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For the mind to be impossible to model digitally, it seems there must be one of two things true about it: it processes on the quantum level, or there is a non-material component to it. I think it's Roger Penrose that argues the former, but from what I remember of his book about it, he was singularly unconvincing (although that doesn't mean he's wrong, he just might have the details wrong while the central idea is correct). The latter is of course the province of the religious, and might be summed up as "machines cannot have a soul."
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Indeed, there has to be something inherently non-deterministic in there for it to be impossible, and so far I'm not aware of anything that qualifies. The continual progress of knowledge should eventually fill in the gaps to a sufficient degree. It also doesn't make sense for consciousness to only be possible using a set of chemical and physical reactions contained in biomass, while a mathematical attempt would suffer from some sort of inherent limitation. I find it unlikely from an evolutionary standpoint that brains evolved to such a large size and then suddenly went "boink" on some quantum level giving rise to consciousness. A sufficiently detailed implementation of the inner workings of the brain should in theory have no problem duplicating all of the behaviour, the "soul" included. Obviously gaining the understanding of how it works is a monumental challenge, but not insurmountable.
Of course, the extension of this is two-fold:
First, the obvious immortality potential. I won't add anything other than I think this is many many years away (easily 100+).
Second, perhaps more troubling to some, you have no free will. I could make a computer model of your brain (i.e. "you"), sync it up, feed it the same inputs (sights, sounds, etc), and get exactly the same response you made in reality. So your "free will" decision to go buy pizza on a whim is based on: the state of your brain, the inputs it receives, and the structure ("programming"), and not anything metaphysical or otherwise unhindered by physics. That may fall under a "possible, but not practical" limitation, but in theory it's possible.