Ideas from 'What Numbers Could Not Be' by Paul Benacerraf [1965], by Theme Structure
[found in 'Philosophy of Mathematics: readings (2nd)' (ed/tr Benacerraf/Putnam) [CUP 1983,0-521-29648-x]].
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
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9912
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There are no such things as numbers
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9901
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Numbers can't be sets if there is no agreement on which sets they are
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
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9151
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Benacerraf says numbers are defined by their natural ordering [Fine,K]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers
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13891
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To understand finite cardinals, it is necessary and sufficient to understand progressions [Wright,C]
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17904
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A set has k members if it one-one corresponds with the numbers less than or equal to k
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17906
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To explain numbers you must also explain cardinality, the counting of things
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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
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9898
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We can count intransitively (reciting numbers) without understanding transitive counting of items
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17903
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Someone can recite numbers but not know how to count things; but not vice versa
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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / g. Applying mathematics
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9897
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The application of a system of numbers is counting and measurement
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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
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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
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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 [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 [Lowe]
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9906
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If ordinal numbers are 'reducible to' some set-theory, then which is which?
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
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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
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9908
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The job is done by the whole system of numbers, so numbers are not objects
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9909
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The number 3 defines the role of being third in a progression
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9911
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Number words no more have referents than do the parts of a ruler
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8925
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Mathematical objects only have properties relating them to other 'elements' of the same structure
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9938
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How can numbers be objects if order is their only property? [Putnam]
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6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
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9910
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Number-as-objects works wholesale, but fails utterly object by object
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6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
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9903
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Number words are not predicates, as they function very differently from adjectives
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
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9904
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The set-theory paradoxes mean that 17 can't be the class of all classes with 17 members
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9. Objects / F. Identity among Objects / 6. Identity between Objects
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9905
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Identity statements make sense only if there are possible individuating conditions
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