So I read that page, and I think the explanation is even worse than what I interpreted it to be the first time.
What they call 'blindly manipulating symbols' I call 'automation'. That is the entire point of computers in the first place, to very quickly perform tasks so that a human doesn't have to. That means describing the solution to those tasks. This is no different than mathematicians using equations. We don't need to do the entire process of solving every problem if they are the same except for changes in some parameters, we can write an equation and plug the parameters in. In software, we can write an algorithm/program that solves some specific problem, so long as the solution is the same and only the inputs are changing. Since programming languages support branching, we can make these algorithms as general as we think is useful.
