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 1983,052129648x]].
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6. Mathematics / A. Nature of Mathematics / 3. 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. Numbers / c. Priority of numbers
9151

Benacerraf says numbers are defined by their natural ordering

6. Mathematics / A. Nature of Mathematics / 3. Numbers / f. Cardinal numbers
13891

To understand finite cardinals, it is necessary and sufficient to understand progressions

17904

A set has k members if it oneone 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 / 3. Numbers / p. Counting
17903

Someone can recite numbers but not know how to count things; but not vice versa

9898

We can count intransitively (reciting numbers) without understanding transitive counting of items

6. Mathematics / A. Nature of Mathematics / 7. Application of Mathematics
9897

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
9899

The successor of x is either x and all its members, or just the unit set of x

9900

For Zermelo 3 belongs to 17, but for Von Neumann it does not

6. Mathematics / B. Foundations for Mathematics / 5. Mathematics as Set Theory / b. Mathematics is not set theory
8697

Disputes about mathematical objects seem irrelevant, and mathematicians cannot resolve them

8304

No particular pair of sets can tell us what 'two' is, just by onetoone correlation

9906

If ordinal numbers are 'reducible to' some settheory, then which is which?

6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / a. Structuralism
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?

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

6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
9910

Numberasobjects 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 settheory 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
