Modelling Magic

How can we make sense of phenomena in the world around us? How can we predict what’s likely to happen in the future, or what will be the consequences of a particular set of actions? In science, the answer is of course, to design a model. But with any model, it’s always important to keep in mind that oft-quoted expression of the twentieth-century statistician George Box, ‘all models are wrong, but some are useful’. Contrary to popular belief, science isn’t about certain facts, for we’re never completely sure about anything. It’s impossible to ever get to the absolute objective ‘truth’ of how the universe works, and impossible to prove with one hundred per cent certainty that an assertion is wrong or right. Science progresses by using observations of the world to reveal ways in which our previous theories are incomplete, and for a model to be useful it has to be capable of making some prediction that can be ‘falsified’ in this way – that is, to be found at odds with what we see in our agreed upon approximate ‘objective’ reality.

Even though our theories may be fundamentally false, this doesn’t mean that we have to despair of ever understanding anything. As long as we’re willing to pragmatically accept certain assumptions and simplifications, the extraordinary complexity of nature can be encapsulated in a language that we can understand. That is the nature of modelling. Models can still be incredibly useful, for two main reasons: the first scientific, and the second practical.

Science is all about understanding the universe for its own sake, so for scientists models are useful simply for the insights that they provide. Of course, the universe doesn’t ‘obey the laws of physics’ in a strict sense. Rather these man-made, abstract principles are approximations that conform, with varying degrees of accuracy, to our own ultimately subjective observations. Therefore, it’s futile for the scientist to aim for perfection at the outset, and account for every conceivable influence of a phenomenon that they want to understand. Instead, science always starts with the simplest models. Beginning with basic assumptions and gliding gently onwards through layers of ascending complexity, scientists create ever more accurate descriptions of how the universe behaves, making increasingly precise predictions that can be tested.

Perhaps counter-intuitively, it’s when our models fail to predict what’s actually observed that things get most exciting, for the limitations of the existing model tell us in which direction better insights lie. A simple model that doesn’t agree at all with what we see would be rejected on the basis that it’s probably inaccurate. But once we hit upon a model with at least some agreement between theory and observation and an initial hypothesis, we can only move on to a more advanced understanding by studying it limitations and challenging, one by one, the initial assumptions that were made.

Often, models become increasingly complicated as their accuracy increases. But occasionally, a ‘paradigm shift’ suddenly occurs and a new, simpler, more beautiful and intuitive picture emerges. Think of the solar system. When everyone believed that Earth was at its centre, increasingly complicated models involving ‘epicycles’ – essentially mysterious knots in the orbits of planets – had to be conjectured to explain increasingly precise observations of the planets’ orbits, a technique that dates back to Ptolemy in the Second Century AD. But in the seventeenth century it was recognised that there was no need for these ugly and distorted orbits if the sun was at the centre instead – a new paradigm shift emerged. In the eighteenth century, Newton’s Law of Gravitation was used to explain the interaction between the sun and planets, but it couldn’t predict the precise orbit of Mercury. Another paradigm shift from Newtonian physics to Einstein’s General Relativity was required for this to eventually be achieved in the twentieth century. On the basis of better agreement with observations, it is assumed that this is the more ‘accurate’ picture of reality, though we can never know for sure.

But the creation of a more accurate model doesn’t mean that all its predecessors become redundant. The second purpose of modelling is practical, to produce useful predictions about the future that have implications for society. Sometimes, it’s not necessary to use the most accurate and recent model to do this; sometimes a simpler model will suffice in spite of its known limitations. When we predict the positions in the sky of the stars and planets, we don’t solve the full equations of General Relativity. Instead, we go back to that Ptolemaic idea of Earth as the centre of the universe, and treat the sky as if it’s rotating about us. This model is not ‘correct’ from the point of view of objective understanding, but it’s much easier to use from a practical perspective.

In other cases, simplification is more of a necessity than a choice. It’s well-known that our weather forecasts are far from perfect. This is because even though we know the Navier-Stokes equations that model very accurately the behaviour of the atmosphere, we just don’t have the computing power to apply them to weather systems smaller than a few kilometres across. Everything smaller has to be approximated by simpler, cruder models by sheer necessity, limiting the accuracy of the forecast. Yet, it’s better to use those simple models than to not be able to make forecasts at all. They may be wrong, but they’re still useful. Furthermore, it can be dangerous to blindly trust in complicated, well-tuned computer models without returning to our basic principles to check that the results make intuitive sense.

In the end, all the models that adorn the crown of human achievement that is modern science are little more than crude approximations to the truth. They can be incredibly useful in making the physical phenomena we interact with more predictable and easier to cope with and understand, but set against the true, wonderful complexity of the universe itself, especially its living inhabitants, even the brightest of these gems pales into dull monochrome. Our vast cosmos will never be entirely comprehensible by any human mind. But our models can at least give us a taste of how our physical universe, planet and societies function. There’s still a whole palette of flavours to be experienced, nuggets of truth to be unveiled. And that, to any scientist, is the most tantalising thought of all.