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 2012,978-0-521-53341-6]].

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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 not-P; bivalence says P is either true or false

5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems

17929

Löwenheim proved his result for a first-order 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. Nature of Numbers / e. Ordinal numbers

17928

Ordinal numbers represent order relations

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite

17923

Intuitionists only accept a few safe infinities

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / j. 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 / 4. Axioms for Number / f. Mathematical induction

17936

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

6. Mathematics / B. Foundations for Mathematics / 6. 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 / 7. 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 / 7. 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 four-colour 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