12 ideas
| 8921 | Structuralism is now common, studying relations, with no regard for what the objects might be [Hellman] |
| Full Idea: With developments in modern mathematics, structuralist ideas have become commonplace. We study 'abstract structures', having relations without regard to the objects. As Hilbert famously said, items of furniture would do. | |
| From: Geoffrey Hellman (Structuralism [2007], §1) | |
| A reaction: Hilbert is known as a Formalist, which suggests that modern Structuralism is a refined and more naturalist version of the rather austere formalist view. Presumably the sofa can't stand for six, so a structural definition of numbers is needed. |
| 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? |
| 8922 | Maybe mathematical objects only have structural roles, and no intrinsic nature [Hellman] |
| Full Idea: There is the tantalizing possibility that perhaps mathematical objects 'have no nature' at all, beyond their 'structural role'. | |
| From: Geoffrey Hellman (Structuralism [2007], §1) | |
| A reaction: This would fit with a number being a function rather than an object. We are interested in what cars do, not the bolts that hold them together? But the ontology of mathematics is quite separate from how you do mathematics. |
| 3061 | Anaxarchus said that he was not even sure that he knew nothing [Anaxarchus, by Diog. Laertius] |
| Full Idea: Anaxarchus said that he was not even sure that he knew nothing. | |
| From: report of Anaxarchus (fragments/reports [c.340 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 09.10.1 |
| 23146 | Motives produce intentions, which lead to actions [Driver] |
| Full Idea: Motives will cause persons for form intentions; it is intentions which more directly guide actions. | |
| From: Julia Driver (The Virtues and Human Nature [1996], 3) | |
| A reaction: This is invites the question of whether there is a sharp distinction between the motive and the action. Detectives look for motives, but law courts look for intentions. |
| 23147 | Good intentions are not necessary for virtue [Driver] |
| Full Idea: I deny the claim that good intentions are necessary for virtue. | |
| From: Julia Driver (The Virtues and Human Nature [1996], 3) | |
| A reaction: Presumably one could continually do the right thing, because it was your duty or your job, without actually being well motivated for it. |
| 23144 | Virtue should be defined by consequences, not by states of mind [Driver] |
| Full Idea: The behavioural aspects of virtue are more important than its phenomenology, because virtue is best defined along consequentialist lines. | |
| From: Julia Driver (The Virtues and Human Nature [1996], Intro) | |
| A reaction: This is the thesis of her paper. Quite persuasive. Consequences are, of course, important in all moral theories (even Kant's). She doesn't rely on human nature. The social virtues vary according to the circumstances, such as gossiping in wartime. |
| 23148 | Virtues are character traits or dispositions which produce good consequences for others [Driver] |
| Full Idea: A moral virtue is a character trait (a disposition or cluster of dispositions) which, generally speaking, produces good consequences for others. | |
| From: Julia Driver (The Virtues and Human Nature [1996], 3) | |
| A reaction: There are self-directed virtues, such as keeping fit and healthy. There are virtues for ways to receive the kindness of others. That said, I like this idea. |
| 23150 | Control of pregnancy and knowledge of paternity have downgraded chastity [Driver] |
| Full Idea: Women now have more control over becoming pregnant. Men can now be more certain of paternity, without the constraint of female chastity. Hence chastity is no longer considered a moral virtue. | |
| From: Julia Driver (The Virtues and Human Nature [1996], 5) | |
| A reaction: A persuasive argument that virtues are defined by their consequences (to which I add my example of gossiping in wartime). Different social situations and crises promote or relegate the status of certain virtues (such as food hoarding). |
| 23149 | If generosity systematically turned recipients into parasites, it wouldn't be a virtue [Driver] |
| Full Idea: If generosity towards the needy in the long run produced [social] parasites, and if generosity did this systematically, then it would not be a moral virtue. | |
| From: Julia Driver (The Virtues and Human Nature [1996], 5) | |
| A reaction: A very persuasive example. Hume has similar views - that we encourage those emotions which have good social outcomes. |