Oi, another repost. With slight editing.
A “number” of a thing presupposes two qualities: separability and interchangeability. The former is an interpretation of sense data, and the second is subjective.
-Separability: We can’t say there are “two” things unless it may be agreed that they are not the same thing or two parts of the same thing.
-Interchangeability: We also must agree that the two things share characteristics of the same sort of thing (for our purposes).
There are, therefore, three sorts of objections a person can make to a numeric categorization: that the objects so categorized are not separable, are not interchangeable, or both. Because this is an abstract sort of thing, I’ll provide some examples of the objections:
The objects so categorized are not separable: Siamese twins. Two people with separate souls and minds (etc.), or one person? Also, two tree trunks with a common root system- two trees or two branches of the same tree? Depends on the definition of tree.
The objects are not interchangeable: This is a word I learned in history class. Apparently, it was a big deal when people learned you could mass-produce rifle stocks that would fit on a bunch of different rifles (see http://en.wikipedia.org/wiki/Interchangeable_parts). Similarly, we can decide (subjectively) that there are “two red cars”, rather than “one red car and one maroon car”.