The title of the last article was almost “Who needs calculus?” Then I remembered my personal moral code—no clickbait titles, not even in the face of Armageddon—and changed it to “Calculus for everyone.” Calculus for everyone is, additionally, the imaginary title of the imaginary textbook for the imaginary course described in that article. There are two parts to the book—a How and a Why of calculus—so I thought, why not a Tao of calculus?
What does that even mean? This is how I look at it: Mathematics is the most valuable tool we have for understanding the world around us. Why not, then, use it to understand the world within us as well? Certain mathematical concepts have an almost spiritual quality. I’ve had professors describe the central idea of a topic as its “moral,” where the pun works completely. I refer to grokking an idea, which Wikipedia defines as
to understand intuitively or by empathy, to establish rapport with — to empathize or communicate sympathetically (with); also, to experience enjoyment
To establish rapport with. It makes itself sound like we’re making friends with the universe. I have shaken the hand of linear algebra.
This is the sort of hippy-dippy almost pseudo-mathematics that can make some readers angry; if you are one such reader, feel free to leave. I don’t propose replacing any curriculum with what I’m about to say. It’s just what my mathematical perspective has told me about the way I sometimes feel.
You know, because we’re friends.
Calculus, in case you didn’t know or don’t think of it this way, is about change. Specifically, you have functions—observable phenomena—dependent on variables. The change in a function given some unit of change in a variable is called a derivative. Studying derivatives, more or less, is the first semester of calculus.
In many settings, the calculus student is completely unconcerned with where a function sits, in other words, its absolute location. The functions , which squares its input, and , which does the same but then adds three, when graphed on a pair of axes sit above and below one another. However, they have the same derivatives with respect to the input .
So the derivative destroys positional information. Why is this important? Consider a car ride: Your sense of the external tells you where you are: I-20 heading out of Mississippi. Your second derivative (i.e. the derivative’s derivative) is your acceleration, and you don’t just see that on the speedometer. You feel that. Something is whispering to you from the dark places where mathematics speaks.
Let’s suppose this is something that generalizes. Position is seen; derivatives are felt. Does this idea extend to other examples? I think it does. I have both a personal example and a more important, broader example to share.
I graduated recently and didn’t go straight into a “proper” job in my field. I work as an adjunct instructor now, but I still do some mostly-unrelated part time work. My situation has been steadily improving since August, and now I am terrified that I will regress at all. Objectively, that’s stupid. If I put slightly less into savings in April than I do in March, why is that so bad? I’m putting something into savings. That’s awesome!
It is because I can feel, or at least feel the threat of, that negative second derivative. A negative second derivative—even for a function with a very high position or even one that’s increasing—means that the increase is slowing, or that the function is about to start tumbling down. A negative second derivative is, in fact, a test for a function’s high point. It has nowhere to go but down.
Here’s the other example, which matters much more than my financial stability. Things are getting worse for white Americans: there is apparently a finite amount of power to go around, and it is being redistributed (albeit at a glacial pace, unfortunately) to racial minorities. Despite still being on top, white America is falling. It could stop falling and still be in a better place than other racial groups in America, but that would not stop the fact that right now, white Americans feel a negative second derivative, and that can be scary. How we respond to the loss of our power is very important.
I think there are other examples that readers can find in their own lives, and it’s a useful idea to keep in mind: derivatives don’t say everything. Look around and see where you are, or so says your friend calculus.