6 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. |
| 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. |
| 10645 | We reach concepts by clarification, or by definition, or by habitual experience [Price,HH] |
| Full Idea: We have three different ways in which we arrive at concepts or universals: there is a clarification, where we have a ready-made concept and define it; we have a combination (where a definition creates a concept); and an experience can lead to a habit. | |
| From: H.H. Price (Review of Aron 'Our Knowledge of Universals' [1946], p.190) | |
| A reaction: [very compressed] He cites Russell as calling the third one a 'condensed induction'. There seems to an intellectualist and non-intellectualist strand in the abstractionist tradition. |
| 10644 | A 'felt familiarity' with universals is more primitive than abstraction [Price,HH] |
| Full Idea: A 'felt familiarity' with universals seems to be more primitive than explicit abstraction. | |
| From: H.H. Price (Review of Aron 'Our Knowledge of Universals' [1946], p.188) | |
| A reaction: This I take to be part of the 'given' of the abstractionist view, which is quite well described in the first instance by Aristotle. Price says that it is 'pre-verbal'. |
| 10646 | Our understanding of 'dog' or 'house' arises from a repeated experience of concomitances [Price,HH] |
| Full Idea: Whether you call it inductive or not, our understanding of such a word as 'dog' or 'house' does arise from a repeated experience of concomitances. | |
| From: H.H. Price (Review of Aron 'Our Knowledge of Universals' [1946], p.191) | |
| A reaction: Philosophers don't use phrases like that last one any more. How else could we form the concept of 'dog' - if we are actually allowed to discuss the question of concept-formation, instead of just the logic of concepts. |
| 468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
| Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice. | |
| From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where? |