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Single Idea 15912
[filed under theme 6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
]
Full Idea
Counting a set produces a well-ordering of it. Conversely, if one has a well-ordering of a set, one can count it by following the well-ordering.
Gist of Idea
Counting results in well-ordering, and well-ordering makes counting possible
Source
Shaughan Lavine (Understanding the Infinite [1994], III.4)
Book Ref
Lavine,Shaughan: 'Understanding the Infinite' [Harvard 1994], p.53
A Reaction
Cantor didn't mean that you could literally count the set, only in principle.
The
32 ideas
with the same theme
[procedure for finding the size of a group of things]:
17861
|
Two men do not make one thing, as well as themselves
[Aristotle]
|
646
|
When we count, are we adding, or naming numbers?
[Aristotle]
|
19584
|
Whoever first counted to two must have seen the possibility of infinite counting
[Novalis]
|
14775
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Numbers are just names devised for counting
[Peirce]
|
9824
|
In counting we see the human ability to relate, correspond and represent
[Dedekind]
|
14424
|
Numbers are needed for counting, so they need a meaning, and not just formal properties
[Russell]
|
14120
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Counting explains none of the real problems about the foundations of arithmetic
[Russell]
|
17424
|
Counting puts an initial segment of a serial ordering 1-1 with some other entities
[Sicha]
|
17425
|
To know how many, you need a numerical quantifier, as well as equinumerosity
[Sicha]
|
9898
|
We can count intransitively (reciting numbers) without understanding transitive counting of items
[Benacerraf]
|
17903
|
Someone can recite numbers but not know how to count things; but not vice versa
[Benacerraf]
|
4045
|
Children may have three innate principles which enable them to learn to count
[Goldman]
|
17447
|
Parsons says counting is tagging as first, second, third..., and converting the last to a cardinal
[Parsons,C, by Heck]
|
17818
|
How many? must first partition an aggregate into sets, and then logic fixes its number
[Yourgrau]
|
17822
|
Nothing is 'intrinsically' numbered
[Yourgrau]
|
16014
|
It is controversial whether only 'numerical identity' allows two things to be counted as one
[Noonan]
|
17812
|
Finite cardinalities don't need numbers as objects; numerical quantifiers will do
[White,NP]
|
3907
|
Could you be intellectually acquainted with numbers, but unable to count objects?
[Scruton]
|
17448
|
In counting, numerals are used, not mentioned (as objects that have to correlated)
[Heck]
|
17455
|
Is counting basically mindless, and independent of the cardinality involved?
[Heck]
|
17456
|
Counting is the assignment of successively larger cardinal numbers to collections
[Heck]
|
10712
|
If set theory didn't found mathematics, it is still needed to count infinite sets
[Potter]
|
7466
|
Mesopotamian numbers applied to specific things, and then became abstract
[Watson]
|
15912
|
Counting results in well-ordering, and well-ordering makes counting possible
[Lavine]
|
17694
|
Some non-count nouns can be used for counting, as in 'several wines' or 'fewer cheeses'
[Laycock]
|
17695
|
Some apparent non-count words can take plural forms, such as 'snows' or 'waters'
[Laycock]
|
23460
|
To count, we must distinguish things, and have a series with successors in it
[Morris,M]
|
23451
|
Counting needs to distinguish things, and also needs the concept of a successor in a series
[Morris,M]
|
23452
|
Discriminating things for counting implies concepts of identity and distinctness
[Morris,M]
|
17439
|
There is no deep reason why we count carrots but not asparagus
[Koslicki]
|
17433
|
We can still count squares, even if they overlap
[Koslicki]
|
17462
|
A single object must not be counted twice, which needs knowledge of distinctness (negative identity)
[Rumfitt]
|