Ideas from 'Introduction to the Philosophy of Mathematics' by Mark Colyvan [2012], by Theme Structure
[found in 'An Introduction to the Philosophy of Mathematics' by Colyvan,Mark [CUP 2012,9780521533416]].
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4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic
17925

Showing a disproof is impossible is not a proof, so don't eliminate double negation

17926

Rejecting double negation elimination undermines reductio proofs

5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
17924

Excluded middle says P or notP; bivalence says P is either true or false

5. Theory of Logic / J. Model Theory in Logic / 3. LöwenheimSkolem Theorems
17929

Löwenheim proved his result for a firstorder sentence, and Skolem generalised it

5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
17930

Axioms are 'categorical' if all of their models are isomorphic

6. Mathematics / A. Nature of Mathematics / 3. Numbers / e. Ordinal numbers
17928

Ordinal numbers represent order relations

6. Mathematics / A. Nature of Mathematics / 4. The Infinite / a. The Infinite
17923

Intuitionists only accept a few safe infinities

6. Mathematics / A. Nature of Mathematics / 4. The Infinite / k. Infinite divisibility
17941

Infinitesimals were sometimes zero, and sometimes close to zero

6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
17922

Reducing real numbers to rationals suggested arithmetic as the foundation of maths

6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / f. Mathematical induction
17936

Transfinite induction moves from all cases, up to the limit ordinal

6. Mathematics / B. Foundations for Mathematics / 5. Mathematics as Set Theory / a. Mathematics is set theory
17940

Most mathematical proofs are using set theory, but without saying so

6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / a. Structuralism
17931

Structuralism say only 'up to isomorphism' matters because that is all there is to it

6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / e. Structuralism critique
17932

If 'in re' structures relies on the world, does the world contain rich enough structures?

14. Science / C. Induction / 6. Bayes's Theorem
17943

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

14. Science / D. Explanation / 2. Types of Explanation / e. Lawlike explanations
17939

Mathematics can reveal structural similarities in diverse systems

14. Science / D. Explanation / 2. Types of Explanation / f. Necessity in explanations
17938

Mathematics can show why some surprising events have to occur

14. Science / D. Explanation / 2. Types of Explanation / m. Explanation by proof
17933

Reductio proofs do not seem to be very explanatory

17935

If inductive proofs hold because of the structure of natural numbers, they may explain theorems

17942

Can a proof that no one understands (of the fourcolour theorem) really be a proof?

17934

Proof by cases (by 'exhaustion') is said to be unexplanatory

15. Nature of Minds / C. Capacities of Minds / 5. Generalisation by mind
17937

Mathematical generalisation is by extending a system, or by abstracting away from it
