I’m not going to teach you calculus here. There are several great YouTube introductions if you’re interested, but I don’t think you need to know calculus to follow this essay.

Calculus deals with change and is fundamentally based on time.

I don’t mean that all calculus problems include time in the equations. Calculus is a way to express how one quantity changes instantaneously with respect to another quantity. Often that second quantity is time, but often it isn’t. You may want to know how the steepness of a hill changes from one spot to another on the hill. The steepness is a function of the location. Time has nothing to do with it.

Still calculus always involves time.

The way we design a calculus equation is through something called a limit. By using the limit concept, we can see what the value of an equation approaches as some quantity approaches a fixed value, usually zero. The beauty of the method is that if we can’t define the equation at some precise point, we can still determine what the equation approaches as we get closer and closer to that point. We then define the equation as having the approached value at the point.

If this seems like a lot of handwaving, you’re in good company. Very smart people have been troubled by this for centuries. However, it works. The answers we get describe the real world around us.

My point is that calculus involves the term “approaches”. This is a temporal term. If you are calculating the steepness of the hill based on where you are on the hill, the calculus equation, though it has no ‘t’ for time in it, is still fundamentally based on the temporal concept of a value approaching another value. You can’t approach anything without time. If there were no time, calculus wouldn’t exist.

This isn’t surprising to me. Calculus calculates change. There is no change without time. In the example of the hill, time is involved in moving from one place to another. The steepness of the hill isn’t changing anywhere. It’s fixed at every point. It’s your experience of steepness that changes as you move on the hill.

Time is fundamental to our experience of our world, even if we aren’t exactly sure what time is (See the later essay, “Duration”). For me, this implies that calculus, more than other parts of mathematics, is based on our personal, human experience of the world. Despite its more forbidding reputation, calculus is actually warmer and fuzzier than arithmetic. Maybe even more friendly.

Hugh Moffatt
Nashville, Tennessee
October 31, 2023