18 ideas
| 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. |
| 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. |
| 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. |
| 6417 | In 1921 Russell abandoned sense-data, and the gap between sensation and object [Russell, by Grayling] |
| Full Idea: In 'The Analysis of Mind' Russell gave up talk of 'sense-data', and ceased to distinguish between the act of sensing and what is sensed. | |
| From: report of Bertrand Russell (The Analysis of Mind [1921]) by A.C. Grayling - Russell Ch.2 | |
| A reaction: This seems to lead towards the modern 'adverbial' account of sensing, where I don't sense 'data', but where qualia (such as redness) are our particular mode of directly perceiving objects, where insects might directly perceive them in a different mode. |
| 6474 | Seeing is not in itself knowledge, but is separate from what is seen, such as a patch of colour [Russell] |
| Full Idea: Undeniably, knowledge comes through seeing, but it is a mistake to regard the mere seeing itself as knowledge; if we are so to regard it, we must distinguish the seeing from what is seen; a patch of colour is one thing, and our seeing it is another. | |
| From: Bertrand Russell (The Analysis of Mind [1921], Lec. VIII) | |
| A reaction: This is Russell's 1921 explanation of why he adopted sense-data (but he rejects them later in this paragraph). This gives a simplistic impression of what he intended, which has three components: the object, the 'sensibile', and the sense-datum. |
| 6476 | We cannot assume that the subject actually exists, so we cannot distinguish sensations from sense-data [Russell] |
| Full Idea: If we are to avoid a perfectly gratuitous assumption, we must dispense with the subject as one of the actual ingredients of the world; but when we do this, the possibility of distinguishing the sensation from the sense-datum vanishes. | |
| From: Bertrand Russell (The Analysis of Mind [1921], Lec. VIII) | |
| A reaction: This is the reason why Russell himself rejected sense-data. It is more normal, I think, to reject them simply as being superfluous. If the subject can simply perceive the sense-data, why can't they just perceive the object more directly? |
| 2792 | It is possible the world came into existence five minutes ago, complete with false memories [Russell] |
| Full Idea: There is no logical impossibility in the hypothesis that the world sprang into being five minutes ago, exactly as it then was, with a population that "remembered" a wholly unreal past. | |
| From: Bertrand Russell (The Analysis of Mind [1921], p.159) | |
| A reaction: One of the great sceptical arguments! At a stroke it undermines forever any dreams that memories are totally certain. This is an extra scepticism, which arises if you decide that current experience IS totally certain. |
| 22326 | Knowledge needs more than a sensitive response; the response must also be appropriate [Russell] |
| Full Idea: Accuracy of response to stimulus does not alone show knowledge, but must be reinforced by appropriateness, i.e. suitability of realising one's purpose. | |
| From: Bertrand Russell (The Analysis of Mind [1921], p.261), quoted by Michael Potter - The Rise of Analytic Philosophy 1879-1930 66 'Rel' | |
| A reaction: The aim of 'realising one's purpose' puts a very pragmatist spin on this. The point is a good one, and seems to apply particularly to Nozick's accurate 'tracking' account of knowledge. |
| 6475 | In perception, the self is just a logical fiction demanded by grammar [Russell] |
| Full Idea: In perception, the idea of the subject appears to be a logical fiction, like mathematical points and instants; it is introduced, not because observation reveals it, but because it is linguistically convenient and apparently demanded by grammar. | |
| From: Bertrand Russell (The Analysis of Mind [1921], Lec. VIII) | |
| A reaction: In 1912, Russell had felt that both the Cogito, and the experience of meta-thought, had confirmed the existence of a non-permanent ego, but here he offers a Humean rejection. His notion of a 'logical fiction' is behaviouristic. I believe in the Self. |
| 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. |