It is an interesting book, and worth reading.

He argues, as you say, that we do things with our brains that can not be reproduced using algorithms. He bases the argument mainly, if I recall correctly, on the incompleteness theorems: we can apprehend certain statements in mathematics as “true” even though they can’t be proven within the system of definitions and axioms in which the statements are made.

However, this conclusion, whether it is ultimately defensible or not, is about replicating the human thought process with algorithms. It does not exclude the creation of machines that can do everything we can do. They just wouldn’t accomplish this through the execution of what we conventionally think of as computer programs.