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
Showing a disproof is impossible is not a proof, so don't eliminate double negation
Rejecting double negation elimination undermines reductio proofs
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
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. L÷wenheim-Skolem Theorems
L÷wenheim proved his result for a first-order sentence, and Skolem generalised it
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
Axioms are 'categorical' if all of their models are isomorphic
6. Mathematics / A. Nature of Mathematics / 3. Numbers / e. Ordinal numbers
Ordinal numbers represent order relations
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / a. The Infinite
Intuitionists only accept a few safe infinities
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / k. Infinite divisibility
Infinitesimals were sometimes zero, and sometimes close to zero
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
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
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
Most mathematical proofs are using set theory, but without saying so
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / a. Structuralism
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
If 'in re' structures relies on the world, does the world contain rich enough structures?
14. Science / C. Induction / 6. Bayes's Theorem
Probability supports Bayesianism better as degrees of belief than as ratios of frequencies
14. Science / D. Explanation / 2. Types of Explanation / d. Lawlike explanations
Mathematics can reveal structural similarities in diverse systems
14. Science / D. Explanation / 2. Types of Explanation / e. Necessity in explanations
Mathematics can show why some surprising events have to occur
14. Science / D. Explanation / 2. Types of Explanation / l. Explanation by proof
Reductio proofs do not seem to be very explanatory
If inductive proofs hold because of the structure of natural numbers, they may explain theorems
Can a proof that no one understands (of the four-colour theorem) really be a proof?
Proof by cases (by 'exhaustion') is said to be unexplanatory
15. Nature of Minds / C. Capacities of Minds / 5. Generalisation by mind
Mathematical generalisation is by extending a system, or by abstracting away from it