by alecto » December 26th, 2008, 10:16 pm
Two questions:
Is it true that a horse is an animal?
Is it true that a cat is an animal?
If both are true, then 1 cat plus 1 horse are 2 animals, and the abstraction 1+1 = 2 from this sentence (and the plethora of others constructed in the same fashion) are true. This is what most people mean by 1+1 = 2. No one expects that the M&M's, pizzas, pets,a nd other things that they count are identical. The meaning of 1+1 = 2 is precisely the abstraction, therefore, of the notion of combination, separated from the issue of non-identity.
Now here are some problems:
The sentence 1x + 1x = 2x where x is a real object (e.g. 1 cat + 1 cat are 2 cats) is necessarily incomplete. I have to not say something about one or more of the cats, otherwise the sentence is false.
True: 1 cat + 1 cat = 2 cats.
Incomplete: Fluffy and Thumper are cats.
The issue of categories has been invoked to solve the problem, but it really just promotes it. What does it mean to say that a certain object is a horse? I happen to think that when it comes right down to it, most sentences are false or incomplete. Are horses large or small? Are swans white? Are people smart?
That having been said, it is impossible to say that no two objects are identical, and there is a possibility that certain phenomena of atomic physics (e.g. the Pauli Exclusion Principle) arise because one cannot force both intrinsic identicalness and locational identicalness on some kinds of objects at the same time. Electrons, for example, are supposed to all be intrinsically identical, therefore it is impossible to stack them in the same place, making their locations identical. This is why we don't fall to the center of the earth and why we cannot pass through walls. On the other hand, certain other kinds of objects ignore this rule, e.g. photons. This is what is called the fermion/boson distinction. (Bosons stack; fermions do not.)
This does not eliminate the problem of non-addibility that is being discussed, except possibly for bosons.
The point I want to finish with is this: unless every noun in your sentence is a proper noun, then your sentence is a kind of abstraction. All of mathematics is abstraction of this kind. Probably, sentences with direct pointers to individuals are implicit abstractions (e.g. "John is my father.") because they rely on understanding of abstract groups (here "fathers") to function. When you say 1+1 = 2 you have to lose definition of the objects involved, but exactly the same thing happens in most sentences, like "John is my father," "Cows are animals", and "God is good."
Sentio ergo est.