Combining Philosophers
Ideas for Herodotus, John von Neumann and Tyler Burge
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10 ideas
6. Mathematics / A. Nature of Mathematics / 2. Geometry
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16901
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The equivalent algebra model of geometry loses some essential spatial meaning [Burge]
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9159
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You can't simply convert geometry into algebra, as some spatial content is lost [Burge]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
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13489
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Von Neumann treated cardinals as a special sort of ordinal [Neumann, by Hart,WD]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
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22716
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Von Neumann defined ordinals as the set of all smaller ordinals [Neumann, by Poundstone]
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12336
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A von Neumann ordinal is a transitive set with transitive elements [Neumann, by Badiou]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
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16902
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Peano arithmetic requires grasping 0 as a primitive number [Burge]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / g. Von Neumann numbers
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18179
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For Von Neumann the successor of n is n U {n} (rather than {n}) [Neumann, by Maddy]
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15925
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Each Von Neumann ordinal number is the set of its predecessors [Neumann, by Lavine]
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18180
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Von Neumann numbers are preferred, because they continue into the transfinite [Maddy on Neumann]
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6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
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13672
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All the axioms for mathematics presuppose set theory [Neumann]
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