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12461 | We believe all mathematical problems are solvable |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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. |