This argument is, however, nonsense. The human capacity for abstract reasoning over mathematical models is in principle a fully general intelligent behaviour, as the scientific revolution has shown: there is no aspect of the natural world which has remained beyond the reach of human understanding, once a sufficient amount of evidence is available. The wave-particle duality of quantum physics, or the 11-dimensional space of string theory may defy human intuition, i.e. our built-in intelligence. But we have proven ourselves perfectly capable of understanding the logical implications of models which employ them. We may not be able to build intuition for how a super-intelligence thinks. Maybe—that's not proven either. But even if that is so, we will be able to reason about its intelligent behaviour in advance, just like string theorists are able to reason about 11-dimensional space-time without using their evolutionarily derived intuitions at all.

This may be retreating to the motte's bailey, so to speak, but I don't think anyone seriously thinks that a superintelligence would be literally impossible to understand. The worry is that there will be such a huge gulf between how superintelligences reason versus how we reason that it would take prohibitively long to understand them.

I think a laptop is a good example. There probably isn't any single human on earth that knows how to build a modern laptop from scratch. There's are computer scientists that know how the operating system is put together--how the operating system is programmed, how memory is written and retrieved from the various buses; there are other computer scientists and electrical engineers who designed the chips themselves, who arrayed circuits efficiently to dissipate heat and optimize signal latency. Even further, there are material scientists and physicists who designed the transistors and chip fabrication processes, and so on.

So, as an individual human, I don't know what it's like to know everything about a laptop all at once in my head, at a glance. I can zoom in on an individual piece and learn about it, but I don't know all the nuances for each piece--just a sort of executive summary. The fundamental objects with which I can reason have a sort of characteristic size in mindspace--I can imagine 5, maybe 6 balls moving around with distinct trajectories (even then, I tend to group them into smaller subgroups). But I can't individually imagine a hundred (I could sit down and trace out the paths of a hundred balls individually, of course, but not all at once).

This is the sense in which a superintelligence could be "dangerously" unpredictable. If the fundamental structures it uses for reasoning greatly exceed a human's characteristic size of mindspace, it would be difficult to tease out its chain of logic. And this only gets worse the more intelligent it gets.

Now, I'll grant you that the lesswrong community likes to sweep under the rug the great competition of timescales and "size"scales that are going on here. It might be prohibitively difficult, for fundamental reasons, to move from working-mind-RAM of size 5 to size 10. It may be that artificial intelligence research progresses so slowly that we never even see an intelligence explosion--just a gently sloped intelligence rise over the next few millennia. But I do think it's a maybe not a mistake but certainly naiive to just proclaim, "Of course we'll be able to understand them, we are generalized reasoners!".

Edit: I should add that this is already a problem for, ironically, computer-assisted theorem proving. If a computer produces a 10,000,000 page "proof" of a mathematical theorem (i.e., something far longer than any human could check by hand), you're putting a huge amount of trust in the correctness of the theorem-proving-software itself.