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## QuoteRef: benaP_1965

topics > all references > ThesaHelp: references a-b

ThesaHelp:
references a-b
Topic:
number as a progression for counting and 1-1 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. 47-73, 1965. Google

#### Other Reference

p. 272-294 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; one-to-one correspondence 274+;;Quote: intransitive counting is repeating number words in right order, as in counting sheep 275 ;;Quote: less-than 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 objects--the 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 'less-than' relation because numeric notation is generated recursively

Related Topics

ThesaHelp: references a-b (396 items)
Topic: number as a progression for counting and 1-1 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)

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