Ideas from 'On the Infinite' by David Hilbert [1925], by Theme Structure

[found in 'Philosophy of Mathematics: readings (2nd)' (ed/tr Benacerraf/Putnam) [CUP 1983,0-521-29648-x]].

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6. Mathematics / A. Nature of Mathematics / 1. Mathematics
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.
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.
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
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.
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.
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.
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / d. Actual infinite
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'.
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / j. Infinite divisibility
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.
6. Mathematics / C. Sources of Mathematics / 7. Formalism
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.
6. Mathematics / C. Sources of Mathematics / 8. Finitism
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
11. Knowledge Aims / B. Certain Knowledge / 1. Certainty
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.