Some more thoughts on numbers

We have already seen how quantity, the "&" and the "or", is a precondition for the existence of mathematical operations.  But quantity, multiplicity, is a vague concept.  Many things exist, but how many? And how do you define a thing? Is it a molecule, an atom, a subatomic particle etc.?  No doubt the act of defining "a thing" determines the quantity, and how we define a thing is up to us.  And according to logical atomism, we define things by assigning names to objects.  It is thus clear that language, the ability to map names to objects, is a precondition for the counting of objects, and mathematical operations in general. And mathematics, the manipulation of numbers, involves one additional marvelous operation.  That of abstraction.   No longer is it necessary to think of an object or a group of objects linked to a name.  The number nine, for instance, may have been born by the act of counting of nine cats.  However, when performing the operation "9 + 3" it is no longer necessary to attach cats to the number nine.  And the result of that operation, "9 + 3" is the same whether we are talking about cats, hedge hogs or asterisks. Thus, the number 9 can be mapped to trillions of different things, so long as we are talking about nine of them.  And the results of these operations appear to be the same throughout the universe.  Thus, numbers themselves appear to have been born by the mapping of names to objects, and the subsequent abstraction of these objects.  And the mapping of names to objects requires reasonably sophisticated linguistic abilities, abilities that only humans seem to possess (though there are probably life forms in other parts of the universe that have such abilities.)