Combining Philosophers

Ideas for David Bostock, Hilary Putnam and Volker Halbach

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15 ideas

6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
9937 I do not believe mathematics either has or needs 'foundations' [Putnam]
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
18156 Modern axioms of geometry do not need the real numbers [Bostock]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
9939 It is conceivable that the axioms of arithmetic or propositional logic might be changed [Putnam]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
18097 The Peano Axioms describe a unique structure [Bostock]
16321 The compactness theorem can prove nonstandard models of PA [Halbach]
16343 The global reflection principle seems to express the soundness of Peano Arithmetic [Halbach]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
13358 Ordinary or mathematical induction assumes for the first, then always for the next, and hence for all [Bostock]
13359 Complete induction assumes for all numbers less than n, then also for n, and hence for all numbers [Bostock]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
18148 Hume's Principle is a definition with existential claims, and won't explain numbers [Bostock]
18145 Many things will satisfy Hume's Principle, so there are many interpretations of it [Bostock]
18149 There are many criteria for the identity of numbers [Bostock]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem
18143 Frege makes numbers sets to solve the Caesar problem, but maybe Caesar is a set! [Bostock]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
16312 To reduce PA to ZF, we represent the non-negative integers with von Neumann ordinals [Halbach]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
18116 Numbers can't be positions, if nothing decides what position a given number has [Bostock]
18117 Structuralism falsely assumes relations to other numbers are numbers' only properties [Bostock]