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### 6. Mathematics / A. Nature of Mathematics / 3. Numbers / p. Counting

#### [procedure for finding the size of a group of things]

48 ideas
 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]
 17612 Arithmetic is just the consequence of counting, which is the successor operation [Dedekind]
 9824 In counting we see the human ability to relate, correspond and represent [Dedekind]
 15916 Frege's one-to-one correspondence replaces well-ordering, because infinities can't be counted [Frege, by Lavine]
 17437 Non-arbitrary division means that what falls under the concept cannot be divided into more of the same [Frege, by Koslicki]
 17438 Our concepts decide what is countable, as in seeing the leaves of the tree, or the foliage [Frege, by Koslicki]
 17428 Frege says counting is determining what number belongs to a given concept [Frege, by Koslicki]
 17446 Counting rests on one-one correspondence, of numerals to objects [Frege]
 9582 Husserl rests sameness of number on one-one correlation, forgetting the correlation with numbers themselves [Frege]
 17444 Husserl said counting is more basic than Frege's one-one correspondence [Husserl, by Heck]
 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]
 12154 Are 'word token' and 'word type' different sorts of countable objects, or two ways of counting [Geach, by Perry]
 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]
 9852 We understand 'there are as many nuts as apples' as easily by pairing them as by counting them [Dummett]
 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]
 17518 Counting 'coin in this box' may have coin as the unit, with 'in this box' merely as the scope [Ayers]
 17516 If counting needs a sortal, what of things which fall under two sortals? [Ayers]
 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]
 17529 Maybe the concept needed under which things coincide must also yield a principle of counting [Wiggins]
 17530 The sortal needed for identities may not always be sufficient to support counting [Wiggins]
 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]
 13867 Instances of a non-sortal concept can only be counted relative to a sortal concept [Wright,C]
 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]
 17450 Understanding 'just as many' needn't involve grasping one-one correspondence [Heck]
 17451 We can know 'just as many' without the concepts of equinumerosity or numbers [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]
 17439 There is no deep reason why we count carrots but not asparagus [Koslicki]
 17427 Frege's 'isolation' could be absence of overlap, or drawing conceptual boundaries [Koslicki]
 17433 We can still count squares, even if they overlap [Koslicki]
 17434 We struggle to count branches and waves because our concepts lack clear boundaries [Koslicki]
 17462 A single object must not be counted twice, which needs knowledge of distinctness (negative identity) [Rumfitt]