7 ideas
| 12221 | 'Corner quotes' (quasi-quotation) designate 'whatever these terms designate' [Quine] |
| Full Idea: A 'quasi-quotation' [corner quotes, Quine quotes] designates that (unspecified) expression which is obtained from the contents of the corners by replacing the Greek letters by the (unspecified) expressions which they designate. | |
| From: Willard Quine (Mathematical Logic (revised) [1940], 1.6) | |
| A reaction: Filed under 'variables', as they seem to be variables that can refer to actual expressions, like algebra. Quine was determined to distinguish clearly between 'mention' and 'use'. 'Half-hearted substitutional quantification', says Fine. |
| 19321 | We might do without names, by converting them into predicates [Quine, by Kirkham] |
| Full Idea: Quine suggests that we can have a language with just predicates and no names. Thus for 'Ralph is red' we say 'x Ralphises and x is red'. | |
| From: report of Willard Quine (Mathematical Logic (revised) [1940]) by Richard L. Kirkham - Theories of Truth: a Critical Introduction 5.6 | |
| A reaction: Kirkham discusses this as a way of getting round the lack of names in Tarski's theory of truth (which just uses objects, predicates and quantifiers). Otherwise you must supplement Tarski with an account of what the names refer to. |
| 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. |
| 468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
| Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice. | |
| From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where? |