6 ideas
| 23505 | Aristotelian logic cannot express 'Everyone loves someone' [White,RM] |
| Full Idea: There is no way within Aristotelian logic that you can give a proper expression for the logical form of such a proposition as 'Everyone loves someone'. | |
| From: Roger M. White (Wittgenstein's 'Tractatus Logico-Philosophicus' [2006], 1 'Frege') | |
| A reaction: This needs a combination of two different quantifiers. |
| 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. |
| 6011 | There is a remote first god (the Good), and a second god who organises the material world [Numenius, by O'Meara] |
| Full Idea: Numenius argues that material reality depends on intelligible being, which depends on a first god - the Good - which is difficult to grasp, but which inspires a second god to imitate it, turning to matter and organizing it as the world. | |
| From: report of Numenius (fragments/reports [c.160]) by Dominic J. O'Meara - Numenius | |
| A reaction: The interaction problem comes either between the two gods, or between the second god and the world. The argument may have failed to catch on for long when people scented an infinite regress lurking in the middle of it. |