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Single Idea 14424

[filed under theme 6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure ]

Full Idea

We want our numbers to be such as can be used for counting common objects, and this requires that our numbers should have a definite meaning, not merely that they should have certain formal properties.

Gist of Idea

Numbers are needed for counting, so they need a meaning, and not just formal properties

Source

Bertrand Russell (Introduction to Mathematical Philosophy [1919], I)

Book Ref

Russell,Bertrand: 'Introduction to Mathematical Philosophy' [George Allen and Unwin 1975], p.10

A Reaction

Why would just having certain formal properties be insufficient for counting? You just need an ordered series of unique items. It isn't just that we 'want' this. If you define something that we can't count with, you haven't defined numbers.

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 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 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]