Enlighten us, if you don't mind, as to which results from complexity theory indicate that these things are not possible. By my reading none of them say anything of the sort.

All the results I know of don't apply to computers any more than they do to humans, who reason no less algorithmically than computers (even if our algorithms are currently more subtle); these results tend to say that certain magic cannot be performed without invoking new classes of computation, which has absolutely no bearing on whether computers can do the type of practical research humans have been engaging in for our entire history.

Then how do humans do it if we're Turing machines?