Ideas from 'Maths as a Science of Patterns' by Michael D. Resnik [1997], by Theme Structure
[found in 'Mathematics as a Science of Patterns' by Resnik,Michael D. [OUP 1999,0-19-825014-2]].
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4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / e. Axioms of PL
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Axioms are often affirmed simply because they produce results which have been accepted
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6. Mathematics / A. Nature of Mathematics / 1. Mathematics
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Mathematical realism says that maths exists, is largely true, and is independent of proofs
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
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Mathematical constants and quantifiers only exist as locations within structures or patterns
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Sets are positions in patterns
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism
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There are too many mathematical objects for them all to be mental or physical
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Maths is pattern recognition and representation, and its truth and proofs are based on these
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Congruence is the strongest relationship of patterns, equivalence comes next, and mutual occurrence is the weakest
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Structuralism must explain why a triangle is a whole, and not a random set of points
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