Ideas from 'Philosophy of Mathematics' by Øystein Linnebo [2017], by Theme Structure

[found in 'Philosophy of Mathematics' by Linnebo,Øystein [Princeton 2017,978-0-691-20229-7]].

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4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / d. Naïve logical sets

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Naïve set theory says any formula defines a set, and coextensive sets are identical

5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic

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In classical semantics singular terms refer, and quantifiers range over domains

5. Theory of Logic / K. Features of Logics / 1. Axiomatisation

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The axioms of group theory are not assertions, but a definition of a structure

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To investigate axiomatic theories, mathematics needs its own foundational axioms

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic

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You can't prove consistency using a weaker theory, but you can use a consistent theory

6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism

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Mathematics is the study of all possible patterns, and is thus bound to describe the world

6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique

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Logical truth is true in all models, so mathematical objects can't be purely logical

6. Mathematics / C. Sources of Mathematics / 7. Formalism

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Game Formalism has no semantics, and Term Formalism reduces the semantics