Infinite Powers: How Calculus Reveals the Secrets of the Universe

by Steven H. Strogatz

CT scanning, is hundreds of times more sensitive than conventional x-ray films. Their precision has revolutionized medicine. The C stands for computerized and the T stands for tomography, meaning the process of visualizing something by cutting it into slices. A CT scan uses x-rays to image an organ or a tissue one slice at a time. When a patient is placed in a CT scanner, x-rays are sent through the person’s body at many different angles and recorded by a detector on the other side. From all that information—from all those views at different angles—it’s possible to reconstruct much more clearly what the x-rays passed through. In other words, CT is not just a matter of seeing; it’s a matter of inferring, deducing, and calculating. Indeed, the most brilliant and revolutionary part of CT is its use of sophisticated mathematics. With the help of calculus, Fourier analysis, signal processing, and computers, the CT software infers the properties of the tissue, organ, or bone through which the x-rays passed and then generates a detailed picture of that part of the body.

Imagine firing a beam of x-rays through a slice of brain tissue. As the x-rays travel, they encounter gray matter, white matter, possibly brain tumors, blood clots, and so on. These tissues absorb the x-rays’ energy to a greater or lesser degree, depending on the type of tissue it is. The goal of CT is to map the absorption pattern in the whole slice. From that information, CT can reveal where tumors or clots may be. CT doesn’t see the brain directly; it sees the x-ray–absorption pattern in the brain.

slicing can be misleading when it makes us forget that the parts belong together, that they’re all part of something bigger. My goal in this book has been to show calculus as a whole, to give a feeling for its beauty, unity, and grandeur.

Every one of your ten trillion or so cells contains about two meters of DNA. If laid end to end, that DNA would reach to the sun and back dozens of times. Still, a skeptic might argue that this comparison is not as impressive as it sounds; it merely reflects how many cells each of us has. A more informative comparison is with the size of the cell’s nucleus, the container that holds the DNA. The diameter of a typical nucleus is about five-millionths of a meter, and it is therefore four hundred thousand times smaller than the DNA that has to fit inside it. That compression factor is equivalent to stuffing twenty miles of string into a tennis ball.

On a linear system like a scale, the whole is equal to the sum of the parts. That’s the first key property of linearity. The second is that causes are proportional to effects. Imagine pulling on the string of an archer’s bow. If it takes a certain amount of force to pull the string back a certain distance, it takes twice as much force to pull it back by twice that distance. Cause and effect are proportional. These two properties—the proportionality between cause and effect, and the equality of the whole to the sum of the parts—are the essence of what it means to be linear. Yet many things in nature are more complicated than this. Whenever parts of a system interfere or cooperate or compete with each other, there are nonlinear interactions taking place. Most of everyday life is spectacularly nonlinear; if you listen to your two favorite songs at the same time, you won’t get double the pleasure. The same goes for consuming alcohol and drugs, where the interaction effects can be deadly. By contrast, peanut butter and jelly are better together. They don’t just add up—they synergize. Nonlinearity is responsible for the richness in the world, for its beauty and complexity and, often, its inscrutability. For example, all of biology is nonlinear; so is sociology. That’s why the soft sciences are hard—and the last to be mathematized. Because of nonlinearity, there’s nothing soft about them.

The great advantage of linearity is that it allows for reductionist thinking. To solve a linear problem, we can break it down to its simplest parts, solve each part separately, and put the parts back together to get the answer.

nonlinearity places limits on human hubris. When a system is nonlinear, its behavior can be impossible to forecast with formulas, even though that behavior is completely determined. In other words, determinism does not imply predictability.

Chaotic systems are finicky. A little change in how they’re started can make a big difference in where they end up. That’s because small changes in their initial conditions get magnified exponentially fast. Any tiny error or disturbance snowballs so rapidly that in the long term, the system becomes unpredictable. Chaotic systems are not random—they’re deterministic and hence predictable in the short run—but in the long run, they’re so sensitive to tiny disturbances that they look effectively random in many respects. Chaotic systems can be predicted perfectly well up to a time known as the predictability horizon. Before that, the determinism of the system makes it predictable. For example, the horizon of predictability for the entire solar system has been calculated to be about four million years. For times much shorter than that, like the single year it takes our Earth to go around the sun, everything behaves like clockwork. But once we move past a few million years, all bets are off. The subtle gravitational perturbations among all the bodies in the solar system accumulate until we can no longer forecast the system accurately.

People always knew that big complex systems like the weather were hard to predict. The surprise was that something as simple as a spinning top or three gravitating bodies was similarly unpredictable. That was a shocker and another blow to Laplace’s naive conflation of determinism with predictability.

the pendulum’s state space had two dimensions because two variables—the pendulum’s angle and its velocity—were necessary and sufficient to predict its future. They gave us exactly the information we needed to predict its angle and velocity an instant later, and an instant after that, on and on into the future. In that sense, the pendulum is an inherently two-dimensional system. It has a two-dimensional state space. The curse of high dimensions arises when we consider systems more complicated than a pendulum. For example, let’s take the problem that gave Newton a headache, the problem of three mutually gravitating bodies. Its state space has eighteen dimensions. To see why, concentrate on one of the bodies. At any instant, it is located somewhere in ordinary three-dimensional physical space. Its location can therefore be specified by three numbers: x, y, z. It can also move in each of those three directions, corresponding to three velocities. So a single body requires six pieces of information: three coordinates for its location plus three for its velocity in the different directions. Those six numbers specify where it is and how it’s moving. Multiply that six by each of the three bodies in the problem and now you have 6 × 3 = 18 dimensions in state space. Thus, in Poincaré’s approach, the changing state of a system of three mutually gravitating bodies is represented by a single abstract point moving around in an eighteen-dimensional space. As time passes, the abstract poin...

although we have an abstract system for doing math in high-dimensional spaces, mathematicians still have trouble visualizing them. Actually, let me be more frank—we can’t visualize them. Our brains just aren’t up to it. We aren’t wired that way. That cognitive limitation deals a serious blow to Poincaré’s program, at least in dimensions higher than three. His approach to nonlinear dynamics depends on visual intuition. If we can’t picture what’s going to happen in four or eighteen or a hundred dimensions, his approach can’t help us all that much. This has become a big obstacle to progress in the field of complex systems, where high-dimensional spaces are exactly what we need to understand if we want to make sense of the thousands of biochemical reactions taking place in a healthy living cell or explain how they go awry in cancer. If we are to have any hope of making sense of cell biology using differential equations, we need to be able to solve those equations with formulas (which Sofia Kovalevskaya showed we cannot) or picture them (which our limited brains won’t allow). So the mathematics of complex nonlinear systems is discouraging. It seems like it will always be hard, if not impossible, for anyone to make headway on the most difficult problems of our time, from the behavior of economies, societies, and cells to the workings of the immune system, genes, brains, and consciousness.

On December 5, 2017, the DeepMind team at Google stunned the chess world with its announcement of a deep-learning program called AlphaZero. The program taught itself chess by playing millions of games against itself and learning from its mistakes. In a matter of hours, it became the best chess player in history. Not only could it easily defeat all the best human masters (it didn’t even bother to try), it crushed the reigning computer world champion of chess. In a hundred-game match against Stockfish, a truly formidable program, AlphaZero scored twenty-eight wins and seventy-two draws. It didn’t lose a single game. The scariest point is that AlphaZero showed insight. It played like no computer ever has, intuitively and beautifully, with a romantic, attacking style. It played gambits and took risks. In some games it paralyzed Stockfish and toyed with it. It seemed malevolent and sadistic. And it was creative beyond words, playing moves no grandmaster or computer would ever dream of making. It had the spirit of a human and the power of a machine. It was humankind’s first glimpse of a terrifying new kind of intelligence.

a hundred million seconds equals 3.17 years, so getting something right to within one part in a hundred million is like planning to snap your fingers exactly 3.17 years from now and timing it right to the nearest second—without the help of a clock or an alarm.

gravity could have a strange effect on time: The passage of time could speed up or slow down as an object moves through a gravitational field. Bizarre as this sounds, it really does occur. It needs to be taken into account in the satellites of the global positioning system as they move high above the Earth. The gravitational field is weaker up there, which reduces the curvature of space-time and causes clocks to run faster than they do on the ground. Without correcting for this effect, the clocks aboard the GPS satellites wouldn’t keep accurate time. They’d get ahead of ground-based clocks by about 45 microseconds per day. That may not sound like much, but keep in mind that the whole global positioning system requires nanosecond accuracy to work properly, and 45 microseconds is 45,000 nanoseconds. Without the correction for general relativity, errors in global positions would accumulate at about ten kilometers each day, and the whole system would become worthless for navigation in a matter of minutes.