Photolysis: actually I think there's no way out of it. It totally destroys the concept of self. It's a beautifully infuriating paradox, largely because the damn thing is actually real. Patently my self does exist; in fact it's the only thing in the world I am sure of! And yet it can't.
I don't actually think your solution addresses the issue; if you analyse the wording it's actually nonsensical. The sentence, 'if I come along later and take half of these out whilst still maintaining the original form of the brain, you would still be you' assumes the existence of the referent; what are you referring to by 'you'? Brain A or brain B? And 'and creating another brain out of these removed parts' has the same problem; you refer to 'the other brain', but the whole problem is that you can't specify if that was brain A or brain B. The two brains will go on to live different lives, and there was perfect symmetry in their formation. Which is you? The former implies that we must choose, the latter implies that we cannot. There is no way out.
The thing about continuity also hides problems underneath. What is exactly meant by continuity? We've already conceded that the material cannot itself be important. What does destroying and then recreating mean? It doesn't actually mean destroying, because if you did that, you would have no way of 'causing' a new brain into existence. In fact the brain is as embodied in physical reality as ever before; it's stored in whatever data storage device you use in the interim. So there is as much a continuous causal link, ion one pushing ion two pushing ion three, as there is in ion one pushing ion two pushing nanobot pushing electron pushing transistor pushing electron pushing nanobot pushing ion three.
As to the Ship of Theseus, I think that can be clarified further. I don't think there's actually a distinction between these problems and the problem we're currently considering. We could for instance consider the ontological status of a computer program, which could be duplicated in the same way, giving the same problems of identity. But the thing is that these problems can be solved, if we just say that the ontological status of an object is just the embodiment of that concept in our brain. A ship isn't 'real', in the sense that fundamentally it's just an array of separate atoms. The concept of a ship as a singular object is just a mental construct. The duplication of the ship isn't 'real' in the atomic sense, it's just that the result is two objects which both create that concept of a ship in our brains. There's no paradox there. But of course, this falls apart when you consider your own consciousness. It doesn't make sense of consciousness to just be something in your consciousness, that's cyclical.
Sornaensis: doesn't work, I address that in the first bit. Your brain is constantly being 'destroyed'. Molecules are constantly entering your body and replacing those in your brain, which exit it. The actual matter of your brain is constantly changing. If you gradually replace each neuron with a functionally equivalent silicon chip, there is no difference. If you concede that your consciousness isn't constantly changing in normal circumstances, you are forced to concede that it won't change when you replace it with silicon.
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