2012 | Introduction to the Philosophy of Mathematics |
1.1.1 | p.5 | 17922 | Reducing real numbers to rationals suggested arithmetic as the foundation of maths |
1.1.3 | p.7 | 17925 | Showing a disproof is impossible is not a proof, so don't eliminate double negation |
1.1.3 | p.7 | 17924 | Excluded middle says P or not-P; bivalence says P is either true or false |
1.1.3 | p.7 | 17923 | Intuitionists only accept a few safe infinities |
1.1.3 | p.8 | 17926 | Rejecting double negation elimination undermines reductio proofs |
1.2.3 n17 | p.12 | 17928 | Ordinal numbers represent order relations |
2.1.2 | p.25 | 17929 | Löwenheim proved his result for a first-order sentence, and Skolem generalised it |
2.1.2 | p.25 | 17930 | Axioms are 'categorical' if all of their models are isomorphic |
3.1.2 | p.40 | 17931 | Structuralism say only 'up to isomorphism' matters because that is all there is to it |
3.1.2 | p.41 | 17932 | If 'in re' structures relies on the world, does the world contain rich enough structures? |
5.2.1 | p.79 | 17933 | Reductio proofs do not seem to be very explanatory |
5.2.1 | p.80 | 17934 | Proof by cases (by 'exhaustion') is said to be unexplanatory |
5.2.1 | p.82 | 17935 | If inductive proofs hold because of the structure of natural numbers, they may explain theorems |
5.2.1 n11 | p.83 | 17936 | Transfinite induction moves from all cases, up to the limit ordinal |
5.2.2 | p.87 | 17937 | Mathematical generalisation is by extending a system, or by abstracting away from it |
6.3.2 | p.115 | 17939 | Mathematics can reveal structural similarities in diverse systems |
6.3.2 | p.115 | 17938 | Mathematics can show why some surprising events have to occur |
7.1.1 | p.119 | 17940 | Most mathematical proofs are using set theory, but without saying so |
7.1.2 | p.121 | 17941 | Infinitesimals were sometimes zero, and sometimes close to zero |
9.1.6 | p.153 | 17942 | Can a proof that no one understands (of the four-colour theorem) really be a proof? |
9.1.8 | p.156 | 17943 | Probability supports Bayesianism better as degrees of belief than as ratios of frequencies |