### Ideas from 'What Numbers Could Not Be' by Paul Benacerraf , 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
 9912 There are no such things as numbers
 9901 Numbers can't be sets if there is no agreement on which sets they are
###### 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
 9151 Benacerraf says numbers are defined by their natural ordering [Fine,K]
###### 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers
 13891 To understand finite cardinals, it is necessary and sufficient to understand progressions [Wright,C]
 17904 A set has k members if it one-one corresponds with the numbers less than or equal to k
 17906 To explain numbers you must also explain cardinality, the counting of things
###### 6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
 9898 We can count intransitively (reciting numbers) without understanding transitive counting of items
 17903 Someone can recite numbers but not know how to count things; but not vice versa
###### 6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / g. Applying mathematics
 9897 The application of a system of numbers is counting and measurement
###### 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
 9900 For Zermelo 3 belongs to 17, but for Von Neumann it does not
 9899 The successor of x is either x and all its members, or just the unit set of x
###### 6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
 8697 Disputes about mathematical objects seem irrelevant, and mathematicians cannot resolve them [Friend]
 8304 No particular pair of sets can tell us what 'two' is, just by one-to-one correlation [Lowe]
 9906 If ordinal numbers are 'reducible to' some set-theory, then which is which?
###### 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
 9908 The job is done by the whole system of numbers, so numbers are not objects
 9907 If any recursive sequence will explain ordinals, then it seems to be the structure which matters
 9909 The number 3 defines the role of being third in a progression
 9911 Number words no more have referents than do the parts of a ruler
 8925 Mathematical objects only have properties relating them to other 'elements' of the same structure
 9938 How can numbers be objects if order is their only property? [Putnam]
###### 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
###### 6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
 9903 Number words are not predicates, as they function very differently from adjectives
###### 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
###### 9. Objects / F. Identity among Objects / 6. Identity between Objects
 9905 Identity statements make sense only if there are possible individuating conditions