8 ideas
| 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. |
| 14286 | In nearby worlds where A is true, 'if A,B' is true or false if B is true or false [Stalnaker] |
| Full Idea: Consider a possible world in which A is true and otherwise differs minimally from the actual world. 'If A, then B' is true (false) just in case B is true (false) in that possible world. | |
| From: Robert C. Stalnaker (A Theory of Conditionals [1968], p.34), quoted by Dorothy Edgington - Conditionals (Stanf) 4.1 | |
| A reaction: This is the first proposal to give a possible worlds semantics for conditional statements. Edgington observes that worlds which are nearby for me may not be nearby for you. |
| 14285 | A possible world is the ontological analogue of hypothetical beliefs [Stalnaker] |
| Full Idea: A possible world is the ontological analogue of a stock of hypothetical beliefs. | |
| From: Robert C. Stalnaker (A Theory of Conditionals [1968], p.34), quoted by Dorothy Edgington - Conditionals (Stanf) 4.1 | |
| A reaction: Sounds neat and persuasive. What is the ontological analogue of a stock of hopes? Heaven! |
| 20696 | We can approach knowledge of God by negative attributes [Maimonides] |
| Full Idea: You will come nearer to the knowledge and comprehension of God by the negative attributes. | |
| From: Moses Maimonides (The Guide of the Perplexed [1190], p.86), quoted by Brian Davies - Introduction to the Philosophy of Religion 2 'Negation' | |
| A reaction: Illustrated by grasping what a ship is by eliminating other categories it might belong to. The assumption is that you have a known and finite list - something like Aristotle's categories. Maimonides fears we know too little for positive attributes. |
| 19085 | Thinking of God as resembling humans results from a bad translation of Genesis 1:26 [Maimonides] |
| Full Idea: Mistranslation of 'image' has been the cause of a crass anthropomorphism because of the verse 'Let us make man in Our image after Our likeness' (Gen.1:26). They think God has the shape and outline of man, ..with face and hands like themselves. | |
| From: Moses Maimonides (The Guide of the Perplexed [1190], I.1) | |
| A reaction: It's interesting that Michelangelo still visualises God as an old man. The idea won't go away, presumably because God is understood as a 'person', in Locke's sense, though of a very special kind. |