[foldr]

>But in this last case there's absolutely no way to tell there's a difference between 2 and 3, so they may as well be just considered as one thing.

Yes, that's the point. But then you lose the transitivity property, since although 2 and 3 are indistinguishable, 3 and 4 are indistinguishable, and 4 and 5 are indistinguishable, 2 and 5 are not. So the kind of operational definition of identity you have in mind yields a relation that's so radically unlike the standard characterization of the identity relation that I don't see any reason to call it "identity" at all.

Here's one way of drawing this out. Suppose that X linearly increases from 2 to 5 over a period of 3 seconds. Do we really want to say that there was no change in the value of X between t=0 and t=1, no change between t=1 and t=2, no change between t=2 and t=3, and yet a change between t=0 and t=3? (?!)

As far as I understand you, you have some kind of positivist skepticism about non-operationalizable notions, and so you want to come up with some kind of stand-in for identity which can play largely the same role in philosophical/scientific discourse as the ordinary, non-operationalizable notion of identity. That's a coherent project, but it rests on assumptions that anyone who's interested in P-zombies is likely to reject.

[TeMPOraL]

> Here's one way of drawing this out. Suppose that X linearly increases from 2 to 5 over a period of 3 seconds. Do we really want to say that there was no change in the value of X between t=0 and t=1, no change between t=1 and t=2, no change between t=2 and t=3, and yet a change between t=0 and t=3? (?!)

Yeah, I get that, but what I meant in my previous comment is that you either limit the domain of t to 0-3 (and X to 2-5) and there is indeed no way to tell the change between t=2 and t=3, or you don't limit yourself to that test and can distinguish the intermedate values by means of the trick I described before. In other words, either you have transitive identity or you have all the reasons to treat non-transitive cases as one (if the identity test is like the one I described in my previous comment).

> positivist skepticism about non-operationalizable notions

I think it's too late in the night for me to understand this, I'll need to come back to it in the morning. Could you ELI5 to me the meaning of "non-operationalizable" in this context?

Again, thanks for making me think and showing me the limits of my understanding.

[foldr]

>Again, thanks for making me think and showing me the limits of my understanding.

Yes this was a fun discussion, thanks.

Your objection stands if you have (and know you have) at least one instance of every value for the quantity. So suppose that we are given a countably infinite set of variables and told that each integer is denoted by at least one of these variables, and then further given a function over pairs of variables f(x,y), such that f(x,y) = 1 if x and y differ by less than 3 and = 0 otherwise. Then, yes, we can figure out which variables are exactly identical to which others.

However, I would regard this as irrelevant scenario in the sense that we could never know, via observation, that we had obtained such a set of variables (even if we allow the possibility of making a countably infinite number of observations). Suppose that we make an infinite series of observations and end up with at least one variable denoting each member of the following set (with the ellipses counting up/down to +/-infinity):

    ...,0,2,3,4,5,6,7,9,...

In other words, we have variables with every integer value except 1 and 8. Then for any variable x with the value 4 and variable y with the value 5, f(x,z) = f(y,z) for all variables z. In other words, there'll be no way to distinguish 4-valued variables from 5-valued variables. It's only in the case where some oracle tells us that we have a variable for every integer value that we can figure out which variables have exactly the same values as which others.