Combining Texts
Ideas for
'Metaphysics', 'Elements of Geometry' and 'Mathematical logic and theory of types'
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10 ideas
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
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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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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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14157
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Modern geometries only accept various parts of the Euclid propositions [Russell on Euclid]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic
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17850
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Each many is just ones, and is measured by the one [Aristotle]
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17851
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Number is plurality measured by unity [Aristotle]
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17843
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The idea of 'one' is the foundation of number [Aristotle]
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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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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
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9793
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Mathematics studies abstracted relations, commensurability and proportion [Aristotle]
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