Combining Philosophers

Ideas for Steven Pinker, A.George / D.J.Velleman and Michael D. Resnik

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11 ideas

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
17899 Second-order induction is stronger as it covers all concepts, not just first-order definable ones [George/Velleman]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
10128 The Incompleteness proofs use arithmetic to talk about formal arithmetic [George/Velleman]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
17902 A successor is the union of a set with its singleton [George/Velleman]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
10133 Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle [George/Velleman]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
10130 Set theory can prove the Peano Postulates [George/Velleman]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
6300 Mathematical constants and quantifiers only exist as locations within structures or patterns [Resnik]
6303 Sets are positions in patterns [Resnik]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism
6302 Structuralism must explain why a triangle is a whole, and not a random set of points [Resnik]
6295 There are too many mathematical objects for them all to be mental or physical [Resnik]
6296 Maths is pattern recognition and representation, and its truth and proofs are based on these [Resnik]
6301 Congruence is the strongest relationship of patterns, equivalence comes next, and mutual occurrence is the weakest [Resnik]