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Single Idea 17906
[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers
]
Full Idea
I would disagree with Quine. The explanation of cardinality - i.e. of the use of numbers for 'transitive counting', as I have called it - is part and parcel of the explication of number.
Clarification
'Transitive counting' is counting objects
Gist of Idea
To explain numbers you must also explain cardinality, the counting of things
Source
Paul Benacerraf (What Numbers Could Not Be [1965], I n2)
Book Ref
'Philosophy of Mathematics: readings (2nd)', ed/tr. Benacerraf/Putnam [CUP 1983], p.275
A Reaction
Quine says numbers are just a progression, with transitive counting as a bonus. Interesting that Benacerraf identifies cardinality with transitive counting. I would have thought it was the possession of numerical quantity, not ascertaining it.
Related Idea
Idea 17905
Any progression will do nicely for numbers; they can all then be used to measure multiplicity [Quine]
The
31 ideas
from Paul Benacerraf
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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
[Benacerraf]
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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
[Benacerraf]
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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
[Benacerraf]
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13415
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An adequate account of a number must relate it to its series
[Benacerraf]
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17927
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Realists have semantics without epistemology, anti-realists epistemology but bad semantics
[Benacerraf, by Colyvan]
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9935
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Mathematical truth is always compromising between ordinary language and sensible epistemology
[Benacerraf]
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9936
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The platonist view of mathematics doesn't fit our epistemology very well
[Benacerraf]
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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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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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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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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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9900
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For Zermelo 3 belongs to 17, but for Von Neumann it does not
[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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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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9910
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Number-as-objects works wholesale, but fails utterly object by object
[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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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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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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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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