Combining Philosophers
Ideas for Hermarchus, John Stuart Mill and Euclid
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8 ideas
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
14157
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Modern geometries only accept various parts of the Euclid propositions [Russell on Euclid]
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22278
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Euclid relied on obvious properties in diagrams, as well as on his axioms [Potter on Euclid]
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8673
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Euclid's parallel postulate defines unique non-intersecting parallel lines [Euclid, by Friend]
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10250
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Euclid needs a principle of continuity, saying some lines must intersect [Shapiro on Euclid]
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10302
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Euclid says we can 'join' two points, but Hilbert says the straight line 'exists' [Euclid, by Bernays]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
8742
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The only axioms needed are for equality, addition, and successive numbers [Mill, by Shapiro]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic
1600
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Euclid's common notions or axioms are what we must have if we are to learn anything at all [Euclid, by Roochnik]
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9800
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Arithmetic is based on definitions, and Sums of equals are equal, and Differences of equals are equal [Mill]
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