37 ideas
23269 | Philosophy must start from clearly observed facts [Galen] |
23248 | Early empiricists said reason was just a useless concept introduced by philosophers [Galen, by Frede,M] |
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] |
12270 | Being is one [Melissus, by Aristotle] |
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] |
23266 | The spirit in the soul wants freedom, power and honour [Galen] |
6003 | Galen showed by experiment that the brain controls the body [Galen, by Hankinson] |
23219 | Stopping the heart doesn't terminate activity; pressing the brain does that [Galen, by Cobb] |
21799 | We just use the word 'faculty' when we don't know the psychological cause [Galen] |
23264 | Philosophers think faculties are in substances, and invent a faculty for every activity [Galen] |
17937 | Mathematical generalisation is by extending a system, or by abstracting away from it [Colyvan] |
23220 | The brain contains memory and reason, and is the source of sensation and decision [Galen] |
23265 | The rational part of the soul is the desire for truth, understanding and recollection [Galen] |
7453 | Galen's medicine followed the mean; each illness was balanced by opposite treatment [Galen, by Hacking] |
6030 | Each part of the soul has its virtue - pleasure for appetite, success for competition, and rectitude for reason [Galen] |
23268 | We execute irredeemable people, to protect ourselves, as a deterrent, and ending a bad life [Galen] |
3059 | There is no real motion, only the appearance of it [Melissus, by Diog. Laertius] |
5100 | The void is not required for change, because a plenum can alter in quality [Aristotle on Melissus] |
456 | Nothing could come out of nothing [Melissus] |