77 ideas
8721 | An 'impredicative' definition seems circular, because it uses the term being defined [Friend] |
8680 | Classical definitions attempt to refer, but intuitionist/constructivist definitions actually create objects [Friend] |
3678 | Reductio ad absurdum proves an idea by showing that its denial produces contradiction [Friend] |
8705 | Anti-realists see truth as our servant, and epistemically contrained [Friend] |
8713 | In classical/realist logic the connectives are defined by truth-tables [Friend] |
8708 | Double negation elimination is not valid in intuitionist logic [Friend] |
8694 | Free logic was developed for fictional or non-existent objects [Friend] |
8665 | A 'proper subset' of A contains only members of A, but not all of them [Friend] |
8672 | A 'powerset' is all the subsets of a set [Friend] |
8677 | Set theory makes a minimum ontological claim, that the empty set exists [Friend] |
8666 | Infinite sets correspond one-to-one with a subset [Friend] |
8682 | Major set theories differ in their axioms, and also over the additional axioms of choice and infinity [Friend] |
8709 | The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false [Friend] |
8711 | Intuitionists read the universal quantifier as "we have a procedure for checking every..." [Friend] |
8675 | Paradoxes can be solved by talking more loosely of 'classes' instead of 'sets' [Friend] |
8674 | The Burali-Forti paradox asks whether the set of all ordinals is itself an ordinal [Friend] |
8667 | The 'integers' are the positive and negative natural numbers, plus zero [Friend] |
8668 | The 'rational' numbers are those representable as fractions [Friend] |
8670 | A number is 'irrational' if it cannot be represented as a fraction [Friend] |
8661 | The natural numbers are primitive, and the ordinals are up one level of abstraction [Friend] |
8664 | Cardinal numbers answer 'how many?', with the order being irrelevant [Friend] |
8671 | The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps [Friend] |
8663 | Raising omega to successive powers of omega reveal an infinity of infinities [Friend] |
8662 | The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega [Friend] |
8669 | Between any two rational numbers there is an infinite number of rational numbers [Friend] |
8676 | Is mathematics based on sets, types, categories, models or topology? [Friend] |
8678 | Most mathematical theories can be translated into the language of set theory [Friend] |
8701 | The number 8 in isolation from the other numbers is of no interest [Friend] |
8702 | In structuralism the number 8 is not quite the same in different structures, only equivalent [Friend] |
8699 | Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)? [Friend] |
8696 | Structuralist says maths concerns concepts about base objects, not base objects themselves [Friend] |
8695 | Structuralism focuses on relations, predicates and functions, with objects being inessential [Friend] |
8700 | 'In re' structuralism says that the process of abstraction is pattern-spotting [Friend] |
8681 | The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts? [Friend] |
8712 | Mathematics should be treated as true whenever it is indispensable to our best physical theory [Friend] |
8716 | Formalism is unconstrained, so cannot indicate importance, or directions for research [Friend] |
8706 | Constructivism rejects too much mathematics [Friend] |
8707 | Intuitionists typically retain bivalence but reject the law of excluded middle [Friend] |
8704 | Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend] |
4266 | Having beliefs involves recognition, expectation and surprise [Scruton] |
4265 | If an animal has beliefs, that implies not only that it can make mistakes, but that it can learn from them [Scruton] |
4264 | Perception (which involves an assessment) is a higher state than sensation [Scruton] |
4271 | There is consciousness whenever behaviour must be explained in terms of mental activity [Scruton] |
4272 | Our concept of a person is derived from Roman law [Scruton] |
4267 | Conditioning may change behaviour without changing the mind [Scruton] |
8685 | Studying biology presumes the laws of chemistry, and it could never contradict them [Friend] |
4269 | An emotion is a motive which is also a feeling [Scruton] |
4270 | Do we use reason to distinguish people from animals, or use that difference to define reason? [Scruton] |
8688 | Concepts can be presented extensionally (as objects) or intensionally (as a characterization) [Friend] |
4284 | All moral life depends ultimately on piety, which is our recognition of our own dependence [Scruton] |
4273 | Kant's Moral Law is the rules rational beings would accept when trying to live by agreement [Scruton] |
4274 | The modern virtues are courage, prudence, wisdom, temperance, justice, charity and loyalty [Scruton] |
4286 | Only just people will drop their own self-interests when faced with an impartial verdict [Scruton] |
4283 | Sympathy can undermine the moral order just as much as crime does [Scruton] |
4290 | That which can only be done by a callous person, ought not to be done [Scruton] |
4285 | As soon as we drop self-interest and judge impartially, we find ourselves agreeing about conflicts [Scruton] |
4287 | Utilitarianism merely guides us (by means of sympathy) when the moral law is silent [Scruton] |
4282 | Morality is not a sort of calculation, it is what sets the limits to when calculation is appropriate [Scruton] |
4281 | Utilitarianism says we can't blame Stalin yet, but such a theory is a sick joke [Scruton] |
4280 | Utilitarianism is wrong precisely because it can't distinguish animals from people [Scruton] |
20595 | You can't distribute goods from behind a veil, because their social meaning is unclear [Walzer, by Tuckness/Wolf] |
20592 | Complex equality restricts equalities from spilling over, like money influencing politics and law [Walzer, by Tuckness/Wolf] |
20549 | Equality is complex, with different spheres of equality where different principles apply [Walzer, by Swift] |
4295 | We favour our own animals over foreign ones because we see them as fellow citizens [Scruton] |
4268 | Animals command our sympathy and moral concern initially because of their intentionality [Scruton] |
4291 | Letting your dog kill wild rats, and keeping rats for your dog to kill, are very different [Scruton] |
4263 | Many of the stranger forms of life (e.g. worms) interest us only as a species, not as individuals [Scruton] |
4276 | An animal has individuality if it is nameable, and advanced animals can respond to their name [Scruton] |
4277 | I may avoid stepping on a spider or flower, but fellow-feeling makes me protect a rabbit [Scruton] |
4278 | Lucky animals are eaten by large predators, the less lucky starve, and worst is death by small predators [Scruton] |
4279 | We can easily remove the risk of suffering from an animal's life, but we shouldn't do it [Scruton] |
4289 | Sheep and cattle live comfortable lives, and die an enviably easy death [Scruton] |
4292 | Concern for one animal may harm the species, if the individual is part of a bigger problem [Scruton] |
4294 | Animals are outside the community of rights, but we still have duties towards them [Scruton] |
4296 | Brutal animal sports are banned because they harm the personality of the watcher [Scruton] |
4288 | Many breeds of animals have needs which our own ancestors planted in them [Scruton] |
4293 | Introducing a natural means of controlling animal population may not be very compassionate [Scruton] |