A useful way to think about problem solving is to think about the diagnosis or description of the problem as coming in different resolutions. And here it is important to remember that it is not always helpful to aim for higher resolution – since what you gain may not be much, and the time and effort it takes to increase resolution may be significant.
An example:
Assume you are asked to determine what letter this is. Does it then matter at all if you have the left-most resolution or the right-most? Probably not, right? Yet, a lot of problem solving, theory building and analysis assumes that the right-most picture is better!
The challenge of resolution is especially acute when you are dealing with complex systems. Here. even a “coarse-grained” model can be extremely useful, since it allows you to see different things and step back from the picture. One of the best examples of this, I think, is the work of Geoffrey West in his book Scale: The Universal Laws of Growth, Innovation, Sustainability, and the Pace of Life in Organisms, Cities, Economies, and Companies (2017).
West notes the usefulness of coarse-grained models in several places, such as this paper. as he describes his collaboration with two outstanding biologists:
In addition to a strong commitment to solve a fundamental long-standing problem that clearly needed close collaboration between physicists and biologists, a crucial ingredient of our success was that Jim and Brian, in addition to being excellent biologists, thought like physicists and were appreciative of the importance of a mathematical framework grounded in ‘first principles’ for addressing problems. Of equal importance was their appreciation that, to varying degrees, all theories and models are approximate; the challenge is to identify the important variables that capture the essential dynamics at each organizational level of a system thereby leading to a calculation of their average properties. This provides a coarse-grained ‘zeroth order’ point of departure for quantitatively understanding specific biosystems, viewed as variations or perturbations around idealized norms due to local environmental conditions or historical evolutionary divergence.
West, G “A theoretical physicist’s journey into biology: from quarks and strings to cells and whales” 2014 Phys. Biol. 11 053013
This idea, achieving a “‘zeroth order’ point of departure” is underestimated and not often used. A first model of any phenomenon is more useful than no model at all – and that is often forgotten.
Then again, of course, there are problems that require at least mid-level resolution. Look at this example of resolution re-construction from Google Brain:
If you are asked to identify a person, the left most low-res image now is useless! So resolution really matters in problems, and in many cases will determine if you succeed in solving a problem or not. A good question to ask oneself, then, becomes – is this a problem that requires a low, mid or high resolution understanding or diagnosis?