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. Numbers / a. Numbers
There are no such things as numbers
Numbers can't be sets if there is no agreement on which sets they are
6. Mathematics / A. Nature of Mathematics / 3. Numbers / c. Priority of numbers
Benacerraf says numbers are defined by their natural ordering
6. Mathematics / A. Nature of Mathematics / 3. Numbers / f. Cardinal numbers
To explain numbers you must also explain cardinality, the counting of things
To understand finite cardinals, it is necessary and sufficient to understand progressions
A set has k members if it one-one corresponds with the numbers less than or equal to k
6. Mathematics / A. Nature of Mathematics / 3. Numbers / p. Counting
We can count intransitively (reciting numbers) without understanding transitive counting of items
Someone can recite numbers but not know how to count things; but not vice versa
6. Mathematics / A. Nature of Mathematics / 7. Application of Mathematics
The application of a system of numbers is counting and measurement
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / a. Axioms for numbers
For Zermelo 3 belongs to 17, but for Von Neumann it does not
The successor of x is either x and all its members, or just the unit set of x
6. Mathematics / B. Foundations for Mathematics / 5. Mathematics as Set Theory / b. Mathematics is not set theory
Disputes about mathematical objects seem irrelevant, and mathematicians cannot resolve them
No particular pair of sets can tell us what 'two' is, just by one-to-one correlation
If ordinal numbers are 'reducible to' some set-theory, then which is which?
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / a. Structuralism
If any recursive sequence will explain ordinals, then it seems to be the structure which matters
The job is done by the whole system of numbers, so numbers are not objects
The number 3 defines the role of being third in a progression
Number words no more have referents than do the parts of a ruler
Mathematical objects only have properties relating them to other 'elements' of the same structure
How can numbers be objects if order is their only property?
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
Number-as-objects works wholesale, but fails utterly object by object
6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
Number words are not predicates, as they function very differently from adjectives
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
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
Identity statements make sense only if there are possible individuating conditions