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That's a summary of Godel's theorem, which nobody disputes, not of Penrose's argument that it implies computers cannot emulate human intelligence.

I'm really not trying to evade further discussion. I just don't think I can sum that argument up. It starts with basically "we can perceive the truth not only of any particular Godel statement, but of all Godel statements, in the abstract, so we can't be algorithms because an algorithm can't do that" but it doesn't stop there. The obvious immediate response is to say "what if we don't really perceive its truth but just fool ourselves into thinking we do?" or "what if we do perceive it but we pay for it by also wrongly perceiving many mathematical falsehoods to be true?". Penrose explored these in detail in the original book and then wrote an entire second book devoted solely to discussing every such objection he was aware of. That is the meat of Penrose' argument and it's mostly about how humans perceive mathematical truth, argued from the point of view of a mathematician. I don't even know where to start with summarising it.

For my part, with a vastly smaller mind than his, I think the counterarguments are valid, as are his counter-counterarguments, and the whole thing isn't properly decided and probably won't be for a very long time, if ever. The intellectually neutral position is to accept it as undecided. To "pick a side" as I have done is on some level a leap of faith. That's as true of those taking the view that the human mind is fundamentally algorithmic as it is of me. I don't dispute that their position is internally consistent and could turn out to be correct, but I do find it annoying when they try to say that my view isn't internally consistent and can never be correct. At that point they are denying the leap of faith they are making, and from my point of view their leap of faith is preventing them seeing a beautiful, consistent and human-centric interpretation of our relationship to computers.

I am aware that despite being solidly atheist, this belief (and I acknowledge it as such) of mine puts me in a similar position to those arguing in favour of the supernatural, and I don't really mind the comparison. To be clear, neither Penrose nor I am arguing that anything is beyond nature, rather that nature is beyond computers, but there are analogies and I probably have more sympathy with religious thinkers (while rejecting almost all of their concrete assertions about how the universe works) than most atheists. In short, I do think there is a purely unique and inherently uncopyable aspect to every human mind that is not of the same discrete, finite, perfectly cloneable nature as digital information. You could call it a soul, but I don't think it has anything to do with any supernatural entity, I don't think it's immortal (anything but), I don't think it is separate from the body or in any sense "non-physical", and I think the question of where it "goes to" when we die is meaningless.

I realise I've gone well beyond Penrose' argument and rambled about my own beliefs, apologies for that. As I say, I struggle to summarise this stuff.