21 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. |
| 7357 | People who control others with fluent language often end up being hated [Kongzi (Confucius)] |
| Full Idea: Of what use is eloquence? He who engages in fluency of words to control men often finds himself hated by them. | |
| From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], V.5) | |
| A reaction: I don't recall Socrates making this very good point to any of the sophists (such as Gorgias). The idea that if you battle or connive your way to dominance over others then you are successful is false. Life is a much longer game than that. |
| 22465 | We see a moral distinction between doing and allowing to happen [Foot] |
| Full Idea: We have an intuition that there is a morally relevant distinction between what we do and what we allow to happen. | |
| From: Philippa Foot (Morality, Action, and Outcome [1985], p.88) | |
| A reaction: She says many deny this distinction, but she defends it. Presumably consequentialists deny the distinction. What is bad if I do it, but OK if I allow it to happen? Neglecting a victim to save others, she suggests. |
| 22466 | We see a moral distinction between our aims and their foreseen consequences [Foot] |
| Full Idea: We have an intuition that there is a moral distinction between what we aim at and what we foresee as a result of what we do. | |
| From: Philippa Foot (Morality, Action, and Outcome [1985], p.88) | |
| A reaction: Cf. Idea 22465. This seems to be the classic doctrine of double effect. It is hard to defend the claim that we are only responsible for what we aim at. A wide assessment of consequences is a moral duty. Well-meaning fools are bad. |
| 22467 | Acts and omissions only matter if they concern doing something versus allowing it [Foot] |
| Full Idea: The difference between acts and omissions is irrelevant to any moral issue except in so far as it corresponds to the distinction between allowing something to happen and being the agent to whom the happening can be ascribed. | |
| From: Philippa Foot (Morality, Action, and Outcome [1985], p.89) | |
| A reaction: The list of anyone's omissions is presumably infinite, but what they 'allow' must be in some way within their power. But what of something I can't now prevent, only because I failed to do some relevant task yesterday? |
| 7358 | All men prefer outward appearance to true excellence [Kongzi (Confucius)] |
| Full Idea: I have yet to meet a man as fond of excellence as he is of outward appearances. | |
| From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], IX.18) | |
| A reaction: Interestingly, this cynical view of the love of virtue is put by Plato into the mouths of Glaucon and Adeimantus (in Bk II of 'Republic', e.g. Idea 12), and not into the mouth of Socrates, who goes on to defend the possibility of true virtue. |
| 7362 | Humans are similar, but social conventions drive us apart (sages and idiots being the exceptions) [Kongzi (Confucius)] |
| Full Idea: In our natures we approximate one another; habits put us further and further apart. The only ones who do not change are sages and idiots. | |
| From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], XVII.2) | |
| A reaction: I find most of Confucius rather uninteresting, but this is a splendid remark about the influence of social conventions on human nature. Sages can achieve universal morality if they rise above social convention, and seek the true virtues of human nature. |
| 22470 | A good moral system benefits its participants, and so demands reciprocity [Foot] |
| Full Idea: It has been suggested that one criterion for a good moral system is that it should be possible to demand reciprocity from every individual because of the good the system renders to him. | |
| From: Philippa Foot (Morality, Action, and Outcome [1985], p.104) | |
| A reaction: Money seems to have this feature, that we mostly conform to the rules for its use, because we value the whole system. Foot accepts this, but says there are also other criteria, such as leaving freedom to live well (ie. not too puritanical). |
| 7360 | Do not do to others what you would not desire yourself [Kongzi (Confucius)] |
| Full Idea: Do not do to others what you would not desire yourself. Then you will have no enemies, either in the state or in your home. | |
| From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], XII.2) | |
| A reaction: The Golden Rule, but note the second sentence. Logically, it leads to the absurdity of not giving someone an Elvis record for Christmas because you yourself don't like Elvis. Kant (Idea 3733) and Nietzsche (Idea 4560) offer good criticisms. |
| 22468 | Virtues can have aims, but good states of affairs are not among them [Foot] |
| Full Idea: Some virtues do give us aims, but nothing from within morality suggests the kind of good state of affairs which it would seem always to be our duty to promote. And why indeed should there be any such thing? | |
| From: Philippa Foot (Morality, Action, and Outcome [1985], p.101) | |
| A reaction: Isn't successful human functioning, such as heath, always to be desired? If honour is a worthy aim, doesn't that make being rightly honoured a desirable state of affairs? She is attacking consequentialism, but I'm not convinced here. |
| 7359 | Excess and deficiency are equally at fault [Kongzi (Confucius)] |
| Full Idea: Excess and deficiency are equally at fault. | |
| From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], XI.16) | |
| A reaction: This is the sort of wisdom we admire in Aristotle (and in any sensible person), but it may also be the deepest motto of conservatism, and it is a long way from romantic philosophy, and the clarion call of Nietzsche to greater excitement in life. |
| 7363 | The virtues of the best people are humility, maganimity, sincerity, diligence, and graciousness [Kongzi (Confucius)] |
| Full Idea: He who in this world can practise five things may indeed be considered Man-at-his-best: humility, maganimity, sincerity, diligence, and graciousness. | |
| From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], XVII.5) | |
| A reaction: A very nice list. Who could resist working with a colleague who had such virtues? Who could go wrong if they married a person who had them? I can't think of anything important that is missing. |
| 22469 | Some virtues imply rules, and others concern attachment [Foot] |
| Full Idea: Virtues such as justice consist mainly in adherence to rules of conduct, while those such as benevolence we might call virtues of attachment. | |
| From: Philippa Foot (Morality, Action, and Outcome [1985], p.101) | |
| A reaction: Not sure about 'attachment'. We should be benevolent towards people to whom we are not particularly attached. Courage doesn't fall into either group. |
| 7361 | Men of the highest calibre avoid political life completely [Kongzi (Confucius)] |
| Full Idea: Men of the highest calibre avoid political life completely. | |
| From: Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE], XIV.37) | |
| A reaction: Plato notes that such people tend to avoid political life (and a left sheltering, as if from a wild storm!), but he thinks they should be dragged into the political arena for the common good. Confucius seems to approve of the avoidance. Plato is right. |
| 23393 | Confucianism assumes that all good developments have happened, and there is only one Way [Norden on Kongzi (Confucius)] |
| Full Idea: The two major limitations of Confucianism are that it assumes that all worthwhile cultural, social and ethical innovation has already occurred, and that it does not recognise the plurality of worthwhile ways of life. | |
| From: comment on Kongzi (Confucius) (The Analects (Lunyu) [c.511 BCE]) by Bryan van Norden - Intro to Classical Chinese Philosophy 3.III | |
| A reaction: In modern liberal terms that is about as conservative as it is possible to get. We think of it as the state of mind of an old person who can only long for the way things were when they were young. But 'hold fast to that which is good'! |