display all the ideas for this combination of philosophers
9 ideas
| 8717 | Hilbert wanted to prove the consistency of all of mathematics (which realists take for granted) [Hilbert, by Friend] |
| Full Idea: Hilbert wanted to derive ideal mathematics from the secure, paradox-free, finite mathematics (known as 'Hilbert's Programme'). ...Note that for the realist consistency is not something we need to prove; it is a precondition of thought. | |
| From: report of David Hilbert (works [1900], 6.7) by Michèle Friend - Introducing the Philosophy of Mathematics | |
| A reaction: I am an intuitive realist, though I am not so sure about that on cautious reflection. Compare the claims that there are reasons or causes for everything. Reality cannot contain contradicitions (can it?). Contradictions would be our fault. |
| 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. |
| 13472 | Hilbert aimed to eliminate number from geometry [Hilbert, by Hart,WD] |
| Full Idea: One of Hilbert's aims in 'The Foundations of Geometry' was to eliminate number [as measure of lengths and angles] from geometry. | |
| From: report of David Hilbert (Foundations of Geometry [1899]) by William D. Hart - The Evolution of Logic 2 | |
| A reaction: Presumably this would particularly have to include the elimination of ratios (rather than actual specific lengths). |
| 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. |