16 ideas
| 17610 | The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres [Maddy] |
| Full Idea: One feature of the Axiom of Choice that troubled many mathematicians was the so-called Banach-Tarski paradox: using the Axiom, a sphere can be decomposed into finitely many parts and those parts reassembled into two spheres the same size as the original. | |
| From: Penelope Maddy (Defending the Axioms [2011], 1.3) | |
| A reaction: (The key is that the parts are non-measurable). To an outsider it is puzzling that the Axiom has been universally accepted, even though it produces such a result. Someone can explain that, I'm sure. |
| 17620 | Critics of if-thenism say that not all starting points, even consistent ones, are worth studying [Maddy] |
| Full Idea: If-thenism denies that mathematics is in the business of discovering truths about abstracta. ...[their opponents] obviously don't regard any starting point, even a consistent one, as equally worthy of investigation. | |
| From: Penelope Maddy (Defending the Axioms [2011], 3.3) | |
| A reaction: I have some sympathy with if-thenism, in that you can obviously study the implications of any 'if' you like, but deep down I agree with the critics. |
| 17605 | Hilbert's geometry and Dedekind's real numbers were role models for axiomatization [Maddy] |
| Full Idea: At the end of the nineteenth century there was a renewed emphasis on rigor, the central tool of which was axiomatization, along the lines of Hilbert's axioms for geometry and Dedekind's axioms for real numbers. | |
| From: Penelope Maddy (Defending the Axioms [2011], 1.3) |
| 17625 | If two mathematical themes coincide, that suggest a single deep truth [Maddy] |
| Full Idea: The fact that two apparently fruitful mathematical themes turn out to coincide makes it all the more likely that they're tracking a genuine strain of mathematical depth. | |
| From: Penelope Maddy (Defending the Axioms [2011], 5.3ii) |
| 17615 | Every infinite set of reals is either countable or of the same size as the full set of reals [Maddy] |
| Full Idea: One form of the Continuum Hypothesis is the claim that every infinite set of reals is either countable or of the same size as the full set of reals. | |
| From: Penelope Maddy (Defending the Axioms [2011], 2.4 n40) |
| 17618 | Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy] |
| Full Idea: Our set-theoretic methods track the underlying contours of mathematical depth. ...What sets are, most fundamentally, is markers for these contours ...they are maximally effective trackers of certain trains of mathematical fruitfulness. | |
| From: Penelope Maddy (Defending the Axioms [2011], 3.4) | |
| A reaction: This seems to make it more like a map of mathematics than the actual essence of mathematics. |
| 17614 | The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy] |
| Full Idea: Ordinary perceptual cognition is most likely involved in our grasp of elementary arithmetic, but ...this connection to the physical world has long since been idealized away in the infinitary structures of contemporary pure mathematics. | |
| From: Penelope Maddy (Defending the Axioms [2011], 2.3) | |
| A reaction: Despite this, Maddy's quest is for a 'naturalistic' account of mathematics. She ends up defending 'objectivity' (and invoking Tyler Burge), rather than even modest realism. You can't 'idealise away' the counting of objects. I blame Cantor. |
| 6402 | In 1927, Russell analysed force and matter in terms of events [Russell, by Grayling] |
| Full Idea: In his 'Analysis of Matter' (1927), Russell sought to analyse the chief concepts of physics, such as force and matter, in terms of events. | |
| From: report of Bertrand Russell (The Analysis of Matter [1927]) by A.C. Grayling - Russell Ch.2 | |
| A reaction: My immediate reaction is that this is not very promising, simply because we can always ask why a particular event occurred, and this seems to point to a deeper level in the analysis. See Idea 4779, for example. |
| 14732 | A perceived physical object is events grouped around a centre [Russell] |
| Full Idea: The physical object, as inferred from perception, is a group of events arranged about a centre. | |
| From: Bertrand Russell (The Analysis of Matter [1927], 23) | |
| A reaction: At least I like the active aspect of this definition. You then have to explain what an event is, without mentioning objects. You'd better no mention properties either, since they will probably depend on the dreaded objects. |
| 14733 | An object produces the same percepts with or without a substance, so that is irrelevant to science [Russell] |
| Full Idea: There may be a substance at the centre of an object, but is no reason to think so, since the group of events making up the object will produce exactly the same percepts; so the substance, if there is one, is an abstract possibility irrelevant to science. | |
| From: Bertrand Russell (The Analysis of Matter [1927], 23) | |
| A reaction: All empiricists (as Russell is in this passage) seem to neglect inference to the best explanation. Things can be indirectly testable, and I would say that there are genuine general entities which are too close to abstraction to ever be testable. |
| 6418 | Russell rejected phenomenalism because it couldn't account for causal relations [Russell, by Grayling] |
| Full Idea: Russell reverted to realism when he recognised that the notion of causality is problematic for phenomenalism; things in the world seem to affect one another causally in ways that are difficult to account for properly by mere reports of sense-experiences. | |
| From: report of Bertrand Russell (The Analysis of Matter [1927]) by A.C. Grayling - Russell Ch.3 | |
| A reaction: This is very interesting, and doesn't seem to have been enough to make A.J. Ayer eschew phenomenalism (Idea 5170). Once your metaphysics becomes realist (like Russell), your account of perception and objects has to change too. |
| 23896 | We see our character as a restricting limit, but also as an unshakable support [Weil] |
| Full Idea: Our character appears to us as a limit by which we do not want to be imprisoned, …but also as a support that we want to believe is unshakable. | |
| From: Simone Weil (On the Concept of Character [1941], p.100) | |
| A reaction: A nice perception. It is fairly easy to criticise, or even laugh at, one's own actions, but extremely hard to criticise our own character. Maybe we all wish we were more determined in our projects, but not much else. |
| 23893 | We don't see character in a single moment, but only over a period of time [Weil] |
| Full Idea: Character is constant over a period of time; the way a person is at a single moment does not at all reflect the character of this person. We do, however, concede that character changes. | |
| From: Simone Weil (On the Concept of Character [1941], p.98) | |
| A reaction: I do think, though, that there are moments in behaviour which are hugely revealing of character, even in a single remark. But I agree that most single moments do not show much. |
| 23894 | The concept of character is at the centre of morality [Weil] |
| Full Idea: We cannot pose a moral problem without putting the concept of character at its centre. | |
| From: Simone Weil (On the Concept of Character [1941], p.98) | |
| A reaction: The question for Aristotle (which I derive from Philippa Foot) is whether moral goodness simply is good character, or whether it is the actions (or even the consequences). Weil is close to modern virtue theory here. |
| 23895 | We modify our character by placing ourselves in situations, or by attending to what seems trivial [Weil] |
| Full Idea: We can modify our character, by putting ourselves in circumstances that will act on us from the outside, …or by the orientation of our attention in the moments that appear most insignificant or indifferent in our lives. | |
| From: Simone Weil (On the Concept of Character [1941], p.99) | |
| A reaction: I've never seen anyone address this question (apart from Aristotle's emphasis on training habits). Choosing your source for current affairs information strikes me as very important. What you read, what you watch, who you spend time with… |
| 21706 | At first matter is basic and known by sense-data; later Russell says matter is constructed [Russell, by Linsky,B] |
| Full Idea: In the beginning Russell's ontology included matter as basic, to be known, however, only by inference from sense-data. By the end he wanted to 'contruct' matter from sense-data. | |
| From: report of Bertrand Russell (The Analysis of Matter [1927]) by Bernard Linsky - Russell's Metaphysical Logic 1 | |
| A reaction: [see also p.133] Russell always seems to have been a robust realist about the external world, but the later view seems a lot less realist than the earlier view. |