60 ideas
18194 | 'Forcing' can produce new models of ZFC from old models [Maddy] |
18195 | A Large Cardinal Axiom would assert ever-increasing stages in the hierarchy [Maddy] |
18191 | Axiom of Infinity: completed infinite collections can be treated mathematically [Maddy] |
18193 | The Axiom of Foundation says every set exists at a level in the set hierarchy [Maddy] |
18169 | Axiom of Reducibility: propositional functions are extensionally predicative [Maddy] |
18168 | 'Propositional functions' are propositions with a variable as subject or predicate [Maddy] |
18190 | Completed infinities resulted from giving foundations to calculus [Maddy] |
18171 | Cantor and Dedekind brought completed infinities into mathematics [Maddy] |
18172 | Infinity has degrees, and large cardinals are the heart of set theory [Maddy] |
18175 | For any cardinal there is always a larger one (so there is no set of all sets) [Maddy] |
18196 | An 'inaccessible' cardinal cannot be reached by union sets or power sets [Maddy] |
18187 | Theorems about limits could only be proved once the real numbers were understood [Maddy] |
18182 | The extension of concepts is not important to me [Maddy] |
18177 | In the ZFC hierarchy it is impossible to form Frege's set of all three-element sets [Maddy] |
18164 | Frege solves the Caesar problem by explicitly defining each number [Maddy] |
18184 | Making set theory foundational to mathematics leads to very fruitful axioms [Maddy] |
18185 | Unified set theory gives a final court of appeal for mathematics [Maddy] |
18183 | Set theory brings mathematics into one arena, where interrelations become clearer [Maddy] |
18186 | Identifying geometric points with real numbers revealed the power of set theory [Maddy] |
18188 | The line of rationals has gaps, but set theory provided an ordered continuum [Maddy] |
18163 | Mathematics rests on the logic of proofs, and on the set theoretic axioms [Maddy] |
18207 | Maybe applications of continuum mathematics are all idealisations [Maddy] |
18204 | Scientists posit as few entities as possible, but set theorist posit as many as possible [Maddy] |
18167 | We can get arithmetic directly from HP; Law V was used to get HP from the definition of number [Maddy] |
18205 | The theoretical indispensability of atoms did not at first convince scientists that they were real [Maddy] |
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] |
18206 | Science idealises the earth's surface, the oceans, continuities, and liquids [Maddy] |
4272 | Our concept of a person is derived from Roman law [Scruton] |
4267 | Conditioning may change behaviour without changing the mind [Scruton] |
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] |
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] |
7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
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] |
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] |
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] |