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'In Defense of Essentialism', 'Logicism Revisited' and 'Elements of Geometry'
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13 ideas
6. Mathematics / A. Nature of Mathematics / 2. Geometry
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6297
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Euclid's geometry is synthetic, but Descartes produced an analytic version of it [Euclid, by Resnik]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
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9603
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An assumption that there is a largest prime leads to a contradiction [Euclid, by Brown,JR]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / m. One
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9894
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A unit is that according to which each existing thing is said to be one [Euclid]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
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8738
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Postulate 2 says a line can be extended continuously [Euclid, by Shapiro]
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6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
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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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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 / 4. Axioms for Number / d. Peano arithmetic
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10058
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No two numbers having the same successor relies on the Axiom of Infinity [Musgrave]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic
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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 / C. Sources of Mathematics / 7. Formalism
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10062
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Formalism seems to exclude all creative, growing mathematics [Musgrave]
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10063
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Formalism is a bulwark of logical positivism [Musgrave]
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