5 ideas
| 19695 | The devil was wise as an angel, and lost no knowledge when he rebelled [Whitcomb] |
| Full Idea: The devil is evil but nonetheless wise; he was a wise angel, and through no loss of knowledge, but, rather, through some sort of affective restructuring tried and failed to take over the throne. | |
| From: Dennis Whitcomb (Wisdom [2011], 'Argument') | |
| A reaction: ['affective restructuring' indeed! philosophers- don't you love 'em?] To fail at something you try to do suggests a flaw in the wisdom. And the new regime the devil wished to introduce doesn't look like a wise regime. Not convinced. |
| 18758 | Validity is provable, but invalidity isn't, because the model is infinite [Church, by McGee] |
| Full Idea: Church showed that logic has a proof procedure, but no decision procedure. If an argument is invalid, there is a model with true premises and false conclusion, but the model will typically be infinite, so there is no way to display it concretely. | |
| From: report of Alonzo Church (A Note on the entscheidungsproblem [1936]) by Vann McGee - Logical Consequence 5 |
| 8698 | Modal structuralism says mathematics studies possible structures, which may or may not be actualised [Hellman, by Friend] |
| Full Idea: The modal structuralist thinks of mathematical structures as possibilities. The application of mathematics is just the realisation that a possible structure is actualised. As structures are possibilities, realist ontological problems are avoided. | |
| From: report of Geoffrey Hellman (Mathematics without Numbers [1989]) by Michèle Friend - Introducing the Philosophy of Mathematics 4.3 | |
| A reaction: Friend criticises this and rejects it, but it is appealing. Mathematics should aim to be applicable to any possible world, and not just the actual one. However, does the actual world 'actualise a mathematical structure'? |
| 9557 | Statements of pure mathematics are elliptical for a sort of modal conditional [Hellman, by Chihara] |
| Full Idea: Hellman represents statements of pure mathematics as elliptical for modal conditionals of a certain sort. | |
| From: report of Geoffrey Hellman (Mathematics without Numbers [1989]) by Charles Chihara - A Structural Account of Mathematics 5.3 | |
| A reaction: It's a pity there is such difficulty in understanding conditionals (see Graham Priest on the subject). I intuit a grain of truth in this, though I take maths to reflect the structure of the actual world (with possibilities being part of that world). |
| 10263 | Modal structuralism can only judge possibility by 'possible' models [Shapiro on Hellman] |
| Full Idea: The usual way to show that a sentence is possible is to show that it has a model, but for Hellman presumably a sentence is possible if it might have a model (or if, possibly, it has a model). It is not clear what this move brings us. | |
| From: comment on Geoffrey Hellman (Mathematics without Numbers [1989]) by Stewart Shapiro - Philosophy of Mathematics 7.3 | |
| A reaction: I can't assess this, but presumably the possibility of the model must be demonstrated in some way. Aren't all models merely possible, because they are based on axioms, which seem to be no more than possibilities? |