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Ideas of Mark Colyvan, by Text

[Australian, fl. 2012, Professor at the University of Sydney.]

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