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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

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 notP; 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 firstorder 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 fourcolour theorem) really be a proof?

9.1.8

p.156

17943

Probability supports Bayesianism better as degrees of belief than as ratios of frequencies
