11 ideas
| 1597 | Thales was the first western thinker to believe the arché was intelligible [Roochnik on Thales] |
| Full Idea: Thales was the first thinker in the west to believe that the arché (the basis of things) was intelligible. | |
| From: comment on Thales (fragments/reports [c.585 BCE]) by David Roochnik - The Tragedy of Reason p.138 |
| 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. |
| 3013 | Nothing is stronger than necessity, which rules everything [Thales, by Diog. Laertius] |
| Full Idea: Necessity is the strongest of things, for it rules everything. | |
| From: report of Thales (fragments/reports [c.585 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 01.2.9 |
| 7810 | The 'Eumenides' of Aeschylus shows blood feuds replaced by law [Aeschylus, by Grayling] |
| Full Idea: The 'Eumenides' of Aeschylus tells how the old rule of revenge and blood feud was replaced by a due process of law before a civil jury. | |
| From: report of Aeschylus (The Eumenides [c.458 BCE]) by A.C. Grayling - What is Good? Ch.2 | |
| A reaction: Compare Idea 1659, where this revolution is attributed to Protagoras (a little later than Aeschylus). I take the rule of law and of society to be above all the rule of reason, because the aim is calm objectivity instead of emotion. |
| 1494 | Thales said water is the first principle, perhaps from observing that food is moist [Thales, by Aristotle] |
| Full Idea: Thales says water is the first principle (which is why he declared the earth is on water); perhaps he concluded this from seeing that all food is moist. | |
| From: report of Thales (fragments/reports [c.585 BCE], A12) by Aristotle - Metaphysics 983b12 |
| 1713 | Thales must have thought soul causes movement, since he thought magnets have soul [Thales, by Aristotle] |
| Full Idea: Thales seems, from what is recorded of him, to have supposed that the soul is something productive of movement, if he really said that the magnet has soul because it produces movement in iron. | |
| From: report of Thales (fragments/reports [c.585 BCE]) by Aristotle - De Anima 405a20 |
| 1742 | Thales said the gods know our wrong thoughts as well as our evil actions [Thales, by Diog. Laertius] |
| Full Idea: When asked whether a man who did wrong could escape the notice of the gods, Thales is said to have replied: 'No, not even if he thinks wrong.' | |
| From: report of Thales (fragments/reports [c.585 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 01.Th.9 |