24 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. |
| 6685 | 'Subjectivism' is an extension of relativism from the social group to the individual [Graham] |
| Full Idea: What is called 'subjectivism' is really just an extension of relativism from the level of the social group to the level of the individual. | |
| From: Gordon Graham (Eight Theories of Ethics [2004], Ch.1) | |
| A reaction: Personally I prefer to stick with 'relativism', at any level. 'Relative' is a two-place predicate, so we should always specify what is relative to what, unless it is obvious from context. Morality might be relative to God, for example. |
| 6699 | The chain of consequences may not be the same as the chain of responsibility [Graham] |
| Full Idea: From a utilitarian point of view, the error of Archduke Ferdinand's driver (he turned up a cul-de-sac) was the worst in history, ...but the chain of consequences may not be the same as the chain of responsibility. | |
| From: Gordon Graham (Eight Theories of Ethics [2004], Ch.7) | |
| A reaction: Can you cause something, and yet not be responsible for it? The driver was presumably fully conscious, rational and deliberate. He must share the responsibility for catastrophe, just as he shares in the causing of all the consequences. |
| 23035 | The good life aims at perfections, or absolute laws, or what is absolutely desirable [Green,TH] |
| Full Idea: The differentia of the good life …is controlled by the consciousness of there being some perfection which has to be fulfilled, some law which has to be obeyed, something absolutely desirable whatever the individual may for the time desire. | |
| From: T.H. Green (Prolegomena to Ethics [1882], p.134), quoted by John H. Muirhead - The Service of the State II | |
| A reaction: The 'perfection' suggests Plato, and the 'law' suggests Kant. The idea that something is 'absolutely desirable' is, I suspect, aimed at the utilitarians, who don't care what is desired. I'm no idealist, but have some sympathy with this idea. |
| 6698 | Negative consequences are very hard (and possibly impossible) to assess [Graham] |
| Full Idea: Negative consequences make the extension of the consequences of our actions indefinite, and this means that it is difficult to assess them; it may make it impossible, since there is now no clear sense to the idea of THE consequences of an action at all. | |
| From: Gordon Graham (Eight Theories of Ethics [2004], Ch.7) | |
| A reaction: The general slogan of 'Do your best' covers most objections to the calculation of consequences. It is no excuse for stealing a wallet that 'at least I wasn't committing genocide'. How easy were the alternative actions to do? |
| 6700 | We can't criticise people because of unforeseeable consequences [Graham] |
| Full Idea: It is unreasonable to say that people have acted badly because of consequences which were not merely unforeseen but unforeseeable. | |
| From: Gordon Graham (Eight Theories of Ethics [2004], Ch.7) | |
| A reaction: Interesting, and it sounds right. A key question in moral philosophy is how much effort people should make to assess the consequences of their actions. We must surely absolve them of the truly 'unforeseeable' consequence. |
| 6704 | Egoism submits to desires, but cannot help form them [Graham] |
| Full Idea: Egoism is inadequate as a guide to good living. Though it tells us what to do, given pre-existent desires, it cannot help us critically form those desires. | |
| From: Gordon Graham (Eight Theories of Ethics [2004], Ch.9) | |
| A reaction: A crucial point in morality. It also applies to utilitarianism (should I change my capacity for pleasure?), and virtue theory (how should I genetically engineer 'human nature'?). I think these problems push us towards Platonism. See Idea 4840. |
| 6701 | Rescue operations need spontaneous benevolence, not careful thought [Graham] |
| Full Idea: If more lives are to be saved in natural disasters, what is needed is spontaneity on the part of the rescuers, a willingness not to stop and think but to act spontaneously. | |
| From: Gordon Graham (Eight Theories of Ethics [2004], Ch.7) | |
| A reaction: This seems right, but must obviously be applied with caution, as when people are drowned attempting hopeless rescues. The most valuable person in an earthquake may be the thinker, not the digger. |
| 6693 | 'What if everybody did that?' rather misses the point as an objection to cheating [Graham] |
| Full Idea: I can object to your walking on the grass by asking 'What if everybody did that?', but the advantages of cheating depend upon the fact that most people don't cheat, so justifying my own cheating must involve special pleading. | |
| From: Gordon Graham (Eight Theories of Ethics [2004], Ch.6) | |
| A reaction: It is, of course, reasonable to ask 'What if everybody cheated?', but it is also reasonable to reply that 'the whole point of cheating is that it exploits the honesty of others'. This shows that Kant cannot simply demolish the 'free rider'. |
| 6691 | It is more plausible to say people can choose between values, than that they can create them [Graham] |
| Full Idea: To say that individuals are free to choose their own values is more naturally interpreted as meaning that they are free to choose between pre-existent values. | |
| From: Gordon Graham (Eight Theories of Ethics [2004], Ch.5) | |
| A reaction: Existentialism seems absurdly individualistic in its morality. Nietzsche was the best existentialist, who saw that most people have to be sheep. Strong personalities can promote or demote the old values on the great scale of what is good. |
| 6688 | Life is only absurd if you expected an explanation and none turns up [Graham] |
| Full Idea: If 'life is absurd' just means 'there is no logical explanation for human existence', we have no reason for anguish, unless we think there should be such an explanation. | |
| From: Gordon Graham (Eight Theories of Ethics [2004], Ch.5) | |
| A reaction: This is aimed at Kierkegaard and Camus. 'Absurd' certainly seems to be a relative notion, and we have nothing to compare life with. However, life does strike us as a bit odd sometimes, don't you think? |
| 6705 | Existentialism may transcend our nature, unlike eudaimonism [Graham] |
| Full Idea: It is the freedom to transcend our nature which eudaimonism seems to ignore and existentialism brings to the fore. | |
| From: Gordon Graham (Eight Theories of Ethics [2004], Ch.9) | |
| A reaction: It is wildly exciting to 'transcend our nature', and very dreary to polish up the nature which is given to us. In this I am a bit conservative. We should not go against the grain, but we shouldn't assume current living is the correct line of the grain. |
| 6690 | A standard problem for existentialism is the 'sincere Nazi' [Graham] |
| Full Idea: A standard problem for existentialism is the 'sincere Nazi'; there were undoubtedly some true believers, who saw in Nazism a creed that they wanted to believe, and who freely chose to endorse it. | |
| From: Gordon Graham (Eight Theories of Ethics [2004], Ch.5) | |
| A reaction: The failing of Nazis was that they were not good citizens. They might have been good members of a faction, but they were (in my opinion) poor citizens of Germany, and (obviously) appalling citizens of Europe. The objection to existentialism is good. |
| 6689 | The key to existentialism: the way you make choices is more important than what you choose [Graham] |
| Full Idea: The chief implication of existentialism is this: what you choose to do, how you choose to spend your life, is not as important as the way you choose it. | |
| From: Gordon Graham (Eight Theories of Ethics [2004], Ch.5) | |
| A reaction: While existentialists place emphasis on some notion of 'pure' choice, this is very close to the virtue theory idea that in a dilemma there may be several different choices which could all be rightly made by virtuous people. Integrity is a central virtue. |
| 23037 | People are improved by egalitarian institutions and habits [Green,TH] |
| Full Idea: Man has bettered himself through institutions and habits which tend to make the welfare of all the welfare of each. | |
| From: T.H. Green (Prolegomena to Ethics [1882], p.180), quoted by John H. Muirhead - The Service of the State II | |
| A reaction: I like this a lot. We underestimate how the best social values are promoted by the existence of enlightened institutions, rather than by preaching and teaching. Schools, law courts and churche embody their values. |
| 23043 | All talk of the progress of a nation must reduce to the progress of its individual members [Green,TH] |
| Full Idea: Our ultimate standard of worth is an ideal of personal worth. All other values are relative to personal values. To speak of any progress of a nation or society or mankind except as relative to some greater worth of persons is to use words without meaning. | |
| From: T.H. Green (Prolegomena to Ethics [1882], p.193), quoted by John H. Muirhead - The Service of the State II | |
| A reaction: Note that, pre-verificationism, a Victorian talks of plausible words actually being meaningless. This is a good statement of the core doctrine of liberalism. |
| 6706 | The great religions are much more concerned with the religious life than with ethics [Graham] |
| Full Idea: The fact is that the great religions of the world are not principally concerned with ethics at all, but with the religious life for its own sake. ..The Sermon on the Mount, for example, is mainly concerned with how to pray and worship. | |
| From: Gordon Graham (Eight Theories of Ethics [2004], Ch.9) | |
| A reaction: This seems to me a highly significant point, given that most people nowadays seem to endorse religion precisely because they wish to endorse morality, and think religion is its essential underpinning. See Idea 336 for the core problem ('Euthyphro'). |
| 6709 | Western religion saves us from death; Eastern religion saves us from immortality [Graham] |
| Full Idea: For Western minds, religion entails the belief and hope that we will be saved from death and live forever, but the belief of Eastern religions is that we do live forever, and it is from this dreadful fate that we must look to spirituality to save us. | |
| From: Gordon Graham (Eight Theories of Ethics [2004], Ch.9) | |
| A reaction: Nice. I have certainly come to prefer the Eastern view, simply on the grounds that human beings have a limited capacity. I quite fancy three hundred years of healthy life, but after that I am sure that any potential I have will be used up. |