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Ideas for
'Thinking About Mathematics', 'Defeasibility Theory' and 'Maths as a Science of Patterns'
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10 ideas
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
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8764
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Categories are the best foundation for mathematics [Shapiro]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / f. Zermelo numbers
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8762
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Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
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6300
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Mathematical constants and quantifiers only exist as locations within structures or patterns [Resnik]
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6303
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Sets are positions in patterns [Resnik]
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8760
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Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro]
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8761
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A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism
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6302
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Structuralism must explain why a triangle is a whole, and not a random set of points [Resnik]
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6295
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There are too many mathematical objects for them all to be mental or physical [Resnik]
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6296
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Maths is pattern recognition and representation, and its truth and proofs are based on these [Resnik]
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6301
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Congruence is the strongest relationship of patterns, equivalence comes next, and mutual occurrence is the weakest [Resnik]
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