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Single Idea 13891
[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers
]
Full Idea
Benacerraf claims that the concept of a progression is in some way the fundamental arithmetical notion, essential to understanding the idea of a finite cardinal, with a grasp of progressions sufficing for grasping finite cardinals.
Gist of Idea
To understand finite cardinals, it is necessary and sufficient to understand progressions
Source
report of Paul Benacerraf (What Numbers Could Not Be [1965]) by Crispin Wright - Frege's Concept of Numbers as Objects 3.xv
Book Ref
Wright,Crispin: 'Frege's Conception of Numbers' [Scots Philosophical Monographs 1983], p.117
A Reaction
He cites Dedekind (and hence the Peano Axioms) as the source of this. The interest is that progression seems to be fundamental to ordianls, but this claims it is also fundamental to cardinals. Note that in the first instance they are finite.
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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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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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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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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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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