Combining Philosophers
Ideas for Hippolytus, Micklethwait,J/Wooldridge,A and David Bostock
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20 ideas
6. Mathematics / C. Sources of Mathematics / 3. Mathematical Nominalism
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18141
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Nominalism about mathematics is either reductionist, or fictionalist [Bostock]
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18157
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Nominalism as based on application of numbers is no good, because there are too many applications [Bostock]
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6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics
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18150
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Actual measurement could never require the precision of the real numbers [Bostock]
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6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
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18158
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Ordinals are mainly used adjectively, as in 'the first', 'the second'... [Bostock]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
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18127
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Simple type theory has 'levels', but ramified type theory has 'orders' [Bostock]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
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18144
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Neo-logicists agree that HP introduces number, but also claim that it suffices for the job [Bostock]
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18147
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Neo-logicists meet the Caesar problem by saying Hume's Principle is unique to number [Bostock]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
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18111
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Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality [Bostock]
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18129
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Many crucial logicist definitions are in fact impredicative [Bostock]
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18146
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If Hume's Principle is the whole story, that implies structuralism [Bostock]
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6. Mathematics / C. Sources of Mathematics / 9. Fictional Mathematics
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18159
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Higher cardinalities in sets are just fairy stories [Bostock]
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18155
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A fairy tale may give predictions, but only a true theory can give explanations [Bostock]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
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18140
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The best version of conceptualism is predicativism [Bostock]
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18138
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Conceptualism fails to grasp mathematical properties, infinity, and objective truth values [Bostock]
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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
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18133
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The usual definitions of identity and of natural numbers are impredicative [Bostock]
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18131
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If abstracta only exist if they are expressible, there can only be denumerably many of them [Bostock]
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18134
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Predicativism makes theories of huge cardinals impossible [Bostock]
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18135
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If mathematics rests on science, predicativism may be the best approach [Bostock]
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18136
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If we can only think of what we can describe, predicativism may be implied [Bostock]
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18132
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The predicativity restriction makes a difference with the real numbers [Bostock]
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