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
Rejecting double negation elimination undermines reductio proofs
Showing a disproof is impossible is not a proof, so don't eliminate double negation
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
Proof by cases (by 'exhaustion') is said to be unexplanatory
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?
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