Our theory provides a way to distinguish between them. Such as predicates P and Y, where Px means "x is purple" and Yx means "x is yellow". Our system also provides the means to do arithmetic with either of them, separately.
Although function symbols don't capture the meaning of functions as they are usually used in modern mathematics (which I'll get to), one can nonetheless decide to use them, and there are even practical reasons for doing so sometimes. In this case, I'll do that -- or, rather, describe a system that does -- because you understand function symbols.
First, consider the purple plus sign and the yellow plus sign.
Both are primitive function symbols of our system.
Both are used in infix position.
The purple plus sign adds purple integers. The yellow plus sign adds yellow integers.
You might wonder if it is really reasonable to have any kind of plus sign in the signature of one's system. After all, surely one can define + in terms of something simpler. This is true, and it is often best to take that approach. But this system (that I am making up) has it as a primitive--or rather, has them, the separate purple and yellow plus signs.
Similarly, we have primitive purple times signs and yellow times signs.
For multiplying purple numbers and yellow numbers respectively.
We can use definite descriptions to express "the purple additive identity, purple 0" and "the yellow additive identity, yellow 0" and "the purple multiplicative identity, purple 1" and "the yellow multiplicative identity, yellow 1" and "the purple additive inverse of x" and "the yellow additive inverse of x" and "the purple multiplicative inverse of x" and "the yellow multiplicative inverse of x".
Sometimes these definite descriptions will fail and so atomic sentences in which they appear as arguments will be false. That's fine. For example, a yellow integer has no purple additive inverse -- or, with respect to the notation we would likely adopt for convenience: it is undefined to apply a purple unary minus sign to a yellow integer.
(Or a yellow unary minus to a purple integer.)
Also, though this situation has arisen due to me not thinking things through well enough, it's actually handy: No yellow integer or purple integer, other than yellow 1, yellow -1, purple 1, and purple -1, has a yellow multiplicative inverse or a purple multiplicative inverse.
(Because this is the integers, so we don't have things like 1/2 of any color.)
Similarly, you cannot take the purple sum of yellow integers or the yellow sum of purple integers.
That's undefined: x + y fails to refer to any thing except when x and y have the same color and the + for that color is being used.
We can express these restrictions with axioms.
The purple sum of yellow integers in this system is undefined in the same sense of "undefined" that division by zero is undefined in the real numbers.
Now, suppose this is the system we use one day while picking apples at an orchard.
And we solve various arithmetic problems involving numbers of apples.
But you use the purple integer and I use the yellow integers.
Every claim you make that just uses purple integers and purple function symbols has a corresponding claim I can make that just uses yellow integers and yellow function symbols.
The arithmetic does not actually use the color.
The knowledge known about the purple integers applies to the yellow integers, and vice versa.
We might disagree about which one is really the integers, but that's sort of missing the point. Knowledge about integers is knowledge about any bunch 'o things and operations on them that behave like integers.
And concretely, in the world, there's plenty of stuff that works like integers, or at least that we can model (in the sense of modeling in the sciences) with integers. That's what makes integers useful and interesting.
If you're trying to express insights about integers, asserting or denying that "the number 7 is yellow" does not do that.
Does my analogy between dimensional analysis and this make sense?
Also, is the number 7 a set?
