Combining Texts
Ideas for
'Defending the Axioms', 'Science without Numbers' and 'Preface to 'Dorian Gray''
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14 ideas
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
8958
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In Field's version of science, space-time points replace real numbers [Field,H, by Szabó]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
17615
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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
18221
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'Metric' axioms uses functions, points and numbers; 'synthetic' axioms give facts about space [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 / 1. Mathematical Platonism / a. For mathematical platonism
8757
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The Indispensability Argument is the only serious ground for the existence of mathematical entities [Field,H]
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6. Mathematics / C. Sources of Mathematics / 3. Mathematical Nominalism
18212
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Nominalists try to only refer to physical objects, or language, or mental constructions [Field,H]
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6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics
10261
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The application of mathematics only needs its possibility, not its truth [Field,H, by Shapiro]
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18218
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Hilbert explains geometry, by non-numerical facts about space [Field,H]
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9623
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Field needs a semantical notion of second-order consequence, and that needs sets [Brown,JR on Field,H]
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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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6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
18215
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It seems impossible to explain the idea that the conclusion is contained in the premises [Field,H]
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6. Mathematics / C. Sources of Mathematics / 9. Fictional Mathematics
18216
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Abstractions can form useful counterparts to concrete statements [Field,H]
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18214
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Mathematics is only empirical as regards which theory is useful [Field,H]
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18210
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Why regard standard mathematics as truths, rather than as interesting fictions? [Field,H]
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