Combining Philosophers

Ideas for David Bostock, Richard Boyd and Paul Benacerraf

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25 ideas

6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
9910 Number-as-objects works wholesale, but fails utterly object by object [Benacerraf]
17927 Realists have semantics without epistemology, anti-realists epistemology but bad semantics [Benacerraf, by Colyvan]
9936 The platonist view of mathematics doesn't fit our epistemology very well [Benacerraf]
6. Mathematics / C. Sources of Mathematics / 3. Mathematical Nominalism
18141 Nominalism about mathematics is either reductionist, or fictionalist [Bostock]
18157 Nominalism as based on application of numbers is no good, because there are too many applications [Bostock]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics
18150 Actual measurement could never require the precision of the real numbers [Bostock]
6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
9903 Number words are not predicates, as they function very differently from adjectives [Benacerraf]
18158 Ordinals are mainly used adjectively, as in 'the first', 'the second'... [Bostock]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
18127 Simple type theory has 'levels', but ramified type theory has 'orders' [Bostock]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
18144 Neo-logicists agree that HP introduces number, but also claim that it suffices for the job [Bostock]
18147 Neo-logicists meet the Caesar problem by saying Hume's Principle is unique to number [Bostock]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
9904 The set-theory paradoxes mean that 17 can't be the class of all classes with 17 members [Benacerraf]
18146 If Hume's Principle is the whole story, that implies structuralism [Bostock]
18111 Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality [Bostock]
18129 Many crucial logicist definitions are in fact impredicative [Bostock]
6. Mathematics / C. Sources of Mathematics / 9. Fictional Mathematics
18159 Higher cardinalities in sets are just fairy stories [Bostock]
18155 A fairy tale may give predictions, but only a true theory can give explanations [Bostock]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
18140 The best version of conceptualism is predicativism [Bostock]
18138 Conceptualism fails to grasp mathematical properties, infinity, and objective truth values [Bostock]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
18131 If abstracta only exist if they are expressible, there can only be denumerably many of them [Bostock]
18134 Predicativism makes theories of huge cardinals impossible [Bostock]
18135 If mathematics rests on science, predicativism may be the best approach [Bostock]
18136 If we can only think of what we can describe, predicativism may be implied [Bostock]
18133 The usual definitions of identity and of natural numbers are impredicative [Bostock]
18132 The predicativity restriction makes a difference with the real numbers [Bostock]