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Ideas of Paul Benacerraf, by Text
[American, b.1931, Professor at Princeton University.]
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1960
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Logicism, Some Considerations (PhD)
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p.164
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p.164
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13411
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If numbers are basically the cardinals (Frege-Russell view) you could know some numbers in isolation
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p.165
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p.165
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13412
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Obtaining numbers by abstraction is impossible - there are too many; only a rule could give them, in order
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p.166
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p.166
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13413
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We must explain how we know so many numbers, and recognise ones we haven't met before
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p.169
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p.169
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13415
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An adequate account of a number must relate it to its series
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1965
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What Numbers Could Not Be
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p.18
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9151
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Benacerraf says numbers are defined by their natural ordering [Fine,K]
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p.83
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8697
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Disputes about mathematical objects seem irrelevant, and mathematicians cannot resolve them [Friend]
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p.117
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13891
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To understand finite cardinals, it is necessary and sufficient to understand progressions [Wright,C]
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p.215
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8304
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No particular pair of sets can tell us what 'two' is, just by one-to-one correlation [Lowe]
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I
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p.274
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9898
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We can count intransitively (reciting numbers) without understanding transitive counting of items
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I
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p.274
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9897
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The application of a system of numbers is counting and measurement
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I
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p.275
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17903
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Someone can recite numbers but not know how to count things; but not vice versa
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I
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p.275
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17904
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A set has k members if it one-one corresponds with the numbers less than or equal to k
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I n2
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p.275
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17906
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To explain numbers you must also explain cardinality, the counting of things
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II
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p.278
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9899
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The successor of x is either x and all its members, or just the unit set of x
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II
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p.278
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9900
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For Zermelo 3 belongs to 17, but for Von Neumann it does not
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II
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p.279
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9901
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Numbers can't be sets if there is no agreement on which sets they are
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II
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p.283
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9903
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Number words are not predicates, as they function very differently from adjectives
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II
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p.284
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9904
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The set-theory paradoxes mean that 17 can't be the class of all classes with 17 members
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III
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p.286
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9905
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Identity statements make sense only if there are possible individuating conditions
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IIIB
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p.290
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9906
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If ordinal numbers are 'reducible to' some set-theory, then which is which?
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IIIC
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p.290
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9908
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The job is done by the whole system of numbers, so numbers are not objects
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IIIC
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p.290
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9907
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If any recursive sequence will explain ordinals, then it seems to be the structure which matters
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IIIC
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p.291
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9909
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The number 3 defines the role of being third in a progression
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IIIC
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p.292
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9911
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Number words no more have referents than do the parts of a ruler
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IIIC
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p.292
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9910
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Number-as-objects works wholesale, but fails utterly object by object
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IIIC
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p.294
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9912
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There are no such things as numbers
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p.285
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p.581
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8925
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Mathematical objects only have properties relating them to other 'elements' of the same structure
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p.301
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p.301
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9938
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How can numbers be objects if order is their only property? [Putnam]
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p.8
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17927
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Realists have semantics without epistemology, anti-realists epistemology but bad semantics [Colyvan]
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Intro
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p.403
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9935
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Mathematical truth is always compromising between ordinary language and sensible epistemology
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III
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p.412
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9936
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The platonist view of mathematics doesn't fit our epistemology very well
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