Ideas of Mark Colyvan, by Theme
[Australian, fl. 2012, Professor at the University of Sydney.]
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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. 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
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?

17933

Reductio proofs do not seem to be very explanatory

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
