29 August 2016

What Remains Unknown

How should we conduct our lives? How should we live? Clichés, I am aware. Nonetheless, they are questions that beset us daily, I think. We each have our ideas on such matters, and we work to not only live out our ideas but to pass them on as well. In his essay “Fate,” Ralph Waldo Emerson says, “In our first steps to gain our wishes we come upon immovable limitations. We are fired with the hope to reform men. After many experiments we find that we must begin earlier,--at school. But the boys and girls are not docile; we can make nothing of them.” Nothing happens, nothing changes. Having spent almost half a century in the classroom, to me Emerson’s observation comes as no surprise. But there are two interpretations of his declaration that “we can make nothing of them.” The first possibility seems to be that in our efforts we cannot really have any effect on the future conduct of our children despite our educational efforts. Like the doctor who tells his patient he must stop smoking or the priest who cautions his congregant that she must stop her profligacy, despite the effort the teacher remains helpless to effect the future behavior of her students. We have but the vaguest idea what they will do or who they might become. (It interests me nonetheless that it is to the teachers that the press returns after a horrific criminal event. “What was (s)he like as a student?” they ask, as if something in the activity of solving geometric proofs or reading The Great Gatsby would have offered some hint to any future behaviors. I, of course, know how to conduct a life (ha!), but there is no way that I can make anyone follow my lead.
     But the second interpretation of Emerson’s remarks intrigues me equally. Because he might mean by “make nothing of them” that we cannot understand them. And I am more and more inclined to accept this ignorance and to establish my practice from it.
     I am neither a mathematician nor physicist. Nor am I a scientist. I would, however, like to think in a relatively technical way about the physical concept of turbulence because I think it has application to the issues of the classroom and our capacity to know our students. The OED offers a definition of turbulence in natural conditions as “stormy or tempestuous state or action,” and turbulent as [a situation] characterized by violent disturbance or commotion; disorderly, troubled.” Turbulence is the chaotic motion of air and water, the wild erratic dance of particles each affected by such factors as speed, viscosity, size, pressure and density. Turbulence characterizes hurricanes and tornadoes and ocean waves breaking onto shore; turbulence is the smoke that rises from a cigarette¾who knows how the smoke rises¾ or the movement of air during airplane flight that causes me to tightly grip the arm rests and to cramp my toes. Turbulence is a “wild erratic dance of fluids,” the novelist Stuart Rojstaczer writes, and turbulence results from the movement of particles comprising the particular substance¾air, earth and water. There would be great benefit to humans if we were able to predict the movement of such flow for such phenomena as hurricanes, tornadoes, and classrooms, etc., for then we could predict their occurrences, paths, and intensities. Such knowledge might have then been able to keep Dorothy from having to travel to Oz! These particles are all in continuous movement and interaction with other particles in motion. Their movements depend on density, velocity, speed, viscosity and air (or even internal) pressure. How to predict the movement of such particles may be approximated but never absolutely defined. Though it might be convenient, we cannot forecast with any high degree of certainty this movement that results in what we know as turbulence. In my experience the weather people are fairly consistently inaccurate: they predict a heavy snowfall and no snow falls; they warn that a hurricane should strike land at a certain hour and alas, with good fortune the storm heads out to sea. I carry my umbrella on the forecaster’s warning but it doesn’t rain; I become rain-drenched on days when the sun was supposed to shine.     
     We like to imagine that everything about our natural world will follow predictable patterns, and science has suggested that formulas can be written to enable accurate prediction. Systems of classroom management offer clear predictive definitions to maintain order and ensure proper functionings. The fictional mathematician Rachela Karnokovitch in Rojstaczer’s novel The Mathemetician’s Shiva to which all the renowned mathematicians in the world have come to visit, says “ . . . to make the physical world around us endurable, we like to emphasize that the gases and liquids that move our oceans, allow our planes to fly, and make our boats sail follow predictable, orderly paths. Often they don’t, and this difference between our wish for an orderly universe and the reality of the calamity of the natural world makes us deny reality.” The problem of turbulence has not yet been solved, and the world continues to suffer its consequences.  “Understanding a physical phenomenon like turbulence ultimately means predicting its behavior, or at the very least understanding just what can and what cannot be predicted over time. Prediction means quantification of the basic physical processes that drive turbulence. But this is something we cannot currently do with any reasonable degree of sophistication.” When asked what he would ask God, Heisenberg is said to have responded, “When I meet God, I am going to ask him two questions: Why relativity? And why turbulence? I really believe he will have an answer for the first.” Heisenberg did not think that turbulence could be understood much less controlled. Turbulence, an occurrence when a particle becomes chaotic and behaves unstably, cannot be accurately predicted. The movement of such particles depends on so many factors: density, velocity, speed, viscosity, pressure. Rojstaczer writes, “From the standpoint of a mathematician . . . the problem of turbulence is fundamentally one of being certain, to prove, that there exists in the universe a set of descriptors, in this case partial differential equations, that can encompass the behavior of fluids as they move at velocities that cause chaos. . . .” But the Nobel Prize winning physicist Richard Feynman has said that turbulence is the last unknown in physics: “. . .  given an arbitrary accuracy, no matter how precise, one can find a time long enough that we cannot make predictions valid for that long a time. Now the point is that this length of time is not very large . . . It turns out that in only a very, very tiny time we lose all our information . . . We can no longer predict what is going to happen.” Feynman says, “If water falls over a dam, it splashes. If we stand nearby, every now and then a drop will land on our nose. This appears to be completely random . . . The tiniest irregularities are magnified in falling, so that we get complete randomness.” Rojstaczer’s topologist protagonist and narrator  says that “Tornados start from small disturbances that don’t mean a thing and almost always dissipate. But somehow one particular random bad event attracts others and all of them grow and attract some nasty stuff.” Tornadoes, hurricanes, and air turbulence cause great tragedy and/or discomfort. If we would understand turbulence, then we could predict its behavior. “Prediction implies quantification of the basic physical processes that drive turbulence but we are as yet incapable of doing so with any reasonable degree of sophistication.” Our methods may discern trends and patterns but certainty eludes us certainly.
     Not that the attempt to predict turbulence (and thus control it) has not been made. The Navier-Stokes equations measure the motion of a fluid—turbulence¾and these formulas have the intention to predict turbulent flow. Navier-Stokes attempts to predict the movement of these substances that are, of course, comprised of ‘particles.’ But in fact, the Navier-Stokes formula, which seems to be based on Newton’s second law of motion, has not been solved: we don’t know that it works to describe the motion of fluids “when they flow as lazily as rivers as when they dance around an airplane wing.” Navier-Stokes occupies a central place in mathematical culture: a solution to the Navier-Stokes formula would earn its solver the Millennium Prize—a Nobel-like prize in mathematics worth one million dollars. The solution to the Navier-Stokes formula would show that the Navier-Stokes formula works in all cases, and that turbulence could be predicted. Thus far, Navier-Stokes remains unsolved.
     In the classroom exists the conditions for the appearance of turbulence. Our students (and ourselves as teachers) are particles in movement and interaction both within and between individuals. The movements in the classroom depend on density, velocity, speed, viscosity and pressure of the various particles. Who are they, I wonder? Emerson writes that “In different hour a man represents each of several of his ancestors, as if there were seven or eight of us rolled up in each man’s skin, seven or eight ancestors at least, and they constitute the variety of notes for that new piece of music which his life is.” How to predict the movement of particles within the classroom may be approximated but never absolutely defined with any degree of certainty. Every day a teacher walks into her classroom with a diagram (a lesson plan) that (like the Navier-Stokes formula) should predict the direction and movement of the mass of particles, but in fact, like the Navier-Stokes these lessons plans have not as yet been proven to effect anything. Who are they, I wonder. And I don’t really know. Nor do I know how to solve this unknown. Charles Fefferman says of Navier-Stokes, “Since we don’t know whether these solutions exist, our understanding is at a very primitive level. Standard methods from PDE appear inadequate to settle the problem. Instead, we probably need some deep, new ideas.” We try to control our lives with rationality, with predictive formulas, but in fact, the Navier Stokes formulas have no solutions. Rojstaczer writes, “[And] this difference between our wish for an orderly universe and the reality of the calamity of the natural world makes us deny reality.” Our classrooms might not need prepare anyone for the real world, but the real world enters our classroom nevertheless, and we deny reality when we ignore the unsolvability of Navier-Stokes.
     And so I stand in the classroom and the weather is calm. I think. But there are particles in motion and I do not know—I cannot know—which particles will attract others and lead to some rather nasty stuff. My lesson plans deny reality because they offer some false picture of the appearance of order. It is, I recognize, a false picture.  But as it is supposed to be my purpose to impose order on disorder, I construct models of reality that assume simple cause and effect relationships, and believe that my work will make correction in lives as easily as changing a lightbulb. But in fact, I am not at all not able to predict the erratic wild dance of either interpersonal or intrapersonal particles. It is not possible to make a theoretical model to describe the behavior of a turbulent flow¾in particular, of its internal structures? At so at this time. I often find myself caught in the midst of turbulence and I tighten my grip on the armrests and curl my toes to grip the soles of my shoes. I stand in the classroom and look out on the horizon for the threat of tornado or the threat of the hurricane or the random drop of water on my nose with only my lesson plan as protection.     


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