ThesaHelp: references ab
Topic: number as a progression for counting and 11 relations
Topic: what is a number
Topic: number as the extension of a class of equinumerous classes
Topic: object and value equivalence
Topic: context
 
Reference
Benacerraf, P.,
"What numbers could not be",
Philosophical Review, 74, pp. 4773, 1965.
Google
Other Reference
p. 272294 in Benacerraf, P., Putnam, H. (eds) Philosophy of mathematics, 2nd ed., Cambridge: Cambridge University Press 1983.
Quotations
274 ;;Quote: transitive counting is measuring the cardinality of a set by counting; onetoone correspondence
 274+;;Quote: intransitive counting is repeating number words in right order, as in counting sheep
 275 ;;Quote: lessthan over natural numbers should be recursive so we can compare numbers and count them
 275 ;;Quote: QuineWV defines the numbers as any progression; but also need transitive counting
 278 ;;Quote: there are many ways to define numbers as sets; each defines relations differently; so number is not a particular set
 281 ;;Quote: number words function like quantification operators; if numbers exist then so should quantifiers
 281+;;Quote: FregeG defined number as an equivalence class, but numbers are neither predicates nor adjectives
 284 ;;Quote: consistent set theories do not include the class of all classes with 17 members; this avoids the paradoxes
 286 ;;Quote: identity statements are senseless outside of a consistent context for individuating things
 286+;;Quote: while FregeG felt numbers needed a reference, it does not make sense to identify '3' with '[[[0]]]'; their contexts differ
 291 ;;Quote: a sequence of number words is just that, a sequence with certain properties; not distinct from numbers
 291+;;Quote: arithmetic concerns the abstract structure of progression, and not particular objectsthe numbers
 293 ;;Quote: the value of a particular expression is the position of a number within a sequence and the corresponding rules for counting and cardinality
 293 ;;Quote: numbers have a recursive 'lessthan' relation because numeric notation is generated recursively

Related Topics
ThesaHelp: references ab (396 items)
Topic: number as a progression for counting and 11 relations (22 items)
Topic: what is a number (55 items)
Topic: number as the extension of a class of equinumerous classes (23 items)
Topic: object and value equivalence (60 items)
Topic: context (8 items)
