38 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] |
| 13007 | Archimedes defined a straight line as the shortest distance between two points [Archimedes, by Leibniz] |
| 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] |
| 7337 | In Mosaic legal theory, crimes are sins and sins are crimes [Johnson,P] |
| 7339 | Because human life is what is sacred, Mosaic law has no death penalty for property violations [Johnson,P] |
| 7353 | The Pharisees undermined slavery, by giving slaves responsibility and status in law courts [Johnson,P] |
| 7340 | Mosaic law was the first to embody the rule of law, and equality before the law [Johnson,P] |
| 7338 | Man's life is sacred, because it is made in God's image [Johnson,P] |
| 7348 | The Jews sharply distinguish human and divine, but the Greeks pull them closer together [Johnson,P] |
| 7336 | A key moment is the idea of a single moral God, who imposes his morality on humanity [Johnson,P] |
| 7341 | Sampson illustrates the idea that religious heroes often begin as outlaws and semi-criminals [Johnson,P] |
| 7342 | Isaiah moved Israelite religion away from the local, onto a more universal plane [Johnson,P] |
| 7355 | The Torah pre-existed creation, and was its blueprint [Johnson,P] |
| 7344 | Judaism involves circumcision, Sabbath, Passover, Pentecost, Tabernacles, New Year, and Atonement [Johnson,P] |
| 7345 | In exile the Jews became a nomocracy [Johnson,P] |
| 7347 | Zoroastrians believed in one eternal beneficent being, Creator through the holy spirit [Johnson,P] |
| 7349 | Immortality based on judgement of merit was developed by the Egyptians (not the Jews) [Johnson,P] |
| 7354 | The main doctrine of the Pharisees was belief in resurrection and the afterlife [Johnson,P] |
| 7356 | Pious Jews saw heaven as a vast library [Johnson,P] |