Combining Philosophers
Ideas for Paul Bernays, Augustin-Louis Cauchy and Leon Horsten
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10 ideas
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
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15364
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English expressions are denumerably infinite, but reals are nondenumerable, so many are unnameable [Horsten]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / k. Infinitesimals
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18085
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Values that approach zero, becoming less than any quantity, are 'infinitesimals' [Cauchy]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / l. Limits
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18084
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When successive variable values approach a fixed value, that is its 'limit' [Cauchy]
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6. Mathematics / B. Foundations for Mathematics / 2. Proof in Mathematics
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10885
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Computer proofs don't provide explanations [Horsten]
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6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
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10881
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The concept of 'ordinal number' is set-theoretic, not arithmetical [Horsten]
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15360
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ZFC showed that the concept of set is mathematical, not logical, because of its existence claims [Horsten]
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15369
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Set theory is substantial over first-order arithmetic, because it enables new proofs [Horsten]
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6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
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10303
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Restricted Platonism is just an ideal projection of a domain of thought [Bernays]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
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10306
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Mathematical abstraction just goes in a different direction from logic [Bernays]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
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15370
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Predicativism says mathematical definitions must not include the thing being defined [Horsten]
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