Calculus is one of those subjects that is so complicated that most people not only don’t understand it, they don’t even know what it is that they don’t understand. But that’s unfortunate, because calculus is one of humanity’s most impressive achievements, an accomplishment that unlocks the secrets of the universe and delivers our most profound and useful technology, from radio and television to GPS navigation and MRI imaging. Calculus is the main protagonist in the story of science, and is a subject every educated person should understand at least conceptually.
Fortunately, you don’t have to trudge through a thousand-page textbook to appreciate the story and power of calculus. Steven Strogatz, in his latest book Infinite Powers, has provided a clear, concise, and fascinating tour of the subject. In fact, if you don’t understand what calculus is all about after reading this book, then the prospects are not great that you ever will. There is simply no better, clearer presentation of the ideas available. Strogatz uses metaphors, illustrations, stories, and examples to guide the reader through the most difficult concepts. While this is not an easy read, it is as reader-friendly as possible; remember, you’re tackling the most sophisticated branch of mathematics, the underlying logic of all science, and a subject that the sharpest mathematical minds in history had to grapple with for thousands of years.
As Strogatz explains, calculus is difficult because it’s tackling the most difficult problems humans encounter, problems that necessitate complex equations, notation, and mathematical manipulation. But behind this computational complexity lies an obsession with simplicity, with breaking down hard problems into easier parts. The special innovation of calculus, as Strogatz explains, is that problems are broken down into infinitely small and manageable parts and then recombined back into the whole.
So what is calculus, exactly? It’s easier to describe calculus by the types of problems it solves than by standard mathematical definitions. When most people hear terms like “infinite series,” “limits,” “derivatives,” and “integrals,” they lose sight of the bigger picture of what calculus is trying to accomplish.
One type of problem calculus can solve is the area under a curved surface. Area is typically quite easy to solve for shapes with straight lines. For rectangles, for example, the area is simply length times width. But what about for shapes with curves where the slope is constantly changing? There is no simple formula to calculate the area in this situation.
You could approximate the area by overlaying rectangular objects over the curved shape (as shown below), but this would only be an approximation as the rectangles would not fit exactly in the curved shape. However, as you made the rectangles smaller (and increased their number) the fit inside the curved shape would keep improving and the approximation would keep getting more accurate. Since you can always keep dividing a number in half (you can always make a number larger or smaller), you can add an infinite number of smaller rectangles into the curved shape. You can never complete this process (which is why the concept of “completed infinity” is logically incoherent), but you could potentially keep adding rectangles forever, which is logically coherent and shows the difference between “completed infinity” and “potential infinity.” As you increase the number of rectangles, you get closer and closer to the area, which is the limit of the infinite series. The area of the curved shape becomes the sum of the infinite series of rectangles. Calculus is the set of equations and procedures to carry out this calculation precisely.
Calculus can also solve problems of motion. Straight-line motion at constant velocity is easy. If you know the speed of an object, then the distance traveled is simply speed times time. But how can you calculate the trajectory of, say, a planet, that not only continuously changes direction in orbit around the sun but that also speeds up or slows down depending on its distance from the sun? This is not so easy, but is solved in a similar way by breaking down the trajectory into infinitely smaller parts and then summing the series. Calculus provides the procedures and notation to carry this out in the most efficient way.
You’ll notice that both examples above solve for problems where some quantity is continuously changing. That means that calculus can solve any problem that involves a quantity that is continuously changing, like the spread of a virus, population growth, or continuously compounding interest in finance. Even without understanding the specific calculations, it’s amazing to contemplate the fact that we can harness the power of infinity to calculate with precision the area under any curved surface, the dynamics of any continuously changing variable, and the trajectory of any object anywhere in the universe!
Of course, this brief sketch is only a description of the subject in its simplest terms; there is much more to the subject and the mechanics of the calculations gets incredibly complex. If you’re interested in diving deeper into the subject, with examples and proofs, Strogatz delivers a nice mixture of pure mathematics, practical examples, and a history of the personalities behind the development of calculus. Of particular interest for me was Strogatz’s solution of Zeno’s Achilles and the Tortoise paradox, a solution that finally made sense to me (in brief, the solution is that an infinite amount of steps can be completed in a finite amount of time).
The Power of Human Cooperation
If you find calculus near impossible to learn, you won’t be happy to know that Isaac Newton invented the subject before he turned 25. But you might find some solace in the fact that Newton did little else; he had few friendships and no romantic relationships, so he had all the time in the world to devote to numbers and experiments.
Newton also couldn’t have done it alone. He was exactly right when he said that he was able to see further by “standing on the shoulders of giants.” As Strogatz explained:
“But again, he [Newton] couldn’t have done any of this without standing on the shoulders of giants. He unified, synthesized, and generalized ideas from his great predecessors: He inherited the Infinity Principle from Archimedes. He learned his tangent lines from Fermat. His decimals came from India. His variables came from Arabic algebra. His representation of curves as equations in the xy plane came from his reading of Descartes. His freewheeling shenanigans with infinity, his spirit of experimentation, and his openness to guesswork and induction came from Wallis. He mashed all of this together to create something fresh, something we’re still using today to solve calculus problems: the versatile method of power series.”
There are at least two lessons here; first, knowledge grows exponentially, not linearly, and there is no limit to what can be discovered. By standing on the shoulders of giants, each generation can build on the developments of the past, as Einstein was able to do by rejecting Newton’s ideas of space and time as absolute. Holding a person, idea, or generation in complete reverence inhibits progress, as when we followed Aristotle for 1,500+ years and maintained the belief that the earth was stationary. The best book I’ve read that elaborates on this point is The Beginning of Infinity by the physicist David Deutsch.
Second, calculus demonstrates the power of human cooperation. No single mind could have developed calculus from scratch. People of diverse origin and circumstance collaborated to find solutions to common, tangible problems, because they didn’t waste their time thinking about arbitrary human divisions and other products of pure imagination, like religion. Newton borrowed from ancient Greek geometry, French analytic geometry, the Indian decimal system, and Arabic algebra. As a result, he discovered the mathematical logic and underlying laws of nature that applied equally to objects anywhere in the universe, thus uniting the entire cosmos. This universality, as Strogatz recognized, sparked the beginning of the Enlightenment.
A final point: in the concluding chapter, Strogatz describes Richard Feynman’s quantum electrodynamics (QED) theory, which, by using calculus, describes the quantum interaction of light and matter. Physicists use the theory to make predictions about the properties of electrons and other particles. As Strogatz said, “by comparing those predictions to extremely precise experimental measurements, they’ve shown that the theory agrees with reality to eight decimal places, better than one part in a hundred million.”
This means that QED is the most accurate theory anyone has ever devised about anything. A prediction with an accuracy of 8 decimal places is like, using Strogatz’s example, planning to snap your fingers exactly 3.17 years from now down to the second, without the help of a clock or alarm. As Strogatz further explains:
“I think this is worth mentioning because it puts the lie to the line you sometimes hear, that science is like faith and other belief systems, that is has no special claim on truth. Come on. Any theory that agrees to one part in a hundred million is not just a matter of faith or somebody’s opinion. It didn’t have to match to eight decimal places.”
You will also often hear that science can’t determine right and wrong actions, which in some sense is correct, but misses the point. The moral element of science does not lie in any particular factual claim; it lies in the orientation to forming beliefs. The scientific mindset is not about clinging on to and forming your identity around a set of unalterable beliefs. The scientific mindset is about curiosity, orientation to discovering truth, intellectual integrity, and revising beliefs in the face of new evidence. It’s also, as I believe calculus shows, about the recognition of the power of human cooperation and the pursuit of knowledge as a collective human endeavor.
Also by Steven Strogatz:
Other great books on applied mathematics:
How Not to Be Wrong: The Power of Mathematical Thinking by Jordan Ellenberg
Algorithms to Live By: The Computer Science of Human Decisions by Brian Christian and Tom Griffiths
Naked Statistics: Stripping the Dread from the Data by Charles Wheelan