Ideas from 'Sets and Numbers' by Penelope Maddy [1981], by Theme Structure

[found in 'Philosophy of Mathematics: anthology' (ed/tr Jacquette,Dale) [Blackwell 2002,0-631-21870-x]].

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4. Formal Logic / F. Set Theory ST / 7. Natural Sets
The master science is physical objects divided into sets
6. Mathematics / B. Foundations for Mathematics / 5. Mathematics as Set Theory / a. Mathematics is set theory
Set theory (unlike the Peano postulates) can explain why multiplication is commutative
Standardly, numbers are said to be sets, which is neat ontology and epistemology
Numbers are properties of sets, just as lengths are properties of physical objects
6. Mathematics / B. Foundations for Mathematics / 5. Mathematics as Set Theory / b. Mathematics is not set theory
Number theory doesn't 'reduce' to set theory, because sets have number properties
Sets exist where their elements are, but numbers are more like universals
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
If mathematical objects exist, how can we know them, and which objects are they?
6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
Number words are unusual as adjectives; we don't say 'is five', and numbers always come first