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'fragments/reports', 'Philosophy of Mathematics' and 'Eliminative Materialism and Prop. Attitudes'
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36 ideas
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
18101
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Each addition changes the ordinality but not the cardinality, prior to aleph-1 [Bostock]
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18100
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ω + 1 is a new ordinal, but its cardinality is unchanged [Bostock]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
18102
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A cardinal is the earliest ordinal that has that number of predecessors [Bostock]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers
18106
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Aleph-1 is the first ordinal that exceeds aleph-0 [Bostock]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
18095
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Instead of by cuts or series convergence, real numbers could be defined by axioms [Bostock]
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18099
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The number of reals is the number of subsets of the natural numbers [Bostock]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
18093
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For Eudoxus cuts in rationals are unique, but not every cut makes a real number [Bostock]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / k. Infinitesimals
18110
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Infinitesimals are not actually contradictory, because they can be non-standard real numbers [Bostock]
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6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
18156
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Modern axioms of geometry do not need the real numbers [Bostock]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
18097
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The Peano Axioms describe a unique structure [Bostock]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
18148
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Hume's Principle is a definition with existential claims, and won't explain numbers [Bostock]
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18145
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Many things will satisfy Hume's Principle, so there are many interpretations of it [Bostock]
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18149
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There are many criteria for the identity of numbers [Bostock]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem
18143
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Frege makes numbers sets to solve the Caesar problem, but maybe Caesar is a set! [Bostock]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
18116
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Numbers can't be positions, if nothing decides what position a given number has [Bostock]
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18117
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Structuralism falsely assumes relations to other numbers are numbers' only properties [Bostock]
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6. Mathematics / C. Sources of Mathematics / 3. Mathematical Nominalism
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
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
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
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
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
18146
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If Hume's Principle is the whole story, that implies structuralism [Bostock]
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18129
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Many crucial logicist definitions are in fact impredicative [Bostock]
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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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6. Mathematics / C. Sources of Mathematics / 9. Fictional Mathematics
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
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
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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18133
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The usual definitions of identity and of natural numbers are impredicative [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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