13 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. |
| 490 | Everything happens by reason and necessity [Leucippus] |
| Full Idea: Nothing happens at random; everything happens out of reason and by necessity. | |
| From: Leucippus (fragments/reports [c.435 BCE], B002), quoted by (who?) - where? |
| 9261 | The 'Ethics' is disappointing, because it fails to try to justify our duties [Prichard] |
| Full Idea: Reading the 'Ethics' is so disappointing, because Aristotle does not try to convince us that we really ought to do what our non-reflective consciousness has hitherto believed we ought to do. | |
| From: H.A. Prichard (Does moral phil rest on a mistake? [1912]) | |
| A reaction: Aristotle didn't speak the language of 'duty' (see Idea 2172), but he could work it into his account if Prichard asked nicely. I take the truly virtuous person to be, above all, a wonderful citizen. Duties are contractual; good deeds flow from virtue. |
| 9262 | The mistake is to think we can prove what can only be seen directly in moral thinking [Prichard] |
| Full Idea: Moral Philosophy rests on the mistake of supposing the possibility of proving what can only be apprehended directly by an act of moral thinking. | |
| From: H.A. Prichard (Does moral phil rest on a mistake? [1912]) | |
| A reaction: This is a beginning of the rebellion against the Enlightenment Project in ethics, which is why Prichard has become popular. At bottom he is offering intuition ('direct moral thinking'), which is a frustratingly thin concept. |
| 9260 | Virtues won't generate an obligation, so it isn't a basis for morality [Prichard] |
| Full Idea: It is untrue to urge that, since courage is a virtue, we ought to act courageously. We feel an obligation to act, but not from a certain desire. The action is done from obligation, so isn't an act of courage. ..In fact, virtue is no basis for morality. | |
| From: H.A. Prichard (Does moral phil rest on a mistake? [1912]) | |
| A reaction: One of the few interesting and direct attacks on virtue theory, before its modern revival. Prichard urges a perception of what is valuable (or good) as the basis for obligation and right action. He is right that values come first, in virtue and elsewhere. |
| 9259 | We feel obligations to overcome our own failings, and these are not relations to other people [Prichard] |
| Full Idea: The relation involved in an obligation need not be a relation to another at all. Thus we should admit that there is an obligation to overcome our natural timidity or greediness, and this involves no relations to others. | |
| From: H.A. Prichard (Does moral phil rest on a mistake? [1912]) | |
| A reaction: An interesting un-Aristotelian and individualistic view of virtue. Why would we want to rid ourselves of timidity or greediness? Either it is self-interested, or we wish to be better citizens. See Richard Taylor on duty. |
| 9258 | If pain were instrinsically wrong, it would be immoral to inflict it on ourselves [Prichard] |
| Full Idea: If the badness of pain were the reason why we ought not to inflict pain on another, it would equally be a reason why we ought not to inflict pain on ourselves; yet, though we would call such behaviour foolish, we wouldn't think it wrong. | |
| From: H.A. Prichard (Does moral phil rest on a mistake? [1912], n4) | |
| A reaction: A very nice point. Note that it will equally well apply to 'benefit' or 'preferences', or any other ideal which utilitarians set out to maximise. It may not be bad to hurt yourself, but it might still be bad to harm yourself. |