OK, I'm wading into some deep philosophical waters here, given the title of this blog post, because I only took one semester of calculus in graduate school, and then only because I was forced to by the powers-that-be in control of the Portland State University Systems Science Ph.D program.
I found calculus to be difficult. By contrast, I've gotten back to reading an engaging book by Stephen Strogatz about calculus, Infinite Powers, which I blogged about back in August, noting that it had some spiritual aspects.
This morning, reading a chapter on "The Vocabulary of Change" before I meditated, I was struck by how mindfulness practice -- which basically involves focusing on what's going on here-and-now versus there-and-then -- resembles what Strogatz calls the Infinity Principle.
Here's how Strogatz described this in the chapter I read today.
Notice that we are using the Infinity Principle here -- we are trying to make a complicated curve simpler by chopping it into infinitesimal straight pieces. This is what we always do in calculus. Curved shapes are hard; straight shapes are easy, even if there are infinitely many of them and even if they are infinitesimally small. Calculating a derivative in this way is a quintessential calculus move and one of the most fundamental applications of the Infinity Principle.
In his book, Strogatz uses a graph of the length of days to talk about how an underlying mathematical function, in this case the complex motions of the Earth as it revolves around the sun, is used to produce a graph such as the one below that I found via some Googling.
What struck me is that while it's impossible to derive a mathematical function of life, we all have the feeling that our life consists of ups and downs.
Our most significant "up," of course, is the moment of our birth. And our most significant "down" is the moment of our death. But in-between those moments we experience a wide variety of joys and sorrows, triumphs and disappointments, pleasures and pains.
It's natural to view the present moment in the context of both remembered previous ups and downs, and of imagined future ups and downs. However, this makes more sense when something is genuinely cyclical, as is the case with the amount of daylight (vertical axis in image above).
In our own life, it's often difficult to predict what is going to happen next. And our memory of what's happened in the past is prone to errors, since we tend to remember most clearly the Big Moments in life, not the Ordinary Moments -- even though the latter are much more common than the former.
So there's a lot to like about mindfulness, which I've noted is my current approach to meditation.
Calculus can deal with curves such as those in the image above because it divides them into an infinity of straight lines. Similarly, mindfulness helps us to deal with the ever-changing circumstances of our life by guiding us to focus on the small things that are happening at this very moment, rather than the largeness of our actual past and potential future.
I'm not saying that mindfulness in any way takes away ups and downs, or pleasures and pains. But just as a portion of a curved line looks almost exactly straight if it is zoomed in enough, so are we able to look upon our life with more equanimity if we're able to be mindful of what is actually occurring to us -- versus what we hope will occur, or are worried will occur.
As a final observation, Strogatz uses Usain Bolt's record-breaking 100 meter dash performance in the 2009 World Championships as an example of how small ups and downs are largely meaningless when looking at a curve that is smooth at a larger scale. He shares a graph based on laser guns that tracked Bolt's speed hundreds of times a second.
The little wiggles on the overall trend represent the ups and downs in speed that inevitably occur during strides. Running, after all, is a series of leapings and landings. Bolt's speed changed a little whenever he landed a foot on the ground and momentarily braked, then propelled himself forward and launched himself airborne again.
...To me, these wiggles hold a larger lesson. i see them as a metaphor, a kind of instructional fable about the nature of modeling real phenomena with calculus. If we try to push the resolution of our measurements too far, if we look at any phenomenon in excruciatingly fine detail in time or space, we will start to see a breakdown of smoothness.
...Yet if what we care about are the overall trends, smoothing out the jitters may be good enough. The enormous insight that calculus has given us into the nature of motion and change in this universe is a testament to the power of smoothness, approximate though it may be.
in the same way, mindfulness requires an appropriate level of resolution. Our life shouldn't be viewed in such excruciating detail as to miss the big picture, nor in such a broad scope as to miss the reality of what is actually happening at any given moment.