Combining Philosophers
Ideas for Bonaventura, L. Jonathan Cohen and Bertrand Russell
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28 ideas
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
14162
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Mathematics doesn't care whether its entities exist [Russell]
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6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
17628
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Arithmetic was probably inferred from relationships between physical objects [Russell]
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6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
5399
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Maths is not known by induction, because further instances are not needed to support it [Russell]
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6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
14465
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Maybe numbers are adjectives, since 'ten men' grammatically resembles 'white men' [Russell]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
13414
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For Russell, numbers are sets of equivalent sets [Russell, by Benacerraf]
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14103
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Pure mathematics is the class of propositions of the form 'p implies q' [Russell]
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6108
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Maths can be deduced from logical axioms and the logic of relations [Russell]
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6423
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We tried to define all of pure maths using logical premisses and concepts [Russell]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
21555
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For 'x is a u' to be meaningful, u must be one range of individuals (or 'type') higher than x [Russell]
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18003
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In 'x is a u', x and u must be of different types, so 'x is an x' is generally meaningless [Russell, by Magidor]
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10418
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Type theory seems an extreme reaction, since self-exemplification is often innocuous [Swoyer on Russell]
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10047
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Russell's improvements blocked mathematics as well as paradoxes, and needed further axioms [Russell, by Musgrave]
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23478
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Type theory means that features shared by different levels cannot be expressed [Morris,M on Russell]
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23457
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Type theory cannot identify features across levels (because such predicates break the rules) [Morris,M on Russell]
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21556
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Classes are defined by propositional functions, and functions are typed, with an axiom of reducibility [Russell, by Lackey]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
21718
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Ramified types can be defended as a system of intensional logic, with a 'no class' view of sets [Russell, by Linsky,B]
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6. Mathematics / C. Sources of Mathematics / 7. Formalism
21570
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Numbers are just verbal conveniences, which can be analysed away [Russell]
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6424
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Formalists say maths is merely conventional marks on paper, like the arbitrary rules of chess [Russell]
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6425
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Formalism can't apply numbers to reality, so it is an evasion [Russell]
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6. Mathematics / C. Sources of Mathematics / 9. Fictional Mathematics
6104
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Numbers are classes of classes, and hence fictions of fictions [Russell]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
6426
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Intuitionism says propositions are only true or false if there is a method of showing it [Russell]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
21559
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We need rules for deciding which norms are predicative (unless none of them are) [Russell]
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18126
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A set does not exist unless at least one of its specifications is predicative [Russell, by Bostock]
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18128
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Russell is a conceptualist here, saying some abstracta only exist because definitions create them [Russell, by Bostock]
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18124
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Vicious Circle says if it is expressed using the whole collection, it can't be in the collection [Russell, by Bostock]
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21568
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A one-variable function is only 'predicative' if it is one order above its arguments [Russell]
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21558
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'Predicative' norms are those which define a class [Russell]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / e. Psychologism
14449
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There is always something psychological about inference [Russell]
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