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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
Two men do not make one thing, as well as themselves [Aristotle]
When we count, are we adding, or naming numbers? [Aristotle]
Whoever first counted to two must have seen the possibility of infinite counting [Novalis]
Numbers are just names devised for counting [Peirce]
Arithmetic is just the consequence of counting, which is the successor operation [Dedekind]
In counting we see the human ability to relate, correspond and represent [Dedekind]
Frege's one-to-one correspondence replaces well-ordering, because infinities can't be counted [Lavine on Frege]
Non-arbitrary division means that what falls under the concept cannot be divided into more of the same [Koslicki on Frege]
Our concepts decide what is countable, as in seeing the leaves of the tree, or the foliage [Koslicki on Frege]
Frege says counting is determining what number belongs to a given concept [Koslicki on Frege]
Counting rests on one-one correspondence, of numerals to objects [Frege]
Husserl rests sameness of number on one-one correlation, forgetting the correlation with numbers themselves [Frege]
Husserl said counting is more basic than Frege's one-one correspondence [Heck on Husserl]
Numbers are needed for counting, so they need a meaning, and not just formal properties [Russell]
Counting explains none of the real problems about the foundations of arithmetic [Russell]
Are 'word token' and 'word type' different sorts of countable objects, or two ways of counting [Perry on Geach]
Counting puts an initial segment of a serial ordering 1-1 with some other entities [Sicha]
To know how many, you need a numerical quantifier, as well as equinumerosity [Sicha]
We understand 'there are as many nuts as apples' as easily by pairing them as by counting them [Dummett]
Someone can recite numbers but not know how to count things; but not vice versa [Benacerraf]
We can count intransitively (reciting numbers) without understanding transitive counting of items [Benacerraf]
Counting 'coin in this box' may have coin as the unit, with 'in this box' merely as the scope [Ayers]
If counting needs a sortal, what of things which fall under two sortals? [Ayers]
Children may have three innate principles which enable them to learn to count [Goldman]
Parsons says counting is tagging as first, second, third..., and converting the last to a cardinal [Heck on Parsons,C]
Maybe the concept needed under which things coincide must also yield a principle of counting [Wiggins]
The sortal needed for identities may not always be sufficient to support counting [Wiggins]
How many? must first partition an aggregate into sets, and then logic fixes its number [Yourgrau]
Nothing is 'intrinsically' numbered [Yourgrau]
It is controversial whether only 'numerical identity' allows two things to be counted as one [Noonan]
Instances of a non-sortal concept can only be counted relative to a sortal concept [Wright,C]
Finite cardinalities don't need numbers as objects; numerical quantifiers will do [White,NP]
Could you be intellectually acquainted with numbers, but unable to count objects? [Scruton]
In counting, numerals are used, not mentioned (as objects that have to correlated) [Heck]
Understanding 'just as many' needn't involve grasping one-one correspondence [Heck]
We can know 'just as many' without the concepts of equinumerosity or numbers [Heck]
Counting is the assignment of successively larger cardinal numbers to collections [Heck]
Is counting basically mindless, and independent of the cardinality involved? [Heck]
If set theory didn't found mathematics, it is still needed to count infinite sets [Potter]
Mesopotamian numbers applied to specific things, and then became abstract [Watson]
Counting results in well-ordering, and well-ordering makes counting possible [Lavine]
Some non-count nouns can be used for counting, as in 'several wines' or 'fewer cheeses' [Laycock]
Some apparent non-count words can take plural forms, such as 'snows' or 'waters' [Laycock]
There is no deep reason why we count carrots but not asparagus [Koslicki]
Frege's 'isolation' could be absence of overlap, or drawing conceptual boundaries [Koslicki]
We can still count squares, even if they overlap [Koslicki]
We struggle to count branches and waves because our concepts lack clear boundaries [Koslicki]
A single object must not be counted twice, which needs knowledge of distinctness (negative identity) [Rumfitt]