### Ideas from 'Introduction to the Philosophy of Mathematics' by Mark Colyvan , 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
 17934 Proof by cases (by 'exhaustion') is said to be unexplanatory
 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?
###### 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