32 ideas
20109 | Hegel inserted society and history between the God-world, man-nature, man-being binary pairs [Hegel, by Safranski] |
7910 | Pursue truth with the urgency of someone whose clothes are on fire [Ashvaghosha] |
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] |
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] |
23274 | World history has no room for happiness [Hegel] |
23275 | The state of nature is one of untamed brutality [Hegel] |
23276 | The soul of the people is an organisation of its members which produces an essential unity [Hegel] |
23272 | The human race matters, and individuals have little importance [Hegel] |
23273 | In a good state the goal of the citizens and of the whole state are united [Hegel] |
23271 | The goal of the world is Spirit's consciousness and enactment of freedom [Hegel] |
23270 | We should all agree that there is reason in history [Hegel] |
7909 | The Eightfold Path concerns morality, wisdom, and tranquillity [Ashvaghosha] |
7908 | At the end of a saint, he is not located in space, but just ceases to be disturbed [Ashvaghosha] |