Combining Philosophers
Ideas for Hermarchus, Kit Fine and David Hilbert
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10 ideas
6. Mathematics / B. Foundations for Mathematics / 2. Proof in Mathematics
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17967
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To decide some questions, we must study the essence of mathematical proof itself [Hilbert]
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6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
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9546
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Euclid axioms concerns possibilities of construction, but Hilbert's assert the existence of objects [Hilbert, by Chihara]
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18742
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Hilbert's formalisation revealed implicit congruence axioms in Euclid [Hilbert, by Horsten/Pettigrew]
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18217
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Hilbert's geometry is interesting because it captures Euclid without using real numbers [Hilbert, by Field,H]
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17965
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The whole of Euclidean geometry derives from a basic equation and transformations [Hilbert]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
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17964
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Number theory just needs calculation laws and rules for integers [Hilbert]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
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12215
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The existence of numbers is not a matter of identities, but of constituents of the world [Fine,K]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
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10529
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If Hume's Principle can define numbers, we needn't worry about its truth [Fine,K]
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10530
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Hume's Principle is either adequate for number but fails to define properly, or vice versa [Fine,K]
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6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
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10560
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Set-theoretic imperialists think sets can represent every mathematical object [Fine,K]
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