48 ideas
| 10702 | Set theory's three roles: taming the infinite, subject-matter of mathematics, and modes of reasoning [Potter] |
| Full Idea: Set theory has three roles: as a means of taming the infinite, as a supplier of the subject-matter of mathematics, and as a source of its modes of reasoning. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], Intro 1) | |
| A reaction: These all seem to be connected with mathematics, but there is also ontological interest in set theory. Potter emphasises that his second role does not entail a commitment to sets 'being' numbers. |
| 10713 | Usually the only reason given for accepting the empty set is convenience [Potter] |
| Full Idea: It is rare to find any direct reason given for believing that the empty set exists, except for variants of Dedekind's argument from convenience. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 04.3) |
| 13044 | Infinity: There is at least one limit level [Potter] |
| Full Idea: Axiom of Infinity: There is at least one limit level. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 04.9) | |
| A reaction: A 'limit ordinal' is one which has successors, but no predecessors. The axiom just says there is at least one infinity. |
| 10708 | Nowadays we derive our conception of collections from the dependence between them [Potter] |
| Full Idea: It is only quite recently that the idea has emerged of deriving our conception of collections from a relation of dependence between them. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 03.2) | |
| A reaction: This is the 'iterative' view of sets, which he traces back to Gödel's 'What is Cantor's Continuum Problem?' |
| 13546 | The 'limitation of size' principles say whether properties collectivise depends on the number of objects [Potter] |
| Full Idea: We group under the heading 'limitation of size' those principles which classify properties as collectivizing or not according to how many objects there are with the property. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 13.5) | |
| A reaction: The idea was floated by Cantor, toyed with by Russell (1906), and advocated by von Neumann. The thought is simply that paradoxes start to appear when sets become enormous. |
| 10707 | Mereology elides the distinction between the cards in a pack and the suits [Potter] |
| Full Idea: Mereology tends to elide the distinction between the cards in a pack and the suits. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 02.1) | |
| A reaction: The example is a favourite of Frege's. Potter is giving a reason why mathematicians opted for set theory. I'm not clear, though, why a pack cannot have either 4 parts or 52 parts. Parts can 'fall under a concept' (such as 'legs'). I'm puzzled. |
| 10704 | We can formalize second-order formation rules, but not inference rules [Potter] |
| Full Idea: In second-order logic only the formation rules are completely formalizable, not the inference rules. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 01.2) | |
| A reaction: He cites Gödel's First Incompleteness theorem for this. |
| 10703 | Supposing axioms (rather than accepting them) give truths, but they are conditional [Potter] |
| Full Idea: A 'supposition' axiomatic theory is as concerned with truth as a 'realist' one (with undefined terms), but the truths are conditional. Satisfying the axioms is satisfying the theorem. This is if-thenism, or implicationism, or eliminative structuralism. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 01.1) | |
| A reaction: Aha! I had failed to make the connection between if-thenism and eliminative structuralism (of which I am rather fond). I think I am an if-thenist (not about all truth, but about provable truth). |
| 17447 | Parsons says counting is tagging as first, second, third..., and converting the last to a cardinal [Parsons,C, by Heck] |
| Full Idea: In Parsons's demonstrative model of counting, '1' means the first, and counting says 'the first, the second, the third', where one is supposed to 'tag' each object exactly once, and report how many by converting the last ordinal into a cardinal. | |
| From: report of Charles Parsons (Frege's Theory of Numbers [1965]) by Richard G. Heck - Cardinality, Counting and Equinumerosity 3 | |
| A reaction: This sounds good. Counting seems to rely on that fact that numbers can be both ordinals and cardinals. You don't 'convert' at the end, though, because all the way you mean 'this cardinality in this order'. |
| 10712 | If set theory didn't found mathematics, it is still needed to count infinite sets [Potter] |
| Full Idea: Even if set theory's role as a foundation for mathematics turned out to be wholly illusory, it would earn its keep through the calculus it provides for counting infinite sets. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 03.8) |
| 17882 | It is remarkable that all natural number arithmetic derives from just the Peano Axioms [Potter] |
| Full Idea: It is a remarkable fact that all the arithmetical properties of the natural numbers can be derived from such a small number of assumptions (as the Peano Axioms). | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 05.2) | |
| A reaction: If one were to defend essentialism about arithmetic, this would be grist to their mill. I'm just saying. |
| 13043 | A relation is a set consisting entirely of ordered pairs [Potter] |
| Full Idea: A set is called a 'relation' if every element of it is an ordered pair. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 04.7) | |
| A reaction: This is the modern extensional view of relations. For 'to the left of', you just list all the things that are to the left, with the things they are to the left of. But just listing the ordered pairs won't necessarily reveal how they are related. |
| 13042 | If dependence is well-founded, with no infinite backward chains, this implies substances [Potter] |
| Full Idea: The argument that the relation of dependence is well-founded ...is a version of the classical arguments for substance. ..Any conceptual scheme which genuinely represents a world cannot contain infinite backward chains of meaning. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 03.3) | |
| A reaction: Thus the iterative conception of set may imply a notion of substance, and Barwise's radical attempt to ditch the Axiom of Foundation (Idea 13039) was a radical attempt to get rid of 'substances'. Potter cites Wittgenstein as a fan of substances here. |
| 13041 | Collections have fixed members, but fusions can be carved in innumerable ways [Potter] |
| Full Idea: A collection has a determinate number of members, whereas a fusion may be carved up into parts in various equally valid (although perhaps not equally interesting) ways. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 02.1) | |
| A reaction: This seems to sum up both the attraction and the weakness of mereology. If you doubt the natural identity of so-called 'objects', then maybe classical mereology is the way to go. |
| 10709 | Priority is a modality, arising from collections and members [Potter] |
| Full Idea: We must conclude that priority is a modality distinct from that of time or necessity, a modality arising in some way out of the manner in which a collection is constituted from its members. | |
| From: Michael Potter (Set Theory and Its Philosophy [2004], 03.3) | |
| A reaction: He is referring to the 'iterative' view of sets, and cites Aristotle 'Metaphysics' 1019a1-4 as background. |
| 4266 | Having beliefs involves recognition, expectation and surprise [Scruton] |
| Full Idea: With the concept of belief (e.g. in animals) comes recognition, expectation and surprise. | |
| From: Roger Scruton (Animal Rights and Wrongs [1996], p.15) | |
| A reaction: A good observation. It is always tempting to see mental faculties in isolation, but each one drags along other capacities with it. Looks a bit holistic. |
| 4265 | If an animal has beliefs, that implies not only that it can make mistakes, but that it can learn from them [Scruton] |
| Full Idea: To say that an animal has beliefs is to imply not just that it can make mistakes, but also that it can learn from them. | |
| From: Roger Scruton (Animal Rights and Wrongs [1996], p.15) | |
| A reaction: A bold claim which is hard to substantiate. Seems right, though. Why would they change a belief? It can't be a belief if it isn't changeable. That would be an instinct. |
| 4264 | Perception (which involves an assessment) is a higher state than sensation [Scruton] |
| Full Idea: Perception is a higher state than sensation: it involves not just a response to the outer world, but also an assessment of it. | |
| From: Roger Scruton (Animal Rights and Wrongs [1996], p.14) | |
| A reaction: This seems to me a simple but really important distinction, even though it wickedly uses the word 'higher', which Greeks like but post-Humeans struggle with. But we all know it is higher, don't we? |
| 4271 | There is consciousness whenever behaviour must be explained in terms of mental activity [Scruton] |
| Full Idea: There is consciousness whenever behaviour must be explained in terms of mental activity. | |
| From: Roger Scruton (Animal Rights and Wrongs [1996], p.23) | |
| A reaction: Not a point that would trouble an eliminativist, as it sounds suspiciously circular or question-begging. |
| 4272 | Our concept of a person is derived from Roman law [Scruton] |
| Full Idea: Our concept of a person is derived from Roman law. | |
| From: Roger Scruton (Animal Rights and Wrongs [1996], p.28) | |
| A reaction: Interesting. I don't believe Roman legislators invented it, so where did it originate? Interesting that it is legalistic - a thing to which rights can accrue. Compare character, to which virtues accrue. |
| 4267 | Conditioning may change behaviour without changing the mind [Scruton] |
| Full Idea: Conditioning involves a change of behaviour, but not necessarily a change of mind. | |
| From: Roger Scruton (Animal Rights and Wrongs [1996], p.16) | |
| A reaction: I am inclined to doubt this. If I was conditioned in some way, I would expect my conscious state to change as well as my behaviour. |
| 4269 | An emotion is a motive which is also a feeling [Scruton] |
| Full Idea: An emotion is a motive which is also a feeling. | |
| From: Roger Scruton (Animal Rights and Wrongs [1996], p.17) | |
| A reaction: What is a motive without feeling? A universalised judgment, perhaps. Which comes first, the motivation or the feeling? |
| 4270 | Do we use reason to distinguish people from animals, or use that difference to define reason? [Scruton] |
| Full Idea: The difficulty of defining reason suggests that while pretending to use it to define the difference between humans and animals, they are actually using that difference to define reason. | |
| From: Roger Scruton (Animal Rights and Wrongs [1996], p.19) | |
| A reaction: Too pessimistic. We are perfectly capable of saying there is no significant difference between us and an alien. We have obvious abilities, which we can partly specify. |
| 4284 | All moral life depends ultimately on piety, which is our recognition of our own dependence [Scruton] |
| Full Idea: The three forms of moral life (respect for persons, the pursuit of virtue and natural sympathy) all depend, in the last analysis, on piety, which is the deep-down recognition of our frailty and dependence. | |
| From: Roger Scruton (Animal Rights and Wrongs [1996], p.56) | |
| A reaction: MacIntyre agrees. 'Piety' is an odd word, which attempts to link the point to religious teachings. 'Dependence' seems an adequate term. But can fully independent creatures dispense with morality? I think not. |
| 4273 | Kant's Moral Law is the rules rational beings would accept when trying to live by agreement [Scruton] |
| Full Idea: We can see the Kantian 'Moral Law' as consisting precisely in those rules which rational beings would accept, when attempting to live by agreement. | |
| From: Roger Scruton (Animal Rights and Wrongs [1996], p.30) | |
| A reaction: If this combines Kantian notions of duty with the obligations of contracts, it is the core of a very powerful moral theory. See the work of Tim Scanlon. Classic problems are still the weak, animals and free riders. |
| 4274 | The modern virtues are courage, prudence, wisdom, temperance, justice, charity and loyalty [Scruton] |
| Full Idea: The antique virtues of courage, prudence, wisdom, temperance and justice, amplified by Christian charity and pagan loyalty, still form the core idea of human excellence. | |
| From: Roger Scruton (Animal Rights and Wrongs [1996], p.33) | |
| A reaction: I always think sense of humour has become a key modern virtue. Where did that come from? Maybe a sense of irony is a good thing. How about efficiency (which is Plato's idea of justice!)? |
| 4286 | Only just people will drop their own self-interests when faced with an impartial verdict [Scruton] |
| Full Idea: Only just people will act on the impartial verdict when their own interests conflict with it. | |
| From: Roger Scruton (Animal Rights and Wrongs [1996], p.61) | |
| A reaction: The Kantian account of the virtues. Virtues are seen in the acceptance of a range of obvious human duties. Very helpful point if one is aiming for one unified theory of morality. |
| 4283 | Sympathy can undermine the moral order just as much as crime does [Scruton] |
| Full Idea: A person who lives by sympathy may undermine the moral order as effectively as the one who lives by crime. | |
| From: Roger Scruton (Animal Rights and Wrongs [1996], p.54) | |
| A reaction: A slightly chilling remark. Presumably one should not feel too much for suffering which is deserved. What about unavoidable suffering? It is certainly important to see that some suffering is morally good (e.g. grief or remorse). |
| 4290 | That which can only be done by a callous person, ought not to be done [Scruton] |
| Full Idea: That which can only be done by a callous person, ought not to be done. | |
| From: Roger Scruton (Animal Rights and Wrongs [1996], p.86) | |
| A reaction: The problem cases all arise in wartime. Ideally we want to show sympathy even when being necessarily ruthless, but in practice we send the callous ones to do the horrible deed. |
| 4285 | As soon as we drop self-interest and judge impartially, we find ourselves agreeing about conflicts [Scruton] |
| Full Idea: As soon as we set our own interests aside and look on human relations with the eye of the impartial judge, we find ourselves agreeing over the rights and wrongs of any conflict. | |
| From: Roger Scruton (Animal Rights and Wrongs [1996], p.59) | |
| A reaction: A nice, and fairly plausible, defence of Kantian ethics. Maybe the UN should actually settle all disputes, instead of just peace-keeping. The idea merely describes the function of the law, and especially an independent judiciary. |
| 4287 | Utilitarianism merely guides us (by means of sympathy) when the moral law is silent [Scruton] |
| Full Idea: Utilitarian thinking does not replace or compete with the moral law, but guides us when the moral law is silent and only sympathy speaks. | |
| From: Roger Scruton (Animal Rights and Wrongs [1996], p.63) | |
| A reaction: If the moral law is silent, it is not quite clear why we should follow sympathy rather than contempt. There is the well-known danger here of the moral law turning out to lack content. |
| 4282 | Morality is not a sort of calculation, it is what sets the limits to when calculation is appropriate [Scruton] |
| Full Idea: It is nearer the truth to see morality as setting the limits to practical reasoning, rather than being a species of it. Moral principles tell us precisely that we must go no further along the path of calculation. | |
| From: Roger Scruton (Animal Rights and Wrongs [1996], p.52) | |
| A reaction: Well said. If you are assessing whether an act of vicious brutality is required, you have probably already gone morally astray. It is not hard, though, to think of counterexamples, especially in wartime. |
| 4281 | Utilitarianism says we can't blame Stalin yet, but such a theory is a sick joke [Scruton] |
| Full Idea: Stalin and Hitler justified their actions in utilitarian terms, ..and no one can accuse them, for who knows what the long-term effects of their actions might be? But a morality which can't pass final judgement on Hitler or Stalin is a kind of sick joke. | |
| From: Roger Scruton (Animal Rights and Wrongs [1996], p.52) | |
| A reaction: A powerful argument against simplistic consequentialism. We can judge an action at any time, even beforehand, and that must be part of morality, which can't just observe the unfolding consequences. |
| 4280 | Utilitarianism is wrong precisely because it can't distinguish animals from people [Scruton] |
| Full Idea: It was precisely the inability of utilitarianism to explain the distinction between animals and people which led to its rejection. | |
| From: Roger Scruton (Animal Rights and Wrongs [1996], p.50) | |
| A reaction: A nice turning of the tables, rejecting the utilitarian pride in incorporating animals into their theory where others (like Kant) reject them. Yet in one respect (suffering) they are inescapably very like us. |
| 4295 | We favour our own animals over foreign ones because we see them as fellow citizens [Scruton] |
| Full Idea: We don't give help to British animals (through the RSPCA) rather than foreign animals because of their nearness or needs, but because of our sense of them as fellow citizens. | |
| From: Roger Scruton (Animal Rights and Wrongs [1996], p.104) | |
| A reaction: A bit strong. It may, in fact, be because we look after them the way we look after the rest of our property. Even Kantians can be sentimental sometimes. |
| 4268 | Animals command our sympathy and moral concern initially because of their intentionality [Scruton] |
| Full Idea: It seems to me that the concept of intentionality introduces the first genuine claim of animals upon our sympathies and our moral concern. | |
| From: Roger Scruton (Animal Rights and Wrongs [1996], p.16) | |
| A reaction: Good. If one's approach to morality is Humean (via sympathy) this seems right. Utilitarianism bases animal rights on qualia (pleasures and pains). |
| 4291 | Letting your dog kill wild rats, and keeping rats for your dog to kill, are very different [Scruton] |
| Full Idea: There is a difference between the person who allows his terrier to kill wild rats, and the person who keeps tame rats for his terrier to kill. | |
| From: Roger Scruton (Animal Rights and Wrongs [1996], p.87) | |
| A reaction: There are areas in the middle, where I encourage pheasants to breed 'wild' on my land. The purchase of a Rottweiller also tests the moral boundaries here. |
| 4263 | Many of the stranger forms of life (e.g. worms) interest us only as a species, not as individuals [Scruton] |
| Full Idea: Most of the stranger forms of life (worms, fleas, locusts etc.) are not really suitors for our moral concern, and interest us primarily as species, and only rarely as individuals. | |
| From: Roger Scruton (Animal Rights and Wrongs [1996], p.13) | |
| A reaction: Interesting, but that seems to reflect on us, rather than cutting nature at the joints. As soon as you look closely, you recognise an individual rather than a member of a species. |
| 4276 | An animal has individuality if it is nameable, and advanced animals can respond to their name [Scruton] |
| Full Idea: An animal has acquired individuality if the gift of a proper name seems appropriate, the high point being reached with animals such as dogs which actually respond to their own name. | |
| From: Roger Scruton (Animal Rights and Wrongs [1996], p.39) | |
| A reaction: Interesting, even though it is rather chauvinistic. I might name the fleas in my circus, but regard a whole section of the human race as indistinguishable and not worth naming. |
| 4277 | I may avoid stepping on a spider or flower, but fellow-feeling makes me protect a rabbit [Scruton] |
| Full Idea: I instinctively recoil from stepping on a spider or a forget-me-knot in my path, but neither of these responses expresses the fellow-feeling that forbids me to step on a rabbit or a mouse. | |
| From: Roger Scruton (Animal Rights and Wrongs [1996], p.41) | |
| A reaction: It is fellow-feeling that makes us prefer mammals to reptiles. It seems wrong to build a moral system purely on empathy, because psychopaths don't even empathise with nice human beings. Externalism in morality. |
| 4278 | Lucky animals are eaten by large predators, the less lucky starve, and worst is death by small predators [Scruton] |
| Full Idea: Lucky animals die in the jaws of a large predator; predators themselves are less lucky, when they die of lingering starvation; least fortunate are those killed by smaller creatures, such as maggots and bacteria. | |
| From: Roger Scruton (Animal Rights and Wrongs [1996], p.43) | |
| A reaction: A nice insight, even if it does slide into claiming that we are simply large predators, and that therefore fox-hunting is a virtue… |
| 4279 | We can easily remove the risk of suffering from an animal's life, but we shouldn't do it [Scruton] |
| Full Idea: It is easy to remove the risk of suffering from an animal's life, but the result is not a life which an animal should lead. | |
| From: Roger Scruton (Animal Rights and Wrongs [1996], p.44) | |
| A reaction: I'm not clear where the "should" derives from here. You can't save them all, and large interventions would destroy the ecosystem. But should we never, say, put a victim out of its misery? |
| 4289 | Sheep and cattle live comfortable lives, and die an enviably easy death [Scruton] |
| Full Idea: Sheep and beef cattle live a quiet and comfortable life among their companions, and are despatched in ways which human beings, if they are rational, must surely envy. | |
| From: Roger Scruton (Animal Rights and Wrongs [1996], p.81) | |
| A reaction: No rational person could envy a premature death, and we don't wait for cattle to be old before eating them. A quick death is little consolation for being murdered, and many people would prefer a slower death (without agony, of course). |
| 4292 | Concern for one animal may harm the species, if the individual is part of a bigger problem [Scruton] |
| Full Idea: Too much concern for individual animals may in fact harm the species, by promoting diseased or degenerate members, or preventing population control. | |
| From: Roger Scruton (Animal Rights and Wrongs [1996], p.87) | |
| A reaction: Okay till we reach human beings, where this principle won't go away, even if further principles about personhood, rationality and deep sympathy enter the picture. We can't be utilitarian about animals, and something else about humans. |
| 4294 | Animals are outside the community of rights, but we still have duties towards them [Scruton] |
| Full Idea: Animals exist outside the web of reciprocal rights and obligations, created by dialogue, but because they have no rights it does not mean that we have no duties towards them. | |
| From: Roger Scruton (Animal Rights and Wrongs [1996], p.97) | |
| A reaction: The modern Kantian view of animals, though Kant struggled to show why we might have any duties to animals. Is mere compassion enough to produce a duty, or is it a luxurious indulgence of our nature? |
| 4296 | Brutal animal sports are banned because they harm the personality of the watcher [Scruton] |
| Full Idea: Dog-fights and bear-baiting are naturally forbidden by law, because they threaten the personality of those who attend them. | |
| From: Roger Scruton (Animal Rights and Wrongs [1996], p.107) | |
| A reaction: Hm. If this is so, it is mainly because it takes place in a closed pen, where we can get a close look at the brutality and blood. It could be said to be more honest than hunting with gun or hounds. 'Go on eyes, look your worst'. |
| 4288 | Many breeds of animals have needs which our own ancestors planted in them [Scruton] |
| Full Idea: Many breeds of animals have needs which our own ancestors planted in them. | |
| From: Roger Scruton (Animal Rights and Wrongs [1996], p.75) | |
| A reaction: He is talking about race horses and St Bernards. This doesn't avoid the moral dilemma, because we could race horses die out if we thought we had created a bad life for them. |
| 4293 | Introducing a natural means of controlling animal population may not be very compassionate [Scruton] |
| Full Idea: It is hard to believe that those who would introduce wolves as a means of controlling the deer population have much sympathy for deer. | |
| From: Roger Scruton (Animal Rights and Wrongs [1996], p.91) | |
| A reaction: Good point. If we assume that culling is required at all, then the decisive human actions which shock us on television may be nicer than the natural deaths that occur during the night. |