43 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] |
22744 | Parts are not parts if their whole is nothing more than the parts [Sext.Empiricus] |
22748 | Some say motion is perceived by sense, but others say it is by intellect [Sext.Empiricus] |
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] |
22746 | If we try to conceive of a line with no breadth, it ceases to exist, and so has no length [Sext.Empiricus] |
22741 | The incorporeal is not in the nature of body, and so could not emerge from it [Sext.Empiricus] |
13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
22747 | A man walking backwards on a forwards-moving ship is moving in a fixed place [Sext.Empiricus] |
20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
22749 | Time doesn't end with the Universe, because tensed statements about destruction remain true [Sext.Empiricus] |
22750 | Time is divisible, into past, present and future [Sext.Empiricus] |
22742 | Socrates either dies when he exists (before his death) or when he doesn't (after his death) [Sext.Empiricus] |
22751 | If the present is just the limit of the past or the future, it can't exist because they don't exist [Sext.Empiricus] |
22730 | All men agree that God is blessed, imperishable, happy and good [Sext.Empiricus] |
22739 | God must suffer to understand suffering [Sext.Empiricus] |
22738 | The Divine must lack the virtues of continence and fortitude, because they are not needed [Sext.Empiricus] |
22734 | God is defended by agreement, order, absurdity of denying God, and refutations [Sext.Empiricus] |
22736 | God's sensations imply change, and hence perishing, which is absurd, so there is no such God [Sext.Empiricus] |
22740 | God without virtue is absurd, but God's virtues will be better than God [Sext.Empiricus] |
22735 | The original substance lacked motion or shape, and was given these by a cause [Sext.Empiricus] |
22732 | The perfections of God were extrapolations from mankind [Sext.Empiricus] |
22728 | Gods were invented as watchers of people's secret actions [Sext.Empiricus] |
22737 | An incorporeal God could do nothing, and a bodily god would perish, so there is no God [Sext.Empiricus] |
22731 | It is mad to think that what is useful to us, like lakes and rivers, are gods [Sext.Empiricus] |