12 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). |
| 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'. |
| 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. |
| 12432 | Explanation of necessity must rest on something necessary or something contingent [Hale] |
| Full Idea: The dilemma is that to give the ultimate source of any necessity, we must either appeal to something which could not have been otherwise (i.e. is itself necessary), or advert to something which could have been otherwise (i.e. is itself merely contingent). | |
| From: Bob Hale (The Source of Necessity [2002], p.301) | |
| A reaction: [Hale is summarising Blackburn's view, and going on to disagree with it] Hale looks for a third way, but Blackburn seems to face us with quite a plausible dilemma. |
| 12434 | Why is this necessary, and what is necessity in general; why is this necessary truth true, and why necessary? [Hale] |
| Full Idea: We must distinguish between explaining particular necessities and explaining necessity in general; and we ought to distinguish between explaining, in regard to any necessary truth, why it is true, and explaining why it is necessary. | |
| From: Bob Hale (The Source of Necessity [2002], p.308) | |
| A reaction: Useful. The pluralist view I associate with Fine says we can explain types of necessity, but not necessity in general. If we seek truthmakers, there is a special case of what adds the necessity to the truth. |
| 12435 | The explanation of a necessity can be by a truth (which may only happen to be a necessary truth) [Hale] |
| Full Idea: My claim is that there are non-transitive explanations of necessities, where what explains is indeed necessary, but what explains the necessity of the explanandum is not the explanation's necessity, but its truth simpliciter. | |
| From: Bob Hale (The Source of Necessity [2002], p.311) | |
| A reaction: The big idea is to avoid a regress of necessities. The actual truths he proposes are essentialist. An interesting proposal. It might depend on how one views essences (as giving identity, or causal power) |
| 12433 | If necessity rests on linguistic conventions, those are contingent, so there is no necessity [Hale] |
| Full Idea: If the alleged necessity, e,g, 2+2=4, really does depend upon a convention governing the use of the words in which we state it, and the existence of that convention is merely a contingent matter, then it can't after all be necessary. | |
| From: Bob Hale (The Source of Necessity [2002], p.302) | |
| A reaction: [Hale is citing Blackburn for this claim] Hale suggests replies, by keeping truth and meaning separate, and involving laws of logic. Blackburn clearly has a good point. |
| 12436 | Concept-identities explain how we know necessities, not why they are necessary [Hale] |
| Full Idea: It seems to me that identity-relations among concepts have more to do with explaining how we know that vixens are female foxes etc., than with explaining why it is necessary, and, more generally, with explaining why some necessities are knowable a priori. | |
| From: Bob Hale (The Source of Necessity [2002], P.313) | |
| A reaction: Hale rejects the conceptual and conventional accounts of necessity, in favour of the essentialist view. This strikes me as a good suggestion of Hale's, since I agree with him about the essentialism. |
| 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. |
| 13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
| Full Idea: In a single day there lies open to men of learning more than there ever does to the unenlightened in the longest of lifetimes. | |
| From: Posidonius (fragments/reports [c.95 BCE]), quoted by Seneca the Younger - Letters from a Stoic 078 | |
| A reaction: These remarks endorsing the infinite superiority of the educated to the uneducated seem to have been popular in late antiquity. It tends to be the religions which discourage great learning, especially in their emphasis on a single book. |
| 20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
| Full Idea: Posidonius defined time thus: it is an interval of motion, or the measure of speed and slowness. | |
| From: report of Posidonius (fragments/reports [c.95 BCE]) by John Stobaeus - Anthology 1.08.42 | |
| A reaction: Hm. Can we define motion or speed without alluding to time? Looks like we have to define them as a conjoined pair, which means we cannot fully understand either of them. |