6 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? |
| 19590 | Empiricists are passive thinkers, given their philosophy by the external world and fate [Novalis] |
| Full Idea: An empiricist is one whose way of thinking is an effect of the external world and of fate - the passive thinker - to whom his philosophy is given. | |
| From: Novalis (Teplitz Fragments [1798], 33) | |
| A reaction: Novalis goes on to enthuse about 'magical idealism', so he rejects empiricism. This is an early attack on the Myth of the Given, found in Sellars and McDowell. |
| 1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
| Full Idea: Archelaus was the first person to say that the universe is boundless. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.3 |
| 5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
| Full Idea: Archelaus wrote that life on Earth began in a primeval slime. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Malcolm Schofield - Archelaus | |
| A reaction: This sounds like a fairly clearcut assertion of the production of life by evolution. Darwin's contribution was to propose the mechanism for achieving it. We should honour the name of Archelaus for this idea. |