In many respects, we can think of our lives as functions over some bounded domain. They are continuous over the domain and undefined before and after the domain (this is arguable; write to me if you have a compelling argument for the afterlife). The function should pass the vertical line test – across any given dimension (i.e. wealth, happiness, health) we cannot be at two points simultaneously. There are plenty of other similarities but I’ll leave these as an exercise for you.
If you buy that we can roughly represent life as a function, you should also be willing to buy the idea that we can exploit mathematical operations to better understand these functions. The most obvious such operation that comes to my mind is taking derivatives, though there are plenty of other interesting ones (Fourier decomposition, anyone?) that I will also leave as a fun exercise for you to think about.
Differentiating this “function of life” can give us all sorts of interesting information about how life is changing at a given moment. Are things getting better \(\left(\frac{df}{dx} > 0\right)\), or worse \(\left(\frac{df}{dx} < 0\right)\)? How quickly are things changing \(\left(\frac{d^2f}{dx^2}\right)\)?
An important but trivial point here: we need not only apply these operations to our own functions, we can use them to understand the functions of those around us too.
But we know that differentiability doesn’t come for free. To differentiate a function, it must be (1) smooth, free of corners and curves and asymptotes, and (2) continuous. I want to argue here that especially in the contexts of understanding other people’s lives and decisions (i.e being able to differentiate their functions) the same requirements apply.
Corners and cusps, we know, occur when there’s a sudden change bringing us from change in one direction abruptly to change in another direction. We can almost interpret vertical asymptotes in the same vein (e.g. winning a $100M lottery or acquiring a harmful addiction (perhaps this is redundant, all addictions are harmful)) which cause our lives to move in the same direction across a dimension for the rest of our domain (time alive). And indeed, sudden changes in the lives of others can make their feelings and decisions more difficult to understand. Even ex-post, understanding what someone was thinking or going through during a period of sudden change is difficult. Indeed, we expect this – the math tells us to! Their functions are difficult, perhaps impossible to differentiate here!
The continuity requirement for differentiability is also interesting in this context. Certainly, there is an argument that certain periods of life are not continuous (i.e. comas and arguably even sleep) and are thus not differentiable. But in the context of the functions of others, discontinuities can be caused by the lack of our presence or observation of their functions. In this sense, differentiability requires presence. We cannot expect to understand and differentiate the functions of others when we haven’t been present in their lives for the period of time before, during, and after the point of differentiation. And whereas we may be unable to control the corners and cusps and asymptotes in the functions of others, we can control our presence.
A brief postamble here. Presence is rarely “opt-out” or the default in most places, except (if you’re lucky) with family, relationships, and in school. Differentiability requires presence, and presence requires intention.