Ideas from 'works' by David Hilbert [1900], by Theme Structure

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6. Mathematics / A. Nature of Mathematics / 1. Mathematics
Hilbert wanted to prove the consistency of all of mathematics (which realists take for granted)
                        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 Michčle 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.
6. Mathematics / C. Sources of Mathematics / 7. Formalism
The grounding of mathematics is 'in the beginning was the sign'
                        Full Idea: The solid philosophical attitude that I think is required for the grounding of pure mathematics is this: In the beginning was the sign.
                        From: David Hilbert (works [1900]), quoted by A.George / D.J.Velleman - Philosophies of Mathematics Ch.6
                        A reaction: Why did people invent those particular signs? Presumably they were meant to designate something, in the world or in our experience.
Hilbert substituted a syntactic for a semantic account of consistency
                        Full Idea: Hilbert replaced a semantic construal of inconsistency (that the theory entails a statement that is necessarily false) by a syntactic one (that the theory formally derives the statement (0 =1 ∧ 0 not-= 1).
                        From: report of David Hilbert (works [1900]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.6
                        A reaction: Finding one particular clash will pinpoint the notion of inconsistency, but it doesn't seem to define what it means, since the concept has very wide application.
6. Mathematics / C. Sources of Mathematics / 8. Finitism
Hilbert aimed to prove the consistency of mathematics finitely, to show infinities won't produce contradictions
                        Full Idea: Hilbert's project was to establish the consistency of classical mathematics using just finitary means, to convince all parties that no contradictions will follow from employing the infinitary notions and reasoning.
                        From: report of David Hilbert (works [1900]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.6
                        A reaction: This is the project which was badly torpedoed by Gödel's Second Incompleteness Theorem.