31 ideas
17925 | Showing a disproof is impossible is not a proof, so don't eliminate double negation [Colyvan] |
17926 | Rejecting double negation elimination undermines reductio proofs [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] |
2667 | A false object might give the same presentation as a true one [Arcesilaus, by Cicero] |
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] |
21241 | Even the fool can hold 'a being than which none greater exists' in his understanding [Anselm] |
21243 | An existing thing is even greater if its non-existence is inconceivable [Anselm] |
21244 | Conceiving a greater being than God leads to absurdity [Anselm] |
21242 | If that than which a greater cannot be thought actually exists, that is greater than the mere idea [Anselm] |
1421 | A perfection must be independent and unlimited, and the necessary existence of Anselm's second proof gives this [Malcolm on Anselm] |
21245 | The word 'God' can be denied, but understanding shows God must exist [Anselm] |
21246 | Guanilo says a supremely fertile island must exist, just because we can conceive it [Anselm] |
21247 | Nonexistence is impossible for the greatest thinkable thing, which has no beginning or end [Anselm] |
1420 | Anselm's first proof fails because existence isn't a real predicate, so it can't be a perfection [Malcolm on Anselm] |