32 ideas
17925 | Showing a disproof is impossible is not a proof, so don't eliminate double negation [Colyvan] |
17926 | Rejecting double negation elimination undermines reductio proofs [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] |
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] |
23529 | Conduct is not isolated from its effect on the moral code [Hart,HLA] |
23530 | The great danger of democracy is that the oppression of the minority becomes unobjectionable [Hart,HLA] |
23522 | In an organised society all actions have some effect on other people [Hart,HLA] |
23528 | The value of liberty allows freedom of action, even if that distresses other people [Hart,HLA] |
23523 | The principle of legality requires crimes to be precisely defined in advance of any action [Hart,HLA] |
23524 | Some private moral issues are no concern of the law [Hart,HLA] |
23521 | Do morals influence law? Is morality an aspect of law? Can law be morally criticised? [Hart,HLA] |
23525 | Is the enforcement of morality morally justifiable? [Hart,HLA] |
23526 | Modern law still suppresses practices seen as immoral, and yet harmless [Hart,HLA] |
23527 | Moral wickedness of an offence is always relevant to the degree of punishment [Hart,HLA] |
5994 | Is the cosmos open or closed, mechanical or teleological, alive or inanimate, and created or eternal? [Robinson,TM, by PG] |