55 ideas
| 17926 | Rejecting double negation elimination undermines reductio proofs [Colyvan] |
| 17925 | Showing a disproof is impossible is not a proof, so don't eliminate double negation [Colyvan] |
| 17924 | Excluded middle says P or not-P; bivalence says P is either true or false [Colyvan] |
| 17929 | Löwenheim proved his result for a first-order sentence, and Skolem generalised it [Colyvan] |
| 17930 | Axioms are 'categorical' if all of their models are isomorphic [Colyvan] |
| 17928 | Ordinal numbers represent order relations [Colyvan] |
| 17923 | Intuitionists only accept a few safe infinities [Colyvan] |
| 17941 | Infinitesimals were sometimes zero, and sometimes close to zero [Colyvan] |
| 17922 | Reducing real numbers to rationals suggested arithmetic as the foundation of maths [Colyvan] |
| 17936 | Transfinite induction moves from all cases, up to the limit ordinal [Colyvan] |
| 17940 | Most mathematical proofs are using set theory, but without saying so [Colyvan] |
| 17931 | Structuralism say only 'up to isomorphism' matters because that is all there is to it [Colyvan] |
| 17932 | If 'in re' structures relies on the world, does the world contain rich enough structures? [Colyvan] |
| 6027 | From the fact that some men die, we cannot infer that they all do [Philodemus] |
| 17943 | Probability supports Bayesianism better as degrees of belief than as ratios of frequencies [Colyvan] |
| 17939 | Mathematics can reveal structural similarities in diverse systems [Colyvan] |
| 17938 | Mathematics can show why some surprising events have to occur [Colyvan] |
| 17934 | Proof by cases (by 'exhaustion') is said to be unexplanatory [Colyvan] |
| 17933 | Reductio proofs do not seem to be very explanatory [Colyvan] |
| 17935 | If inductive proofs hold because of the structure of natural numbers, they may explain theorems [Colyvan] |
| 17942 | Can a proof that no one understands (of the four-colour theorem) really be a proof? [Colyvan] |
| 17937 | Mathematical generalisation is by extending a system, or by abstracting away from it [Colyvan] |
| 22241 | Don't fear god or worry about death; the good is easily got and the terrible easily cured [Philodemus] |
| 20489 | Human beings can never really flourish in a long-term state of nature [Wolff,J] |
| 20483 | Collective rationality is individuals doing their best, assuming others all do the same [Wolff,J] |
| 20532 | Should love be the first virtue of a society, as it is of the family? [Wolff,J] |
| 20490 | For utilitarians, consent to the state is irrelevant, if it produces more happiness [Wolff,J] |
| 20493 | Social contract theory has the attracton of including everyone, and being voluntary [Wolff,J] |
| 20494 | Maybe voting in elections is a grant of legitimacy to the winners [Wolff,J] |
| 20500 | We can see the 'general will' as what is in the general interest [Wolff,J] |
| 20497 | How can dictators advance the interests of the people, if they don't consult them about interests? [Wolff,J] |
| 20506 | 'Separation of powers' allows legislative, executive and judicial functions to monitor one another [Wolff,J] |
| 20530 | Political choice can be by utility, or maximin, or maximax [Wolff,J] |
| 20487 | A realistic and less utopian anarchism looks increasingly like liberal democracy [Wolff,J] |
| 20488 | It is hard for anarchists to deny that we need experts [Wolff,J] |
| 20529 | Utilitarianism probably implies a free market plus welfare [Wolff,J] |
| 20510 | A system of democracy which includes both freedom and equality is almost impossible [Wolff,J] |
| 20511 | Democracy expresses equal respect (which explains why criminals forfeit the vote) [Wolff,J] |
| 20502 | Democracy has been seen as consistent with many types of inequality [Wolff,J] |
| 20496 | A true democracy could not tolerate slavery, exploitation or colonialism [Wolff,J] |
| 20498 | We should decide whether voting is for self-interests, or for the common good [Wolff,J] |
| 20499 | Condorcet proved that sensible voting leads to an emphatically right answer [Wolff,J] |
| 20509 | Occasional defeat is acceptable, but a minority that is continually defeated is a problem [Wolff,J] |
| 20524 | Market prices indicate shortages and gluts, and where the profits are to be made [Wolff,J] |
| 20518 | Liberty principles can't justify laws against duelling, incest between siblings and euthanasia [Wolff,J] |
| 20531 | Either Difference allows unequal liberty, or Liberty makes implementing Difference impossible [Wolff,J] |
| 20526 | Utilitarians argue for equal distribution because of diminishing utility of repetition [Wolff,J] |
| 20528 | Difference Principle: all inequalities should be in favour of the disadvantaged [Wolff,J] |
| 20503 | Political equality is not much use without social equality [Wolff,J] |
| 20512 | Standard rights: life, free speech, assembly, movement, vote, stand (plus shelter, food, health?) [Wolff,J] |
| 20513 | If natural rights are axiomatic, there is then no way we can defend them [Wolff,J] |
| 20514 | If rights are natural, rather than inferred, how do we know which rights we have? [Wolff,J] |
| 20522 | Utilitarians might say property ownership encourages the best use of the land [Wolff,J] |
| 20534 | Rights and justice are only the last resorts of a society, something to fall back on [Wolff,J] |
| 20492 | Following some laws is not a moral matter; trivial traffic rules, for example [Wolff,J] |