Combining Philosophers
Ideas for Hermarchus, Samuel Scheffler and Willard Quine
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14 ideas
6. Mathematics / C. Sources of Mathematics / 3. Mathematical Nominalism
21696
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Nominalism rejects both attributes and classes (where extensionalism accepts the classes) [Quine]
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6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
17738
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Quine blurs the difference between knowledge of arithmetic and of physics [Jenkins on Quine]
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6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics
9556
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Nearly all of mathematics has to quantify over abstract objects [Quine]
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18198
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Mathematics is part of science; transfinite mathematics I take as mostly uninterpreted [Quine]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
8993
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If mathematics follows from definitions, then it is conventional, and part of logic [Quine]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
21557
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Russell confused use and mention, and reduced classes to properties, not to language [Quine, by Lackey]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
1613
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Logicists cheerfully accept reference to bound variables and all sorts of abstract entities [Quine]
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9004
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If set theory is not actually a branch of logic, then Frege's derivation of arithmetic would not be from logic [Quine]
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1635
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Mathematics reduces to set theory (which is a bit vague and unobvious), but not to logic proper [Quine]
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6. Mathematics / C. Sources of Mathematics / 7. Formalism
1616
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Formalism says maths is built of meaningless notations; these build into rules which have meaning [Quine]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
1615
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Intuitionism says classes are invented, and abstract entities are constructed from specified ingredients [Quine]
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8466
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For Quine, intuitionist ontology is inadequate for classical mathematics [Quine, by Orenstein]
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8467
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Intuitionists only admit numbers properly constructed, but classical maths covers all reals in a 'limit' [Quine, by Orenstein]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
1614
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Conceptualism holds that there are universals but they are mind-made [Quine]
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