Ideas from 'Ontology and Mathematical Truth' by Michael Jubien [1977], by Theme Structure
[found in 'Philosophy of Mathematics: anthology' (ed/tr Jacquette,Dale) [Blackwell 2002,063121870x]].
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4. Formal Logic / F. Set Theory ST / 1. Set Theory
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'Impure' sets have a concrete member, while 'pure' (abstract) sets do not

5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
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A model is 'fundamental' if it contains only concrete entities

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / d. Natural numbers
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There couldn't just be one number, such as 17

6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
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The subjectmatter of (pure) mathematics is abstract structure

6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
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How can pure abstract entities give models to serve as interpretations?

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If we all intuited mathematical objects, platonism would be agreed

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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
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The empty set is the purest abstract object
