The case for the humanities is easy:
- Science is about how the physical world works.
- Engineering is about how to get the physical world to do what you want.
- The humanities are about deciding what you want in the first place.
Say life is about finding the shortest path through a graph. Science tells you what the edges of the graph are -- what nodes are connected to what other nodes. Engineering gives you a shortest-path algorithm (say, Dijkstra's). The humanities tell you what node to find the shortest path to
A control-theoretic perspective:
More generally, the world has a state x(t), and science gives us the transition model -- the function f such that,
dx/dt = f(x,u)
where u(t) is our control input to the world at time t. This is Newton's Theory of Universal Gravitation, or Einstein's Theory of General Relativity, or whatever.
Engineering is about the following problem: Given a functional V that takes a state trajectory and returns a cost for that trajectory (this summarizes our opinion about what we want and what we do not want), solve (or approximately solve) the following optimization problem:
dx/dt = f(x,u)
x(0) = x0
In other words: Having decided what I want (V), and having figured out how the universe will react to my actions (f), figure out how to make the universe do what I want.
The humanities are about deciding what functional V to use. Science can't give it to you: It's an input to this whole thing.
The above formulation can be tweaked a little -- for instance, there is no uncertainty involved -- but it captures the gist of things.