Combining Philosophers
Ideas for David Bostock, Richard Boyd and Paul Benacerraf
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22 ideas
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
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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 / a. Axioms for numbers
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9900
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For Zermelo 3 belongs to 17, but for Von Neumann it does not [Benacerraf]
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9899
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The successor of x is either x and all its members, or just the unit set of x [Benacerraf]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
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18097
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The Peano Axioms describe a unique structure [Bostock]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
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13358
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Ordinary or mathematical induction assumes for the first, then always for the next, and hence for all [Bostock]
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13359
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Complete induction assumes for all numbers less than n, then also for n, and hence for all numbers [Bostock]
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
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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
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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 / 6. Mathematics as Set Theory / b. Mathematics is not set theory
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8697
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Disputes about mathematical objects seem irrelevant, and mathematicians cannot resolve them [Benacerraf, by Friend]
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8304
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No particular pair of sets can tell us what 'two' is, just by one-to-one correlation [Benacerraf, by Lowe]
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9906
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If ordinal numbers are 'reducible to' some set-theory, then which is which? [Benacerraf]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
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13415
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An adequate account of a number must relate it to its series [Benacerraf]
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9907
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If any recursive sequence will explain ordinals, then it seems to be the structure which matters [Benacerraf]
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9908
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The job is done by the whole system of numbers, so numbers are not objects [Benacerraf]
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9909
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The number 3 defines the role of being third in a progression [Benacerraf]
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9911
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Number words no more have referents than do the parts of a ruler [Benacerraf]
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8925
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Mathematical objects only have properties relating them to other 'elements' of the same structure [Benacerraf]
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9938
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How can numbers be objects if order is their only property? [Benacerraf, by Putnam]
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
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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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