Combining Texts
Ideas for
'Defending the Axioms', 'works' and 'Foundations of Geometry'
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7 ideas
6. Mathematics / A. Nature of Mathematics / 2. Geometry
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 / 5. The Infinite / g. Continuum Hypothesis
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Every infinite set of reals is either countable or of the same size as the full set of reals [Maddy]
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6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
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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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Hilbert's formalisation revealed implicit congruence axioms in Euclid [Hilbert, by Horsten/Pettigrew]
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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 / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
17618
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Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy]
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6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
17614
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The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy]
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