Combining Philosophers
Ideas for Herodotus, Michael D. Resnik and John P. Burgess
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12 ideas
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
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10185
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Set theory is the standard background for modern mathematics [Burgess]
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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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10184
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Structuralists take the name 'R' of the reals to be a variable ranging over structures, not a structure [Burgess]
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10189
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There is no one relation for the real number 2, as relations differ in different models [Burgess]
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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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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
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10186
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If set theory is used to define 'structure', we can't define set theory structurally [Burgess]
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10187
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Abstract algebra concerns relations between models, not common features of all the models [Burgess]
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10188
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How can mathematical relations be either internal, or external, or intrinsic? [Burgess]
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