36 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] |
| 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] |
| 13195 | To explain a house we must describe its use, as well as its parts [Leibniz] |
| 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] |
| 13193 | Active force is not just potential for action, since it involves a real effort or striving [Leibniz] |
| 21075 | The state of nature always involves the threat of war [Kant] |
| 20569 | Kant made the social contract international and cosmopolitan [Kant, by Oksala] |
| 21079 | The a priori general will of a people shows what is right [Kant] |
| 21077 | Each nation should, from self-interest, join an international security constitution [Kant] |
| 21078 | A constitution must always be improved when necessary [Kant] |
| 21076 | Equality is where you cannot impose a legal obligation you yourself wouldn't endure [Kant] |
| 20570 | There is now a growing universal community, and violations of rights are felt everywhere [Kant] |
| 20571 | There are political and inter-national rights, but also universal cosmopolitan rights [Kant] |
| 21073 | Hiring soldiers is to use them as instruments, ignoring their personal rights [Kant] |
| 21074 | Some trust in the enemy is needed during wartime, or peace would be impossible [Kant] |
| 13194 | God's laws would be meaningless without internal powers for following them [Leibniz] |
| 13196 | All qualities of bodies reduce to forces [Leibniz] |
| 13192 | Power is passive force, which is mass, and active force, which is entelechy or form [Leibniz] |