A Conversation with Mario Livio on Symmetry

Q: Describe the evolution of this project. How did you come to write this book?

A: I’ve been fascinated by the Frenchman Évariste Galois ever since high school. Here’s this guy—a true visionary—who died at age 20 in a duel, and yet he was responsible for one of the most remarkable accomplishments in the history of mathematics: the creation of group theory, which is recognized today as the “official” language of symmetry. During my undergraduate years, my fascination grew as I studied both mathematics and physics, because group theory embraces all the symmetries of the universe. In fact, it permeates every discipline you could possibly name, from the visual arts and music to psychology and the natural sciences, so its significance cannot be overemphasized. And yet, there has never been a popular book on the subject. I thought this is the time to do it, and that led to The Equation that Couldn’t Be Solved.

Q: How is it different from your previous bestseller The Golden Ratio?

A: This is much more of an all-embracing type of book. At the end of the day, The Golden Ratio is about one number. It’s an extraordinarily interesting number, and one that appears in natural phenomena, the arts, and so forth, but it’s still just one number. This book is about an entire language. Group theory is the language of symmetry in the same way that arithmetic is the language of the financial world. If you need to compare companies, you need numbers. If you want to talk about any type of symmetry—in science, music and the arts; even in mate selection and sexual attraction—you have to talk group theory. This book addresses something much more fundamental to the world around us than the golden ratio.

Q: What is symmetry and why is it important?

A: It’s important because it is one of the most essential tools in deciphering nature’s design, and it sits right at the intersection of science, art, and perceptual psychology. Our faces have almost precise bilateral symmetry, as do most animals around us. (If you reflect the left half in a mirror you obtain something that is almost identical to the right half.) Then there are objects that have rotational symmetry. Imagine passing a pin through the center of a six-cornered snowflake. Every time you rotate that snowflake by sixty degrees it looks exactly the same. Palindromes—such as “Madam, I’m Adam”—have a particular symmetry under back-to-front reading. Now think of wallpaper or a street of row houses: If you shift something by a certain distance in a certain direction you’ll see the same thing. That’s symmetry under translation. Many experiments have shown that symmetry is very important for perception. If you look at something that’s symmetric, you’re able to recognize it much faster and reproduce it from memory more easily.

Q: How did this ability develop?

A: There were a number of factors. One is the predator-avoidance mechanism built into all of us. Imagine you’re a primitive human living in the ancient world. If you see something that is bilaterally symmetrical, what is it likely to be? It’s likely to be another animal. You need to recognize this quickly, because if that other animal is a predator looking to have you for dinner, it could mean the difference between life and death. Then there’s mate selection. An animal or person who exhibits precise symmetry, or very close to it, is showing that they have conquered all kinds of debilitating mutations and are probably relatively healthy. Think of the peacock’s tail. A large, perfectly symmetrical tail announces to a potential mate loud and clear, “I am healthy.”

Q: What is so significant about the equation that couldn’t be solved?

A: The significance is in what evolved out of the fact that it couldn’t be solved. Equations come in degrees. There are linear equations—such as 2x +1 = 3—in which x appears in the first power. Then there are quadratic equations in which x appears squared, cubic equations which contain x to the third power, and quartic equations which contain x to the fourth. It took mathematicians thousands of years—up through the mid-1500s—to solve these progressively more difficult equations. And since each was solved by a formula, it was generally assumed that as math advanced, mathematicians would eventually learn how to solve the next level—the quintic equation—which contains x to the fifth power. Those assumptions were wrong. Two-hundred-and-fifty years passed, during which all the mathematical power of an entire continent was thrown at the quintic and yet no one succeeded in solving it. And then something truly incredible happened; something that had no precedent in the history of math. These two young people—Niels Henrik Abel, a Norwegian, and the Frenchman Évariste Galois—working separately, proved mathematically that quintic equations couldn’t be solved by a formula. It was a stunning revolution in thinking because it amounted to a new beginning in the history of mathematics. Galois then topped himself, as he set out to prove what he was saying, by inventing group theory—the “language” of symmetry. That’s why I say there is great importance in the equation itself but even greater importance in what the equation has led to.

Q: Why do you consider Abel and Galois the most tragic figures in the history of science?

A: Here you have two young mathematicians who happen to live at almost exactly the same time, even though they never meet. Abel, slightly older than Galois, lives in Norway and is poor as a church mouse. Despite being recognized as a genius he fails to get a university position and remains poor. Since people back then didn’t marry unless they had enough money to support a household, he ends up not marrying the woman to whom he’s engaged. He lives in miserable conditions and poverty his entire life, gets tuberculosis and dies at age 26.

Galois is born in post-revolutionary France, a tumultuous time to say the least. French society is thoroughly polarized with the liberals and republicans on the left and the ultraroyalists on the right. People in the streets carry guns wherever they go and it seems like everyone is against everyone else. Although recognized as one of the most brilliant geniuses of his day, he fails twice to get into the École polytechnique—the high school for the sciences—because the examiners can’t understand his unorthodox methods. He writes three masterful manuscripts—including the one in which he invents group theory—and submits them in succession to the Academy of Science. The first one is ignored completely; the second one is lost by a member of the Academy; and the third is rejected. He is such a firebrand that he manages to get imprisoned twice. He falls in love with a 17-year-old who doesn’t love him and sends him heart-piercing letters of rejection, and then he dies in a duel at age 20 over the same girl. If that’s not tragic what is?

Q: You also described this as one of the most tumultuous sagas in the history of mathematics. Why?

A: There are some parts of this story that are extraordinarily tumultuous. For example, I tell the tale of 16th century Italian mathematician Scipione dal Ferro, the first person ever to solve a cubic equation. The practice of keeping mathematical discoveries secret was quite common at the time so dal Ferro did not rush to publish. However, he revealed the solution to his son-in-law and to one of his students, a rather mediocre mathematician named Fiore. Upon dal Ferro’s death, Fiore also chose not to publish. Instead he treated the cubic solution as if it were his to be exploited. He sat back and waited for the right moment—one that would allow him to make a name for himself.

In the day and age, if you wanted to show how clever you were—in math, science, the arts, theology or any subject—you’d challenge someone to a public debate. And the stakes were high. One’s tenure at a university, the renewal of a teaching appointment, and the salary you earned were all largely determined by how well you did. Having a secret weapon at your disposal could mean the difference between professional survival and perishing. When another mathematician, Nicolo Tartaglia, claimed he knew how to solve cubic equations, Fiore saw his chance. He challenged Tartaglia to a public debate in the form of a problem-solving contest. Tartaglia creamed him. He blasted through all of the problems in just two hours while Fiore failed to solve even one.

The next player to enter the story was a mathematician named Cardano. Determined to get the secret of the cubic, he schmoozed Tartaglia in every way imaginable. Eventually Tartaglia agreed to give him the solution but only after Cardano took a solemn oath never to publish what was being revealed to him. Cardano then tracked down dal Ferro’s son-in-law and got from him the original solution to the cubic. Figuring he was now freed from his obligation to Tartaglia, Cardano went ahead and published. Tartaglia was furious and rushed out a book of his own. In it he directly accused Cardano of perjury, using the most insulting language imaginable. What followed was a long public exchange of insults between Tartaglia and Cardano’s student, who rose to his mentor’s defense. Eventually the student and Tartaglia met in debate. Only this time it was Tartaglia who suffered an agonizing and humiliating defeat. To round out the story I should also mention that the student was eventually murdered by his sister so that she could inherit his money and property (much of which came as a result of his skyrocketing career after winning the Tartaglia debate).

Q: What surprised you most as you did your research for this book?

A: There were elements of the story of Galois’s death that I did not know, including many of the conspiracy theories that had grown up around it. That was a surprise for me. I was also quite amazed at the amount of research that has been done on human perception and symmetry, because it’s not an area in which I normally work. I was particularly unaware of the vast amount of research that has been done in the realm of evolutionary psychology: the study of how our minds evolved so as to have the tools to respond to symmetry and how that evolution relates to the way we view the universe. That surprised me, and turned into what I think is a very important contribution of this book: an exploration of the relationship between the symmetry of the universe and the symmetry of the human mind. I don’t think there has ever been a study that seriously looked at whether symmetry is indeed fundamental to the universe or whether our minds are somehow fine-tuned to latch onto the universe’s symmetric aspects. That’s one question I attempt to answer here.

Q: And what answer did you come up with?

A: There’s no question that our minds are fine-tuned to latch onto symmetry. There’s also no question in my mind that symmetry continues to be extraordinarily fruitful in understanding the laws of nature. Whether it will turn out to be the most fundamental thing in understanding the laws of nature remains an open question.

Q: What do you think will most surprise readers of this book?

A: I think readers will be surprised at how important symmetry is, at the extent to which it pervades every part of our lives—science, psychology, the arts, and so on—and also at how group theory is this incredible language that describes symmetries wherever they are found. In addition, I believe readers who never heard of Galois will find his story extremely surprising.

Q: This is a book one has to pay attention to when reading. When dealing with complex subjects like this how do you walk the fine line between making it too esoteric and telling a rich story in an entertaining way?

A: I’ll be first to admit it’s not easy. Fortunately, as a theoretical astrophysicist who works with the Hubble Space Telescope, I give many talks to the public about various aspects of science. That background has given me an appreciation of where that fine line is located. You never want to bore the people who know some science, but you also don’t want to turn off those people to whom science is a confusing mystery. I also happen to be an art fanatic. I own more than a thousand books on art—a subject very dear to my heart. So combining science with art, and moving from one to the other comes very naturally to me. I never need to search for examples from the arts to illustrate science. They come to my head all the time.

Q: What was the most difficult challenge in writing this book?

A: The hardest part was explaining the details of what Galois discovered: that the quintic equation couldn’t be solved by a formula. Although I could have written the book without getting into the somewhat difficult mathematics of his brilliant proof, I thought there was something to be gained by understanding its essence. It shows what a genius he was. You’re left wondering, how could he have thought of this? How did he get from this equation to the symmetry of solutions that he didn’t know? In the history of science, even great discoveries can usually be traced to something that was “in the air” at the time. These are ideas whose time has come. Most physicists would agree, for instance, that had Einstein not suggested his theory of special relativity, from which the famous E = mc2 emerged, someone else would have sooner or later come up with the same idea. (The same is not true, by the way, of Einstein’s general relativity.) But Galois’s insight was nothing short of a miracle.

The other significant challenge was in trying to more or less summarize all of physics in one chapter. My goal wasn’t to truly explain what all of physics is about but rather to help readers understand how the whole of physics relies on symmetry.

Q: What do you want readers to get out of this book?

A: I hope they’ll come away from it with a sense of wonder about symmetry, which is all around us; about the language we use to describe it; and about the extraordinary people who came to invent that language.

All content © Mario Livio unless noted otherwise.