I was unaware until yesterday that there was such thing as the "Thermodynamic Imperative." It's been given in different form, but here are a few:
Bayliss' Free Energy Imperative
"Waste not Free Energy; treasure it and make the best use of it."
Lindsey's Thermodynamic Imperative
"While we do live, we ought always to act in all things in such a way as to produce as much order in our environment as possible."
Both the Free Energy Imperative and the Thermodynamic Imperative seem to be a little bit off, but generally close to what we actually should be doing. What I like about the first version is that we should not waste free energy. The problem is that it doesn't tell you how to make best use of "free energy", i.e. exergy. The second version tells you how to make use of free energy (to produce order.) But as we know, there is no thermodynamic quantity for "order." In fact, there's no agreement on what "order" is. So, the second version leaves us hanging. Worse is the fact that the real motivation behind the second version is that Lindsey wants us to minimize the production of entropy in the universe. (Since then, there's been all sorts of pseudo-science trying to link entropy with pollution or waste exergy with pollution. This is just silly. One life-form's "pollution" is another life-form's "cornupopia." There is no thermodynamic variable that corresponds to "pollution" because there is no thermodynamic variable that changes value depending on whether we are talking about humans or vultures or bacteria.)
In general, my issue with the Thermodynamic Imperative (as given above) is that it doesn't mention what the real goal of life is:
The universe is currently not-in-thermodynamic equilibrium, though it is in the process of getting closer to equilibrium with every irreversible process (such as diffusion of heat, mass, charge, etc...). Life is a means of increasing the entropy of the universe at a faster rate than if there were no life. Increasing the entropy of the universe brings the universe closer to equilibrium. Since this is the goal of life, then it too should be the goal of human life.
The problem is that there is no way to calculate how to maximize the production of entropy in systems far-from-equilibrium (such as the Earth and the universe in general.) This means that there is no real Thermodynamic Imperative, since we don't know how to maximize the production of entropy. All we can do is come up with algorithms that helps us reach the goal.
And therefore, there is no way to prove that one algorithm is better than another.
The Thermodynamic Imperative is one algorithm for how to live life, but it doesn't seem to be a very good one because it suggests the opposite of what we should be doing. It asks us to slow down the increase of entropy in the universe.
The algorithm that seems the best at maximizing the entropy production in the universe is:
"Maximize the average rate of return on work invested."
Although this doesn't seem related to maximizing entropy production, what humans have found out is that maximizing work invested causes an exponential increase in the amount of work available, and along the way, an exponential increase in the amount of entropy generated, even though at every step along the way we tried to minimize entropy production so as to maximize the work done by the system (e.g. power plant.)
This is a Catch-22 situation. In order to maximize the global entropy production, we need to minimize the production of entropy at each step of the way to produce work. All the while, we continue to invest the work we generated into producing more work.
So, in conclusion, if we find sources of "free energy" (i.e. exergy or money), we should not waste it. Instead, we should put that "free energy" (i.e. exergy or money) to use in building a device that can convert "free energy" (i.e. exergy or money) into even more "free energy" (i.e. exergy or money). And then repeat over and over again.
This sounds simple. The problem is that there's no way to calculate the best route to maximize the average rate of return on work invested (ARROWI). So, we are left with having to make simplifications to the equation so that the optimization calculations don't consume more work than can be derived from the maximization.
Here's the recursive paradox.
The equation for maximizing the "average rate of return on work invested" calls itself as a function because that act of calculating consumes work and must be included in the equation.
One way to get around this paradox is to not include the work spent in calculating how to maximize ARROWI. This might be safe in some cases (such as powerplant design), but it highlights the fact that there is no right way to calculate how to maximize the entropy production rate in the universe.
Any sufficiently powerful algebra (that is capable of referring back to itself can not be both consistent and complete. (Godel's incompleteness theorem...paraphrased)
Similarly, any sufficiently power algorithm for maximizing ARROWI or entropy production rate is capable of referring back to its, and therefore can't be solved exactly.