The author doesn't seem to be very familiar with the Church side of the Church-Turing thesis. That is where the "limits of computation" notion comes from.
"All total functions are recursively enumerable"
For a long while I've had the opinion that the major conceit about Turing machines is, yes they can do things that humans suck at. However it's likely they are a pretty hackneyed way to accomplish what neuron based systems are really good at.