Combining Philosophers
Ideas for Archimedes, Michael Morris and Michael D. Resnik
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11 ideas
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 [Resnik]
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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
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Counting needs to distinguish things, and also needs the concept of a successor in a series [Morris,M]
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To count, we must distinguish things, and have a series with successors in it [Morris,M]
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Discriminating things for counting implies concepts of identity and distinctness [Morris,M]
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6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
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Archimedes defined a straight line as the shortest distance between two points [Archimedes, by Leibniz]
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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 [Resnik]
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Sets are positions in patterns [Resnik]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism
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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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There are too many mathematical objects for them all to be mental or physical [Resnik]
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Maths is pattern recognition and representation, and its truth and proofs are based on these [Resnik]
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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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