Ideas from 'Ontology and Mathematical Truth' by Michael Jubien [1977], 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 / 1. Set Theory
'Impure' sets have a concrete member, while 'pure' (abstract) sets do not
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
A model is 'fundamental' if it contains only concrete entities
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / d. Natural numbers
There couldn't just be one number, such as 17
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
The subject-matter of (pure) mathematics is abstract structure
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
How can pure abstract entities give models to serve as interpretations?
If we all intuited mathematical objects, platonism would be agreed
Since mathematical objects are essentially relational, they can't be picked out on their own
9. Objects / A. Existence of Objects / 2. Abstract Objects / c. Modern abstracta
The empty set is the purest abstract object