Ideas from 'Maths as a Science of Patterns' by Michael D. Resnik [1997], by Theme Structure
[found in 'Mathematics as Science of Pattern Recognition' by Resnik,Michael D. [OUP 1999,0198250142]].
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4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / e. Axioms of PL
6299

Axioms are often affirmed simply because they produce results which have been accepted

6. Mathematics / A. Nature of Mathematics / 1. Mathematics
6304

Mathematical realism says that maths exists, is largely true, and is independent of proofs

6. Mathematics / A. Nature of Mathematics / 5. Geometry
6297

Euclid's geometry is synthetic, but Descartes produced an analytic version of it

6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / a. Structuralism
6300

Mathematical constants and quantifiers only exist as locations within structures or patterns

6303

Sets are positions in patterns

6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / d. Platonist structuralism
6296

Maths is pattern recognition and representation, and its truth and proofs are based on these

6301

Congruence is the strongest relationship of patterns, equivalence comes next, and mutual occurrence is the weakest

6302

Structuralism must explain why a triangle is a whole, and not a random set of points

6295

There are too many mathematical objects for them all to be mental or physical
