I'm writing this just to try to get my own thoughts straight, on a topic I know nothing about. And because commenter Ram asked me to. Read at your own risk.
I have a theory about water levels. According to my theory, the level of water in Lake Ontario will always be exactly the same on both sides of the Canada/US border.
Suppose it weren't the same. Suppose it were higher on the Canadian side. Then pressure would be higher on the Canadian side, and water would flow from high to low pressure, which would equalise the levels.
That was a counterfactual conditional statement (actually four counterfactual conditional statements). According to my theory, there never has been a time and never will be a time at which the water level is higher on the Canadian side of the border. I am asking what would happen if something were to happen which my theory says has never happened and will never happen. And my counterfactual conditional statements explain why it has never happened and will never happen. Because if it started to happen it would immediately stop happening and go back to where it was before.
But it is only a counterfactual relative to my theory. In the real world, there may indeed be times when the water level is higher on the Canadian side, due to wind or waves or rainfall, or something else that my simple theory assumes away. But my counterfactual conditional statements assume away wind and waves and rainfall too. It is my theory, and not the real world, that says it is counterfactual.
And yet, at the same time, that counterfactual conditional thought-experiment is an integral part of my theory. It explains why my theory is true. It explains why the water cannot be higher on the Canadian side, by supposing it were higher. Which is a bit of a paradox.
I stole this example from Michael Parkin, who used it as an example of a stability experiment ages ago. I've always liked it. And anybody reading this will also probably see what I am doing as checking to see if an equilibrium (same water level both sides of the border) is stable (does it return towards equilibrium if it's initially away from equilibrium).
Most people think of "equilibrium" as meaning "at rest", or "constant", or "balance of opposing forces", or something like that. But economists nowadays don't think of "equilibrium" like that. We don't even (necessarily) mean "quantity supplied equals quantity demanded". We don't even (necessarily) mean "an attractor towards which things will move regardless of initial conditions". There can be moving, non-market-clearing, history dependent, equilibria. All we really mean is: "that which the theory says will happen". The word "equilibrium" is almost redundant. But it's not totally redundant.
It's a useful word to use if we are checking to see whether something is or is not compatible with the theory: "No, that can't be an equilibrium, because there are $100 bills nobody is picking up" or "because there would be an excess demand for peanuts".
And it's a useful word if we are asking about stability. Because there we ask a counterfactual conditional question about what would happen if we were out of equilibrium.
But stability thought-experiments, like all counterfactual conditionals, are paradoxical. The theory itself denies the antecedent. So how can it be part of the theory? I think this is why some economists refuse to talk about stability. But understanding why Lake Ontario is the same level both sides of the border is as important as understanding that it is the same level both sides of the border. Stability is about what keeps it the same level both sides of the border. It's about what's stopping deviations from equilibrium from even starting, not what is happening when there is a deviation from equilibrium. Because there never are any deviations, according to the theory. It's about what is stopping incipient deviations from becoming actual deviations from equilibrium.