6 ideas
3299 | In logic identity involves reflexivity (x=x), symmetry (if x=y, then y=x) and transitivity (if x=y and y=z, then x=z) [Baillie] |
Full Idea: In logic identity is an equivalence relation, which involves reflexivity (x=x), symmetry (if x=y, then y=x), and transitivity (if x=y and y=z, then x=z). | |
From: James Baillie (Problems in Personal Identity [1993], Intr p.4) |
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 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. |
10113 | The grounding of mathematics is 'in the beginning was the sign' [Hilbert] |
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. |
10115 | Hilbert substituted a syntactic for a semantic account of consistency [Hilbert, by George/Velleman] |
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. |
10116 | Hilbert aimed to prove the consistency of mathematics finitely, to show infinities won't produce contradictions [Hilbert, by George/Velleman] |
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. |
3447 | All theory is against free will, and all experience is in favour of it [Johnson,S] |
Full Idea: All theory is against free will, and all experience is in favour of it. | |
From: Samuel Johnson (works [1770]), quoted by PG - Db (ideas) |