The Mathematician’s Shiva (4 page)

K
önigsberg was not only the birthplace of my childhood “buddy” Leonhardt Euler but, as fortune would have it, also was home to one of the greatest mathematicians of the twentieth century, David Hilbert. Whether he, like little Leo, avoided crossing bridges twice is unknown. Probably not. Hilbert was a thoroughly purposeful man who perhaps, even as a child, would frown upon such a frivolous use of the mind. My father, certainly, never introduced Hilbert during his lessons to me. Even he wasn’t inclined to dwell on killjoys. But he should have made an exception for Hilbert. My father knew, after all, that my mother had spent many years working on two of Hilbert’s unsolved problems, one of them being the monstrously difficult Hilbert’s sixth.

Hilbert would spend his entire career at the University of Göttingen. Hilbert’s gift consisted not only of the ability to solve immensely difficult problems, but also the insight to identify what was
worth solving
in mathematics. He laid a road map for the work of future mathematicians that is still used today.

Even I, a mere user of mathematics, am under Hilbert’s influence. Every student who wishes to understand the mathematics behind physical processes still passes through him or, more precisely, through Hilbert and his student Courant’s textbook,
Methods of Mathematical Physics
. A textbook that is still useful ninety years after it is first published is unheard of. Yet I have Courant and Hilbert’s textbook on my shelf. My students have this textbook on their shelves. My mother also had it, in the original German, on hers.

I open up this textbook now, dear reader. It’s been decades since I’ve looked at even one page. The book is made of the good stuff: a sewn binding, thick acid-free paper, and a utilitarian gray canvas cloth cover with black-and-red lettering. It isn’t an inviting text. Instead its appearance says “I know I’m important. Who the hell are you?”

I open a page at random, read the words, and look at the equations. Oh my, I had to struggle mightily to comprehend this material, but it was well worth the effort. It changed how I thought about math forever. I fully understood, in a way that I previously knew only vaguely, that math was about the ability to transform symbolism into palpable mental images. This is what my father had been trying to teach me when I was young. This is where genius lay. Hilbert had this ability. Whoever has the best visceral understanding of what seems to most to be abstract and obtuse wins in the battle to solve seemingly impossible mathematical problems.

For example, consider this equation, a formula that guides me in virtually everything I study:

Yes, OK, reader, I know you are probably sweating almost instantly at the sight of such a thing. You are thinking perhaps, “Why does this author show us such opaque symbolism? Forget this book by this middle-aged man raised by eccentric mathematicians (as if there are any other kind). One of his parents is already dead in this story and she was probably the most interesting character of the lot.”

Why am I making your life difficult? Because while maybe math is shit to you, it isn’t to me, and it wasn’t to my mother or father. It is like breathing to us, and to ignore math in this story would be akin to listening to Frank Zappa without ever having taken hallucinogens, an incomplete experience.

Dear reader, don’t panic. Newton was barely past twenty when he invented calculus. It’s pure adolescent whimsy at work. Think of the language of mathematics as shorthand that has been around for centuries, the equivalent of teenage texting, but for geeks. Yes, I know you don’t know half the text abbreviations that your teenage children use, but you can figure out their argot if pressed, can you not? You can figure out this one as well.

As my father noted to the priest at the hospital, I study the movement of air and water in our atmosphere. To study it, I fly into hurricanes and make measurements, which actually isn’t as dangerous as it sounds. It’s certainly less harrowing than experiencing the full blast of a hurricane on land.

The equation above has a name, it’s so famous: Navier-Stokes. People like me routinely use this equation to describe the chaotic motion of air and water like that found in hurricanes. You’ve seen videos and pictures of such natural calamities, no doubt. Perhaps—and this would be unfortunate—you’ve experienced such danger firsthand.

Let’s say we have a hurricane and we want to make some predictions as to where this horrible thing will move and how nasty its winds will be. If you live near a coastline, you, no doubt, would love to have such a prediction. Right now, unfortunately, I can say with complete certainty that I will never be able to provide one, although perhaps the students of the students of my students will be successful. Right now I am trying to understand what happens within a 10 inch by 10 inch by 10 inch piece of a hurricane. That may sound depressing, the idea of studying such a little speck of a big nasty storm, but the good news is that I’m making progress! To be of use to you, we will have to get much better at understanding my little 1,000-cubic-inch box, and then eventually move up to boxes 1,000 cubic miles in size. It will be awhile.

In the meantime, the National Hurricane Center makes predictions that essentially assume a hurricane is a solid cork floating in a swirling but well-behaved soup of water-loaded air. How well do these cork predictions work? Not well at all. Hurricanes aren’t corks. They are not floating in a well-behaved soup. The soup is a mess, as is the hurricane, which is not surprising, since trying to make a distinction between the hurricane and what surrounds it is never precise.

I wish to describe and predict where a fluid particle inside my 1,000-cubic-inch box inside a hurricane will go. So everything in that bit of geek texting above called the Navier-Stokes equation relates to either the nature of the fluid or how fast it’s moving. ρ is how dense the fluid is,
p
is its pressure,
v
is its velocity, and
t
is time. The equation states that if you want to know how the velocities of a fluid change with time, you need to keep track of how the fluid’s pressure changes with space and the stresses,
T
, and external forces,
f
, on that fluid.

It is routine for upper-class undergraduates who are concerned with fluid dynamics to derive the Navier-Stokes equation, known for more than 150 years, on their own. It is simple to do this, actually. You don’t even have to know anything about fluid dynamics. Just start out assuming Newton is right and apply Newton’s fundamental laws to fluids. Voilà. But please don’t misunderstand and think that just because it is easy to do, this work is trivial. It took more than 150 years to go from Newton to Navier-Stokes, and the work involved the best and the brightest mathematicians of Europe, including Leonhardt Euler.

I am happy to use the Navier-Stokes equation. But I am not a mathematician. I’m a user of Navier-Stokes, not an inspector of its correctness. David Hilbert was, however, an inspector. In 1900, he announced to the world that there were twenty-three major problems in mathematics that awaited solutions. One of those problems, number six, expresses the need to prove the fundamental correctness of using equations like Navier-Stokes to describe the physical behavior of materials. Then, in 2000, one hundred years after Hilbert, a group of mathematicians examined the future of mathematics again. Many of Hilbert’s problems had in fact been solved at least partially over the interim years. But Hilbert’s sixth remained a complete enigma. In a nutshell, the new committee dramatically reduced the scope of Hilbert’s original problem to something more manageable than all equations used to describe physical processes. They chose just one equation—perhaps the most important and certainly one of the most baffling—Navier-Stokes.

The fact is that when your geeky college niece or nephew or my students derive this equation from Newton’s laws (with the help of Euler’s work), they are making assumptions about the behavior of fluids that are so naïve as to be ridiculous.

Think of a hurricane, the water and air violently going every which way with a mixture of order and chaos. We call this mixture of movement, this wildly erratic dance of fluids, turbulence. You know this word from air travel, and it never means anything good. But to me, it means something beautiful. Without it, understanding hurricanes would be boring. Mathematicians’ worries about this equation would be the height of neurosis.

Add turbulence and mathematicians’ concerns are quite sane. The committee of 2000 thought they were so appropriate that they offered $1 million, a Millennium Prize, to anyone who could show that the Navier-Stokes equation was indeed appropriate for all conditions, even turbulent ones. You can do a lot with $1 million, of course. But the money associated with a Millennium Prize isn’t really an incentive. It’s used as a symbol of importance, to show that a community views the solution of a problem to be so enormously difficult that it is willing to offer a ridiculous amount of money to anyone who can solve it.

Even before this million-dollar prize was offered it was recognized just how near impossible this problem was. Hilbert knew it. Kolmogorov, who had studied turbulence, knew it. My mother, who studied turbulence with Kolmogorov and unlike him continued her research in this field, of course knew it as well.

It is one thing to derive this equation and another to truly understand it. When I see the Navier-Stokes equation, it comes to life for me. It’s not just an abstract combination of symbols that will eventually produce numbers. It is a living and breathing description of fluids dancing in space to whatever may be whipping them around. I can see the fluids. I can imagine clearly the forces upon them as they move in a way that is semi-ordered but ultimately unpredictable.

Actually, it’s more than this. The big
D
in the Navier-Stokes equation is called the material derivative, and it refers to watching velocities of fluid change not from a fixed reference frame but from one in which you are riding with the storm. When I think of Navier-Stokes, sometimes I imagine myself as a Lilliputian in a tiny canoe that has been lifted up and tossed high into the hurricane. The wind is so strong that it lifts my cheeks, distorting them as if the storm were a real-life version of a fun-house mirror. I watch as the fluids careen against and flow around me.

As well as I can visualize these fluids dancing, I know that my mother could see them even better. She couldn’t describe what she saw to me in any concrete way, but I know from her work just how much more ornate, nuanced, and ultimately more accurate her vision was than mine. That’s why she was a mathematician, the best of her generation. That’s, obviously, why I am not.

The problem of the universal appropriateness of the use of the Navier-Stokes equation was the problem that my mother was rumored to have cheated death to solve. Where does such a crazy idea come from? I submit to you the following evidence: my mother, a seventy-year-old woman, ill, taking medications several times a day that made her throw up, weak, unable to realistically command anywhere near the concentration required to solve such a problem, and far too old, even if healthy, to have the freshness of mind to make any headway on such a task. Who could possibly think my mother could somehow capture the magic necessary to answer a question that had baffled mathematicians, the greatest and brightest minds, for more than a century? Here is one simple answer. Mathematicians can think like this. Impossible problems perhaps require impossible scenarios. Since no one young and healthy had solved this problem, perhaps someone old and sick, by sheer will, could.

Here is another answer. Only crazy people can think like this. And mathematicians are, as my uncle would say,
szalency
, inherently crazy dreamers. They have no real sense of what can and can’t be done. They work on impossible problems because they are impossible people. As I’ve noted, there are likely no reasonable geniuses in this world. While there are, I know from personal experience, some reasonable mathematicians, they are not at the forefront of their profession but mere worker bees. Even most worker bees in mathematics are hopeless as fully functioning human beings.

For better or worse, I’m stuck with them. They are all I’ve ever known. I grew up around them. They are my family. I love them wholeheartedly. After a brief and unsuccessful effort at trying to pull away from this intensely dysfunctional culture obsessed with abstraction, I even ended up marrying into it.

Yes, I have, with one exception, avoided any discussion of my own relationships, past and present. I certainly haven’t forgotten about them, especially my life with my ex. You don’t forget even two years of marriage. In my August years, my memories of that time are, strangely, far more vivid than they were thirty years ago. Oh Catherine. Perhaps you will pick up this book if only to see where you are mentioned. Here. This is the first real time. Catherine. Scan this text and find your name. You were my first, and for many years I thought you’d be my only, true romantic love.