Isn't this interesting? Intuitively, elsewhere in math, you might think that qualitative indicernibility always suffices for numerical identity. Well, perhaps this needs discussion. In some sense the imaginary numbers ι and -ι are qualitatively indiscernible, but numerically distinct. However, arguably they're not *fully* qualitatively indiscernible. They don't both bear all the same relations to ι for instance. But then, if we include numerical identity as a relation, then `ycell` and `xcell` don't both bear all the same relations to `ycell`, either. Yet there is still a useful sense in which they can be understood to be qualitatively equal---at least, at a given stage in a computation.
Isn't this interesting? Intuitively, elsewhere in math, you might think that qualitative indicernibility always suffices for numerical identity. Well, perhaps this needs discussion. In some sense the imaginary numbers ι and -ι are qualitatively indiscernible, but numerically distinct. However, arguably they're not *fully* qualitatively indiscernible. They don't both bear all the same relations to ι for instance. But then, if we include numerical identity as a relation, then `ycell` and `xcell` don't both bear all the same relations to `ycell`, either. Yet there is still a useful sense in which they can be understood to be qualitatively equal---at least, at a given stage in a computation.