Consider this weird thought experiment I was reading about in Peter Van Inwagen's book, Material Beings.
Imagine yourself as a freshly fertilized egg. Call it A. The single cell then becomes two cells, A and B. Now imagine that B fails to make it and A goes on to divide making A and C and so on. After grieving the loss of B it is reasonable to assume that A is you. But what if rather than A surviving, B survives instead? It would be equally reasonable to think that B was you. But now suppose that both A and B survive but that they become detached and go on to be identical twins. It appears that you would be both A and B. But that is absurd.
Why is it weird? Surely whichever one survives is you and so what's the problem? Well, consider Leibniz law. Leibniz law states that for any objects X and Y, if X is identical to Y then X has all the properties that Y has and Y has all the properties that X has. Furthermore, the law implies that X is always going to have to have the same properties as Y and vice versa. However, the problem seems to show that this law is not true or leads to bizarre results.
Okay, so you are back to being a single cell, A. Now, before becoming A and B and C, both B and C were identical to A (A=B, A=C). After A becomes A and B, A was not identical to B (~A=B), so even though A and B were at one time identical they became non-identical, so Leibniz law is false. But that doesn't seem right either. Surely something that is identical is always identical.
Here is a possible solution: "one might suggest that B and C did not really exist before the split, that they came into existence then. If they did not exist before the split, then they were not A before the split. So it's not the case that B=C before the split."(Logic by Graham Priest, 90).
Priest doesn't think that works either. Apart from the obvious and distasteful conclusion that you cease to exist and then begin to exist continuously whenever there is a change of properties, there is no reason to say that B and C are the new you. Rather, one thinks of identity in terms of somesuch entity whose properties change over time. We rely upon an assumption that every object is identical to itself. The identity relation is so obviously true: every object bears the identity relation to itself and nothing else. If identity relations do not hold when there are changes in properties then are no identity relations. And that is equally absurd.
Another solution might be to say that B and C are actually just parts of A. Thus when cells divide they are really just parts of a whole separating. But single cells don't have parts that are cells so that doesn't work either.
Alternatively, one might suggest that in the case of a single cell human the cell in question has a corresponding soul. A soul is not divisible and therefore endures without the complications facing cells. Though this might count as a way out for our human situation try the same experiment with an amoeba or some other single cell organism.
Van Inwagen supposes that the problem invokes absurdities because identity is a vague notion. What is it for some one thing to maintain identity through time invokes vagueness. To say that something is vague is not necessarily to merely punt, but it is to say that some metaphysical reality escapes human conceptual grasp; it has what's called "intrinsic vagueness" attached to it. One is justified in appealing to vagueness if, though one is applying reasonable assumptions to a problem (such as the law of identity), there just seems to be no non-vague answer.
On a slightly conjectural note: is reality easy to understand? On the one hand, perhaps we're inclined to say yes in the sense that we appear to be able to understand what we need. Scientists are often optimistic about understanding the physical nature of reality according to some theory. On the other hand, people say that while physical reality is easy to understand any kind of non-physical reality is incomprehensible. That is naive. Reality--physical and non--is much more mysterious than we first appreciate. I think this points to a transcendent reality that stands behind reality, a divine mind that comprehends reality perfectly. His capacity to comprehend so outstrips ours that to him nothing is intrinsically vague. Though he designs us to comprehend some of reality he also designs us with inbuilt limitations. When we reach them we should utter praise to the one who makes us and our world so gloriously deep!