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'Causation', 'Philosophies of Mathematics' and 'works'
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20 ideas
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
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10106
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Rational numbers give answers to division problems with integers [George/Velleman]
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10102
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The integers are answers to subtraction problems involving natural numbers [George/Velleman]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
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10107
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Real numbers provide answers to square root problems [George/Velleman]
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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / g. Applying mathematics
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9946
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Logicists say mathematics is applicable because it is totally general [George/Velleman]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / d. Actual infinite
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10125
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The classical mathematician believes the real numbers form an actual set [George/Velleman]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
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17899
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Second-order induction is stronger as it covers all concepts, not just first-order definable ones [George/Velleman]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
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10128
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The Incompleteness proofs use arithmetic to talk about formal arithmetic [George/Velleman]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
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17902
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A successor is the union of a set with its singleton [George/Velleman]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
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10133
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Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle [George/Velleman]
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6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
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10130
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Set theory can prove the Peano Postulates [George/Velleman]
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6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
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10089
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Talk of 'abstract entities' is more a label for the problem than a solution to it [George/Velleman]
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6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
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10131
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If mathematics is not about particulars, observing particulars must be irrelevant [George/Velleman]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
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10092
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In the unramified theory of types, the types are objects, then sets of objects, sets of sets etc. [George/Velleman]
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10094
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The theory of types seems to rule out harmless sets as well as paradoxical ones. [George/Velleman]
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10095
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Type theory has only finitely many items at each level, which is a problem for mathematics [George/Velleman]
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17901
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Type theory prohibits (oddly) a set containing an individual and a set of individuals [George/Velleman]
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6. Mathematics / C. Sources of Mathematics / 8. Finitism
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10114
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Bounded quantification is originally finitary, as conjunctions and disjunctions [George/Velleman]
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10134
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Much infinite mathematics can still be justified finitely [George/Velleman]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
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10123
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The intuitionists are the idealists of mathematics [George/Velleman]
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10124
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Gödel's First Theorem suggests there are truths which are independent of proof [George/Velleman]
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