Yep. Penrose's argument goes something like this (oversimplified): 1) Turing machines can't solve the halting problem. 2) Humans can solve the halting problem.

Yep. Penrose's argument goes something like this (oversimplified): 1) Turing machines can't solve the halting problem. 2) Humans can solve the halting problem. 3) Therefore, humans can't be Turing machines. There are two problems with this line of reasoning. The first problem is that 2) is unsubstantiated. A human can see immediately that 15 = 3 x 5, but this doesn't mean that we have a magical ability to factor integers instantly. By the frivolous theorem of arithmetic, almost all natural numbers are very, very, very large. We have some ability to analyze formal systems, but the systems we have managed to understand are very simple in the grand scheme of things. Almost all formal systems are very, very, very complex, and there is no evidence that we can understand them all and/or that our understanding process actually is correct (consistent) when extended indefinitely. Penrose's actual argument is of course more technical, and others have made more precise rebuttals. The second problem is that the argument does not solve the problem of consciousness. Scott Aaronson put it nicely: "Even if we supposed the brain was solving a hard computational problem, it's not clear why that would bring us any closer to understanding consciousness. If it doesn't feel like anything to be a Turing machine, then why does it feel like something to be a Turing machine with an oracle for the halting problem?"