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Originally Posted by asaris
But kd4, could the quantities represented by those numbers be such that 3 + 2 = 6?
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Sure. As long as other additions keep the system consistent. And you can define consistency however you like, so you can really do anything at all.
Modern math considers numbers as we've all learned about them to be a particular example of a much more general type of object. For example, the integers form a
ring, and the real numbers are a type of
field. If you follow those two links and some of the pages linked from them, you'll start to get an idea of the types of things that you could come up with if you wanted to. Of course, you could say that those definitions aren't general enough either, but then you're really better off not using + and * signs anymore if you want anybody to read what you're doing.
All of these concepts are supposed to be completely abstract. The fact that we commonly use numerical quantities (along with the standard addition and multiplication operations) in describing the physical world is a completely separate philosophical problem.