Combining Texts
Ideas for
'Parmenides', 'Foundations of Geometry' and 'Continuity and Irrational Numbers'
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9 ideas
6. Mathematics / A. Nature of Mathematics / 2. Geometry
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13472
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Hilbert aimed to eliminate number from geometry [Hilbert, by Hart,WD]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
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17611
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We want the essence of continuity, by showing its origin in arithmetic [Dedekind]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
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10572
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A cut between rational numbers creates and defines an irrational number [Dedekind]
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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / f. Arithmetic
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17612
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Arithmetic is just the consequence of counting, which is the successor operation [Dedekind]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / l. Limits
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18087
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If x changes by less and less, it must approach a limit [Dedekind]
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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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6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
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16150
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One is, so numbers exist, so endless numbers exist, and each one must partake of being [Plato]
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