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