12 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. |
7535 | If all beliefs are propositional, then belief and judgement are the same thing [Monk] |
Full Idea: Whether the words 'belief' and 'judgement' mean the same thing is a moot point. Traditionally, a judgement is the assent of mind to a proposition. If one thinks that all beliefs are propositional, then beliefs and judgements are the same thing. | |
From: Ray Monk (Bertrand Russell: Spirit of Solitude [1996], Ch.19 n6) | |
A reaction: If I think I have put a bit too much toothpaste on my brush, that strikes me as a non-propositional judgement, even though it could be spelled out as a proposition. But it also strikes me as a belief. |
16764 | The soul conserves the body, as we see by its dissolution when the soul leaves [Toletus] |
Full Idea: Every accident of a living thing, as well as all its organs and temperaments and its dispositions are conserved by the soul. We see this from experience, since when that soul recedes, all these dissolve and become corrupted. | |
From: Franciscus Toletus (Commentary on 'De Anima' [1572], II.1.1), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 24.5 | |
A reaction: A nice example of observing a phenemonon, but not being able to observe the dependence relation the right way round. Compare Descartes in Idea 16763. |