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### Ideas of Paul Benacerraf, by Text

#### [American, b.1931, Professor at Princeton University.]

 1960 Logicism, Some Considerations (PhD)
 p.164 p.164 13411 If numbers are basically the cardinals (Frege-Russell view) you could know some numbers in isolation
 p.165 p.165 13412 Obtaining numbers by abstraction is impossible - there are too many; only a rule could give them, in order
 p.166 p.166 13413 We must explain how we know so many numbers, and recognise ones we haven't met before
 p.169 p.169 13415 An adequate account of a number must relate it to its series
 1965 What Numbers Could Not Be
 p.18 9151 Benacerraf says numbers are defined by their natural ordering [Fine,K]
 p.83 8697 Disputes about mathematical objects seem irrelevant, and mathematicians cannot resolve them [Friend]
 p.117 13891 To understand finite cardinals, it is necessary and sufficient to understand progressions [Wright,C]
 p.215 8304 No particular pair of sets can tell us what 'two' is, just by one-to-one correlation [Lowe]
 I p.274 9898 We can count intransitively (reciting numbers) without understanding transitive counting of items
 I p.274 9897 The application of a system of numbers is counting and measurement
 I p.275 17904 A set has k members if it one-one corresponds with the numbers less than or equal to k
 I p.275 17903 Someone can recite numbers but not know how to count things; but not vice versa
 I n2 p.275 17906 To explain numbers you must also explain cardinality, the counting of things
 II p.278 9900 For Zermelo 3 belongs to 17, but for Von Neumann it does not
 II p.278 9899 The successor of x is either x and all its members, or just the unit set of x
 II p.279 9901 Numbers can't be sets if there is no agreement on which sets they are
 II p.283 9903 Number words are not predicates, as they function very differently from adjectives
 II p.284 9904 The set-theory paradoxes mean that 17 can't be the class of all classes with 17 members
 III p.286 9905 Identity statements make sense only if there are possible individuating conditions
 IIIB p.290 9906 If ordinal numbers are 'reducible to' some set-theory, then which is which?
 IIIC p.290 9908 The job is done by the whole system of numbers, so numbers are not objects
 IIIC p.290 9907 If any recursive sequence will explain ordinals, then it seems to be the structure which matters
 IIIC p.291 9909 The number 3 defines the role of being third in a progression
 IIIC p.292 9911 Number words no more have referents than do the parts of a ruler
 IIIC p.292 9910 Number-as-objects works wholesale, but fails utterly object by object
 IIIC p.294 9912 There are no such things as numbers
 p.285 p.581 8925 Mathematical objects only have properties relating them to other 'elements' of the same structure
 p.301 p.301 9938 How can numbers be objects if order is their only property? [Putnam]
 1973 Mathematical Truth
 p.8 17927 Realists have semantics without epistemology, anti-realists epistemology but bad semantics [Colyvan]
 Intro p.403 9935 Mathematical truth is always compromising between ordinary language and sensible epistemology
 III p.412 9936 The platonist view of mathematics doesn't fit our epistemology very well