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Single Idea 9901
[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
]
Full Idea
The fact that Zermelo and Von Neumann disagree on which particular sets the numbers are is fatal to the view that each number is some particular set.
Gist of Idea
Numbers can't be sets if there is no agreement on which sets they are
Source
Paul Benacerraf (What Numbers Could Not Be [1965], II)
Book Ref
'Philosophy of Mathematics: readings (2nd)', ed/tr. Benacerraf/Putnam [CUP 1983], p.279
A Reaction
I agree. A brilliantly simple argument. There is the possibility that one of the two accounts is correct (I would vote for Zermelo), but it is not actually possible to prove it.
The
24 ideas
from 'What Numbers Could Not Be'
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9151
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Benacerraf says numbers are defined by their natural ordering
[Benacerraf, by Fine,K]
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13891
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To understand finite cardinals, it is necessary and sufficient to understand progressions
[Benacerraf, by Wright,C]
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8697
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Disputes about mathematical objects seem irrelevant, and mathematicians cannot resolve them
[Benacerraf, by Friend]
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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
[Benacerraf, by Lowe]
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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
[Benacerraf]
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9898
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We can count intransitively (reciting numbers) without understanding transitive counting of items
[Benacerraf]
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17903
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Someone can recite numbers but not know how to count things; but not vice versa
[Benacerraf]
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9897
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The application of a system of numbers is counting and measurement
[Benacerraf]
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17906
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To explain numbers you must also explain cardinality, the counting of things
[Benacerraf]
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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
[Benacerraf]
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9900
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For Zermelo 3 belongs to 17, but for Von Neumann it does not
[Benacerraf]
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9901
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Numbers can't be sets if there is no agreement on which sets they are
[Benacerraf]
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9903
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Number words are not predicates, as they function very differently from adjectives
[Benacerraf]
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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
[Benacerraf]
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9905
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Identity statements make sense only if there are possible individuating conditions
[Benacerraf]
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9906
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If ordinal numbers are 'reducible to' some set-theory, then which is which?
[Benacerraf]
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9908
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The job is done by the whole system of numbers, so numbers are not objects
[Benacerraf]
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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
[Benacerraf]
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9909
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The number 3 defines the role of being third in a progression
[Benacerraf]
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9911
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Number words no more have referents than do the parts of a ruler
[Benacerraf]
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9912
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There are no such things as numbers
[Benacerraf]
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9910
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Number-as-objects works wholesale, but fails utterly object by object
[Benacerraf]
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8925
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Mathematical objects only have properties relating them to other 'elements' of the same structure
[Benacerraf]
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9938
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How can numbers be objects if order is their only property?
[Benacerraf, by Putnam]
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