7 ideas
| 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? |
| 21386 | We should accept as explanations all the plausible ways in which something could come about [Epicurus] |
| Full Idea: The phases of the Moon could happen in all the ways [at least four] which the phenomena in our experience suggest for the explanation of this kind of thing - as long as one is not so enamoured of unique explanations as to groundlessly reject the others. | |
| From: Epicurus (Letter to Pythocles [c.292 BCE], 94) | |
| A reaction: Very interesting, for IBE. While you want to embrace the 'best', it is irrational to reject all of the other candidates, simply because you want a single explanation, if there are no good grounds for the rejection. |
| 3031 | The greatest good is not the achievement of desire, but to desire what is proper [Menedemus, by Diog. Laertius] |
| Full Idea: Hearing someone assert that the greatest good was to succeed in everything that one desires, he said,"It is a much greater good to desire what is proper". | |
| From: report of Menedemus (fragments/reports [c.310 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.16.12 |
| 14051 | A cosmos is a collection of stars and an earth, with some sort of boundary, movement and shape [Epicurus] |
| Full Idea: A cosmos is a circumscribed portion of the heavens containing stars and an earth; it is separated from the unlimited, with a boundary which is rare or dense; it is revolving or stationary; it is round or triangular, or some shape. All these are possible. | |
| From: Epicurus (Letter to Pythocles [c.292 BCE], 88) | |
| A reaction: Notice that there seem to exist the 'heavens' which extend beyond the cosmos. See Idea 14036, saying that there are many other cosmoi in the heavens. |
| 1828 | God does not intervene in heavenly movements, but is beyond all action and perfectly happy [Epicurus] |
| Full Idea: Let us beware of making the Deity interpose in heavenly movements, for that being we ought to suppose exempt from all occupation and perfectly happy. | |
| From: Epicurus (Letter to Pythocles [c.292 BCE]), quoted by Diogenes Laertius - Lives of Eminent Philosophers 10.25 |