essentially so, but a lump if clay is not essentially a statue. Another case where 'necessary' would do instead of 'essentially'. |
| 16081 | The constitution view gives a unified account of the relation of persons/bodies, statues/bronze etc [Rudder Baker] |
| Full Idea: Constitution-without-identity is superior to constitution-as-identity in that it provides a unified view of the relation between persons and bodies, statues and pieces of bronze, and so on. | |
| From: Lynne Rudder Baker (Why Constitution is not Identity [1997], IV) | |
| A reaction: I have a problem with the intrinsic dualism of this whole picture. Clay needs shape, statues need matter - there aren't two 'things' here which have a 'relation'. |
| 16082 | Statues essentially have relational properties lacked by lumps [Rudder Baker] |
| Full Idea: The statue has relational properties which the lump of clay does not have essentially. | |
| From: Lynne Rudder Baker (Why Constitution is not Identity [1997], V) | |
| A reaction: She has in mind relations to the community of artistic life. I don't think this is convincing. Is something only a statue if it is validated by an artistic community? That sounds like relative identity, which she doesn't like. |
| 3756 | Perception, introspection, testimony, memory, reason, and inference can give us knowledge [Bernecker/Dretske] |
| Full Idea: The basic sources of knowledge and justification are perception, introspection, testimony, memory, reason, and inference. | |
| From: Bernecker / Dretske (Knowledge:Readings in Cont.Epist [2000], Pt.V Int) |
| 3757 | Causal theory says true perceptions must be caused by the object perceived [Bernecker/Dretske] |
| Full Idea: The causal theory of perceptions says that to perceive an object is to have a sense-datum caused by that object; it is not enough for the world to be the way we perceive it; the world must cause the perception. | |
| From: Bernecker / Dretske (Knowledge:Readings in Cont.Epist [2000], Pt.V Int) | |
| A reaction: All causal theories seem dubious to me; what causes something is not the same was what it means, or refers to, or what justifies it. The hallmark of successful perception is truth. I would perceive a tree if God planted the perception in me. |
| 3759 | You can acquire new knowledge by exploring memories [Bernecker/Dretske] |
| Full Idea: You can first come to know by remembering, as in learning how many windows there were in your childhood home by imagining a tour. | |
| From: Bernecker / Dretske (Knowledge:Readings in Cont.Epist [2000], Pt.V Int) |
| 3752 | Justification can be of the belief, or of the person holding the belief [Bernecker/Dretske] |
| Full Idea: There is a distinction between a person being justified in holding a belief, and the belief itself being justified. | |
| From: Bernecker / Dretske (Knowledge:Readings in Cont.Epist [2000], Pt.II Int) | |
| A reaction: This is the crucial and elementary distinction which even the most sophisticated of epistemologists keep losing sight of. Epistemology is about persons. All true beliefs are justified - by the facts! |
| 3753 | Foundationalism aims to avoid an infinite regress [Bernecker/Dretske] |
| Full Idea: The driving force behind foundationalism has always been the threat of an infinite regress. | |
| From: Bernecker / Dretske (Knowledge:Readings in Cont.Epist [2000], Pt.III Int) | |
| A reaction: You could just live with the regress (Peter Klein), or say that the regress fades away, or that it is cut off by social epistemological convention, or the regress circles round and rejoins. |
| 3754 | Infallible sensations can't be foundations if they are non-epistemic [Bernecker/Dretske] |
| Full Idea: If sense experiences are non-epistemic they may be infallible, but they are unsuitable for providing the foundations for other beliefs. | |
| From: Bernecker / Dretske (Knowledge:Readings in Cont.Epist [2000], Pt.III Int) | |
| A reaction: If we experience flashing lights in the retina, or an afterimage, we don't think we are seeing objects, so why is normal perception different? Ans: because it is supported by judgement. |
| 3755 | Justification is normative, so it can't be reduced to cognitive psychology [Bernecker/Dretske] |
| Full Idea: The concept of justification is absolutely central to epistemology; but this concept is normative (i.e. it lays down norms), so epistemology can't be reduced to factual cognitive psychology. | |
| From: Bernecker / Dretske (Knowledge:Readings in Cont.Epist [2000], Pt.III Int) | |
| A reaction: A simple rejection of the 'epistemology naturalised' idea. Best to start with slugs rather than people. You can confuse a slug, so it has truth or falsehood, but what is slug normativity? This is an interesting discussion point, not an argument. |
| 3761 | Modern arguments against the sceptic are epistemological and semantic externalism, and the focus on relevance [Bernecker/Dretske] |
| Full Idea: In modern epistemology the three strategies to rebut the sceptic are 1) epistemological externalism, 2) the 'relevant alternative account of knowledge' (that scepticism is too extreme to be relevant), and 3) semantic externalism. | |
| From: Bernecker / Dretske (Knowledge:Readings in Cont.Epist [2000], Pt.IV Int) |
| 3760 | Predictions are bound to be arbitrary if they depend on the language used [Bernecker/Dretske] |
| Full Idea: The new riddle of induction ('grue') seems to demonstrate that sound inductive inferences are arbitrary because they depend on the actual language people use to formulate predictions. | |
| From: Bernecker / Dretske (Knowledge:Readings in Cont.Epist [2000], Pt.V Int) |
| 3758 | Semantic externalism ties content to the world, reducing error [Bernecker/Dretske] |
| Full Idea: Semantic externalism ties our mental content down to our actual environment so there is no possibility of massive error. | |
| From: Bernecker / Dretske (Knowledge:Readings in Cont.Epist [2000], Pt.V Int) | |
| A reaction: This sounds more prescriptive than descriptive. People do make massive errors in their concepts. Maybe educated people are more externalist (respectful of experts) than uneducated people? |
| 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. |