27 ideas
| 12461 | We believe all mathematical problems are solvable [Hilbert] |
| Full Idea: The thesis that every mathematical problem is solvable - we are all convinced that it really is so. | |
| From: David Hilbert (On the Infinite [1925], p.200) | |
| A reaction: This will include, for example, Goldbach's Conjecture (every even is the sum of two primes), which is utterly simple but with no proof anywhere in sight. |
| 12456 | I aim to establish certainty for mathematical methods [Hilbert] |
| Full Idea: The goal of my theory is to establish once and for all the certitude of mathematical methods. | |
| From: David Hilbert (On the Infinite [1925], p.184) | |
| A reaction: This is the clearest statement of the famous Hilbert Programme, which is said to have been brought to an abrupt end by Gödel's Incompleteness Theorems. |
| 9633 | No one shall drive us out of the paradise the Cantor has created for us [Hilbert] |
| Full Idea: No one shall drive us out of the paradise the Cantor has created for us. | |
| From: David Hilbert (On the Infinite [1925], p.191), quoted by James Robert Brown - Philosophy of Mathematics | |
| A reaction: This is Hilbert's famous refusal to accept any account of mathematics, such as Kant's, which excludes actual infinities. Cantor had laid out a whole glorious hierarchy of different infinities. |
| 12460 | We extend finite statements with ideal ones, in order to preserve our logic [Hilbert] |
| Full Idea: To preserve the simple formal rules of ordinary Aristotelian logic, we must supplement the finitary statements with ideal statements. | |
| From: David Hilbert (On the Infinite [1925], p.195) | |
| A reaction: I find very appealing the picture of mathematics as rooted in the physical world, and then gradually extended by a series of 'idealisations', which should perhaps be thought of as fictions. |
| 12462 | Only the finite can bring certainty to the infinite [Hilbert] |
| Full Idea: Operating with the infinite can be made certain only by the finitary. | |
| From: David Hilbert (On the Infinite [1925], p.201) | |
| A reaction: See 'Compactness' for one aspect of this claim. I think Hilbert was fighting a rearguard action, and his idea now has few followers. |
| 12455 | The idea of an infinite totality is an illusion [Hilbert] |
| Full Idea: Just as in the limit processes of the infinitesimal calculus, the infinitely large and small proved to be a mere figure of speech, so too we must realise that the infinite in the sense of an infinite totality, used in deductive methods, is an illusion. | |
| From: David Hilbert (On the Infinite [1925], p.184) | |
| A reaction: This is a very authoritative rearguard action. I no longer think the dispute matters much, it being just a dispute over a proposed new meaning for the word 'number'. |
| 12457 | There is no continuum in reality to realise the infinitely small [Hilbert] |
| Full Idea: A homogeneous continuum which admits of the sort of divisibility needed to realise the infinitely small is nowhere to be found in reality. | |
| From: David Hilbert (On the Infinite [1925], p.186) | |
| A reaction: He makes this remark as a response to Planck's new quantum theory (the year before the big works of Heisenberg and Schrödinger). Personally I don't see why infinities should depend on the physical world, since they are imaginary. |
| 12459 | The subject matter of mathematics is immediate and clear concrete symbols [Hilbert] |
| Full Idea: The subject matter of mathematics is the concrete symbols themselves whose structure is immediately clear and recognisable. | |
| From: David Hilbert (On the Infinite [1925], p.192) | |
| A reaction: I don't think many people will agree with Hilbert here. Does he mean token-symbols or type-symbols? You can do maths in your head, or with different symbols. If type-symbols, you have to explain what a type is. |
| 18112 | Mathematics divides in two: meaningful finitary statements, and empty idealised statements [Hilbert] |
| Full Idea: We can conceive mathematics to be a stock of two kinds of formulas: first, those to which the meaningful communications of finitary statements correspond; and secondly, other formulas which signify nothing and which are ideal structures of our theory. | |
| From: David Hilbert (On the Infinite [1925], p.196), quoted by David Bostock - Philosophy of Mathematics 6.1 |
| 9636 | My theory aims at the certitude of mathematical methods [Hilbert] |
| Full Idea: The goal of my theory is to establish once and for all the certitude of mathematical methods. | |
| From: David Hilbert (On the Infinite [1925], p.184), quoted by James Robert Brown - Philosophy of Mathematics Ch.5 | |
| A reaction: This dream is famous for being shattered by Gödel's Incompleteness Theorem a mere six years later. Neverless there seem to be more limited certainties which are accepted in mathematics. The certainty of the whole of arithmetic is beyond us. |
| 12157 | Kant gave form and status to aesthetics, and Hegel gave it content [Kant, by Scruton] |
| Full Idea: Kant gave form and status to aesthetics, and Hegel endowed it with content. | |
| From: report of Immanuel Kant (Critique of Judgement I: Aesthetic [1790]) by Roger Scruton - Recent Aesthetics in England and America p.3 |
| 20346 | The aesthetic attitude is a matter of disinterestedness [Kant, by Wollheim] |
| Full Idea: The aesthetic attitude is defined by Kant in terms of disinterestedness. | |
| From: report of Immanuel Kant (Critique of Judgement I: Aesthetic [1790]) by Richard Wollheim - Art and Its Objects 54 | |
| A reaction: This is presumably, mainly, to explain our enjoyment of the miseries of tragedy. We just give ourselves up to a merry jig by Haydn. |
| 18547 | Only rational beings can experience beauty [Kant, by Scruton] |
| Full Idea: Kant is surely right that the experience of beauty, like the judgements in which it issues, is the prerogative of rational beings. | |
| From: report of Immanuel Kant (Critique of Judgement I: Aesthetic [1790]) by Roger Scruton - Beauty: a very short introduction 1 | |
| A reaction: I'm not sure how Scruton can say that Kant is 'surely right'. It is an interesting speculation. Are we to dogmatically affirm that bees get no aesthetic thrill when they spot a promising flower? Something in their little brains attracts them. |
| 24172 | It is hard to see why we would have developed Kant's 'disinterested' aesthetic attitude [Cochrane on Kant] |
| Full Idea: The Kantian notion of disinterest isolated aesthetic value from the rest of our lives. It is hard to understand why we should have developed a tendency that is detached from our everyday practical purposes. | |
| From: comment on Immanuel Kant (Critique of Judgement I: Aesthetic [1790], §2) by Tom Cochrane - The Aesthetic Value of the World 1.4 | |
| A reaction: Cochrane always seeks an evolutionary framework for accounts of aesthetics, and I agree with him. That doesn't devalue them. The best things in life, like piano music, are obviously offshoots of things which evolved for other reasons. |
| 20408 | With respect to the senses, taste is an entirely personal matter [Kant] |
| Full Idea: With regard to the agreeable, the principle Everyone has his own taste (of the senses) is valid. | |
| From: Immanuel Kant (Critique of Judgement I: Aesthetic [1790], CUP 7 5:212), quoted by Elizabeth Schellekens - Immanuel Kant (aesthetics) 1 | |
| A reaction: This is a preliminary concession, and he goes on to defend more objective views of taste. |
| 20409 | When we judge beauty, it isn't just personal; we judge on behalf of everybody [Kant] |
| Full Idea: It is ridiculous if someone justifies his tast by saying 'this object is beautiful for me'. . .If he pronounces that something is beautiful, then he expects the very same satisfaction of others: he judges not merely for himself, but for everyone. | |
| From: Immanuel Kant (Critique of Judgement I: Aesthetic [1790], CUP 7 5:213), quoted by Elizabeth Schellekens - Immanuel Kant (aesthetics) 1 | |
| A reaction: For Kant this would also be the hallmark of rationality - that we expect, or hope for, a consensus when we express a rational judgement. But this expectation is far less in cases of beauty. We do not expect total agreement from very tasteful people. |
| 20411 | Saying everyone has their own taste destroys the very idea of taste [Kant] |
| Full Idea: To say thast 'Everyone has his special taste' would be to dismiss the very possibility of aesthetic taste, and to deny that there could be aesthetic judgement 'that could make a rightful claim to the assent of everyone'. | |
| From: Immanuel Kant (Critique of Judgement I: Aesthetic [1790], CUP 7 5:213), quoted by Elizabeth Schellekens - Immanuel Kant (aesthetics) 2.2 | |
| A reaction: I am a great believer in the objectivity of taste (within sensible reason). But the great evidence against it is the shifting standards of taste over the centuries. Nineteenth century collectors wasted fortunes on inferior works, it seems to us. |
| 24170 | Kant thinks beauty ignores its objects, because it is only 'form' engaging with mind [Cochrane on Kant] |
| Full Idea: Kant thinks that the ideal of beauty requires no concept of what the object is. Universality demands that appreciation be purely a matter of the way the form of the object fits one's cognitive machinery. | |
| From: comment on Immanuel Kant (Critique of Judgement I: Aesthetic [1790]) by Tom Cochrane - The Aesthetic Value of the World 1.3 | |
| A reaction: This confirms further my increasingly negative view of Kant. Everything in him points to idealism (despite denials by his fans), and via Hegel we arrive at the idea that our values are all 'cultural constructs', rather than responses to reality. |
| 22711 | The beautiful is not conceptualised as moral, but it symbolises or resembles goodness [Kant, by Murdoch] |
| Full Idea: Kant insists that the beautiful must not be tainted with the good (that is, not conceptualised in any way which would bring it into the sphere of moral judgement) yet he says that the beautiful symbolises the good, it is an analogy of the good. | |
| From: report of Immanuel Kant (Critique of Judgement I: Aesthetic [1790]) by Iris Murdoch - The Sublime and the Good p.209 | |
| A reaction: Kant evidently wanted a very pure view of the aesthetic experience, drained of any overlapping feelings or beliefs. I'm not sure I understand how the beautiful can symbolise or be analogous to the good, while being devoid of it. |
| 4025 | Kant saw beauty as a sort of disinterested pleasure, which has become separate from the good [Kant, by Taylor,C] |
| Full Idea: Kant, in his third critique, defined beauty in terms of a certain kind of disinterested pleasure;….this is the basis for a declaration of independence of the beautiful relative to the good. | |
| From: report of Immanuel Kant (Critique of Judgement I: Aesthetic [1790]) by Charles Taylor - Sources of the Self §23.1 | |
| A reaction: This is a rebellion against the Greeks, especially Plato, and prepares the ground for the idea of 'art for art's sake'. Personally, I'm with Plato. |
| 20412 | Beauty is only judged in pure contemplation, and not with something else at stake [Kant] |
| Full Idea: If the question is whether something is beautiful, one does not want to know whether there is something that is or that could be at stake, for us or for someone else, in the existence of the thing, but rather how we judge it in mere contemplation. | |
| From: Immanuel Kant (Critique of Judgement I: Aesthetic [1790], CUP 2 5:204), quoted by Elizabeth Schellekens - Immanuel Kant (aesthetics) 2.3 | |
| A reaction: This evidently denies that function has anything to do with beauty, and seems to be a prelude to 'art for art's sake'. But a running cheetah cannot be separated from the sheer efficiencey and focus of the performance. |
| 22046 | The mathematical sublime is immeasurable greatness; the dynamical sublime is overpowering [Kant, by Pinkard] |
| Full Idea: Kant distinguished the 'mathematical' and 'dynamical' sublime. The former involves immeasurable greatness (or smallness) such that we cannot even present them to ourselves. The latter is of something large and overpowering, which we can morally resist. | |
| From: report of Immanuel Kant (Critique of Judgement I: Aesthetic [1790]) by Terry Pinkard - German Philosophy 1760-1860 13 | |
| A reaction: Presumably Cantor revealed the full extent of the mathematical sublime ('heaven', according to Hilbert). We await the comet that destroys the Earth to fully experience the other one. |
| 21458 | The sublime is a moral experience [Kant, by Gardner] |
| Full Idea: The sublime is understood by Kant as a moral experience. | |
| From: report of Immanuel Kant (Critique of Judgement I: Aesthetic [1790], 28-9) by Sebastian Gardner - Kant and the Critique of Pure Reason 09 'Judgment' | |
| A reaction: Gardner give the source in Kant. I can't accept that the initial experience of the sublime is moral in character. It could easily acquire a moral character after contemplation by someone who had such inclinations. |
| 5643 | Aesthetic values are not objectively valid, but we must treat them as if they are [Kant, by Scruton] |
| Full Idea: The 'Critique of Judgement' argues, then, not for the objective validity of aesthetic values, but for the fact that we must think of them as objectively valid. | |
| From: report of Immanuel Kant (Critique of Judgement I: Aesthetic [1790]) by Roger Scruton - Short History of Modern Philosophy §11.7 | |
| A reaction: The trouble with these transcendental arguments of Kant is that they render you powerless to discuss the question of whether values are actually objective. We are all trapped in presuppositions, instead of testing suppositions. |
| 20410 | The judgement of beauty is not cognitive, but relates, via imagination, to pleasurable feelings [Kant] |
| Full Idea: In order to understand whether or not something is beautiful, we do not relate the representation by means of understanding to the object for cognition, but relate it by means of the imagination ..to the subject and its feeling of pleasure or displeasure. | |
| From: Immanuel Kant (Critique of Judgement I: Aesthetic [1790], CUP 1 5:203), quoted by Elizabeth Schellekens - Immanuel Kant (aesthetics) 2.1 | |
| A reaction: This is to distinguish the particular type of judgement which counts as 'aesthetic' - the point being that it is not cognitive - it is not a matter of knowledge and facts, but a cool judgement made about a warm feeling of pleasure. I think. |
| 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. |