Mathematician Eugenia Cheng explores the uses of math beyond the classroom. Read more columns here.
Many of us have experienced grief recently, having lost loved ones to the pandemic or suffered other tragedies. It is often said that grief isn’t linear; it doesn’t plod in a predictable straight line, but takes terrible twists and turns, rearing up without warning.
I can think of few things as unmathematical as grief, yet we use this oddly mathematical term “linear” to discuss it. Linear algebra is essentially the math of things that move in straight lines. It is a very stringent requirement. Like grief, most things are not linear. But there are other types of behavior we can look for in mathematical functions and real life.
If a function keeps going in generally the same direction, though not in a straight line, it’s called monotonic. Exponentials and logarithms are monotonic but not linear. Some people argue that rates of taxation should be monotonic with income, so that someone who earns more doesn’t pay a lower effective tax rate; however, this is often not the case in practice. People who want to lose or gain weight often get frustrated that body weight is not monotonic across time, but fluctuates up and down with things like fluid retention. Sometimes a monotonic function can be derived, despite fluctuations, by taking averages across chunks of time, say a week, or a month.
“Charts are not just about drawing pretty pictures, but finding ways to understand key features about a function quickly.”
The behavior of functions is much easier to recognize visually than in a formula, so we draw graphs to help us. Turning a function into a graph is an amazing process of translating something abstract and invisible into something that allows us to invoke our visual intuition. When we draw a function out on a graph we can immediately see other features beyond linearity and monotonicity, such as whether or not it has gaps or sharp corners. If it has no gaps it is called continuous, and if it has no corners it is called smooth.
Calculus provides techniques for figuring what the graph is going to look like beyond plotting points, because even if you plot millions of points you might miss some feature in between those points. The idea is to understand how abstract features give rise to the visual features (corners, gaps and so on): It’s not just about drawing pretty pictures, but finding ways to understand key features about a function quickly.
We might also do the reverse and take some data, plot the graph, and then try and fit a function to it. We can then use the function to predict what will happen in the future. This is what is done with data in the pandemic: Case numbers provide data that can be plotted in a graph, and then a function can be found that approximates those numbers. As the function can be applied to points in time beyond the data we have now, this gives us a way of estimating what will happen in the future. It’s not an exact science, because many non-linear functions that start off with the same shape behave differently later.
Reverse-engineering a function to fit some experimental data can also give us insight into causality. The structure of the function may indicate to us some basic principles at work, such as the laws of physics. Tracking the motion of planets enabled mathematicians to fit their path to an ellipse shape, with the motion dependent on a planet’s position relative to the sun at any moment. This is just a mathematical formula, but hints at the sun’s physical role in influencing the movement of the planets.
Saying grief isn’t linear is a severe understatement, as it’s not even monotonic, continuous, or smooth. Math gives us ways to carefully distinguish between different scenarios and sequences to delineate any situation. This doesn’t make grief go away, but it can be cathartic to have ways to depict its unpredictable trajectory.
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