The Innocent Pleasures of Simple ModelsOctober 26, 2004
From Mathematics in Nature: Modeling Patterns in the Natural World.
Philip J. Davis
Mathematics in Nature: Modeling Patterns in the Natural World. By John A. Adam, Princeton University Press, Princeton, New Jersey, 2003, 360 pages, including a full bibliography, $39.50.
I suppose that to the average person "nature" suggests the birds, the bees, and the flowers. To John A. Adam, nature includes these things, certainly, but it also takes in mud cracks, river meanders, and the natural frequencies of shaking trees. This is as it should be. To me, with my pluralist tendencies, nature encompasses everything there is. But having said that, I pull back a bit and ask myself whether nature includes products of the human imagination, such as the rules of baseball, the song Love Me or Leave Me, or even mathematics itself.
Adam, a professor of mathematics at Old Dominion University, is a long-time devoted collector and skillful expositor of simple mathematical models of natural phenomena. I infer that two of his criteria in selecting the models for this fine book were that the phenomena (a) have a visual aspect (such as a tidal bore coming up the Bay of Fundy) and (b) can be modeled with mathematics that is no more advanced than the elementary portions of partial differential equations. To put this last in another way: The basic mathematics in the collection should have been developed no later than the 19th century.
Mathematics in Nature is thus appropriate as a source of supplementary undergraduate material. But it can be more than this: Adam has laced his mathematical models with popular descriptions of the phenomena selected. One example: He lays out the "loves, life, and times" of water striders (the pond skater bug), of whose existence I've been aware since childhood, but of whose hydro-dynamical aspects I admit to a shameful ignorance. Mathematics in Nature can accordingly be read for pleasure and instruction by the select laity who are not afraid of reading between the lines of equations.
Adam has a soft spot for meteorological optical phenomena. Accordingly, halos, rainbows, glories, iridescences of various sorts are given pride of place. Then come clouds, sand dunes, and hurricanes. The book winds up with models of the processes by which the leopard and the butterfly get their spots.
It is refreshing to see how much can be accomplished with simple mathematical means. Focusing on the art of modeling, Adam devotes an introductory chapter to an explanation of the nature and methodology of mathematical modeling, and the limitations of models both simple and complex. I now know that meteorologists currently employ at least a dozen different models of hurricanes, exhibiting different levels and varieties of complexity and serving different purposes. Modeling is no easy matter, but simple modeling is a necessary platform from which the process of mathematization gets off the ground.
The "simplicity" of Adam's examples, in the sense of the preceding paragraph, leads me to muse quite generally about the notion of simplicity. But first, a few odd thoughts generated by the book.
I had supposed that random walk theory would tell me how long it takes for a spoon of unstirred sugar to disperse uniformly through a cup of tea. Before reading this book, I wasn't aware that it takes three months. (I plan to check this out experimentally.)
Though there are six major trees of fifty or more feet around my house, and numerous minor trees, I never knew that the natural frequency of their vibration in the wind is inversely proportional to the square root of their height. Adam presents a model that is the platform for aeroelastic computations that still plague theorists.
Have you ever speculated on how large an elephant or a sequoia can grow? You'll learn here how the solution to Airy's differential equation provides an answer.
All the assumptions built into a model could hardly be made explicit; many are made unconsciously. When mathematics and physics blow the whistle and say "no way" to a hundred-foot elephant, how firm is the "no"? An evolutionary biologist of my acquaintance once told me that despite the roadblocks placed by physicists and applied mathematicians in the way of certain putative developments, nature can do what she wants.
"After all," my friend explained, "nature didn't make the laws of nature. Man made them."
"You mean to say," I teased him, "that nature could produce a Narragansett Bay crab that comes out of the sand and whistles Yankee Doodle on the Fourth of July, and only on that day?"
"Absolutely. And factor in the leap years as well."
That ended our interchange, for I sensed that I had inadvertently moved the discussion from biophysics to metaphysics.
Adam has included two dozen beautiful color photos of patterns in nature. Why is the human mind so keen to pick out certain visual aspects of nature, call them patterns, hang them on the wall, and then create mathematical equations to describe them? Is there a mathematical model that explains such passions in terms of the hard-wiring of our brains?
The idea of simplicity is itself by no means simple. What is simplicity? Can it be defined? Is it logical? Is it psychological? Can it be measured and compared? Does it make sense to ask whether it is simpler to roll out of bed in the morning than to boil a four-minute egg? Is simplicity objective or subjective? Is it time varying or culture varying?
There is a story---possibly apocryphal---that the late Stan Ulam did some of the early A-bomb computations on the back of an envelope. This leads me to wonder why certain phenomena can be explained or modeled with simple means and others cannot.
In science, simplicity is often related to truth and beauty. Adam uses the word when he mentions the "beautiful mathematics" that George Greenhill employed in 1881 to show that the flagpole in Kew Gardens could not exceed three hundred feet in height. I've heard physicists opine that it is the simple theories (short formulas, few arbitrary elements) that are most likely to be true. At the same time, they point out that for 25 years particle physics has been dominated by the "standard model." This model agrees with experiment-and yet it has something like 17 arbitrary constants and the whole thing has such a patched-together look that nobody believes it to be in its final form. Is this a call for the development of new mathematics?
The almost impenetrable geometry in which Newton couched many of his conceptions seems to us to be screaming for analytic reformulations to be let out. I have on my desk a monograph on string theory, which is said to unify general relativity and quantum theory to produce quantum gravity. The book is 700 pages long. Before tackling this tome, you'd better brush up, for starters, on your Ricci tensors and Lie group theory, your pseudo-symplectic spinors, and your supersymmetry algebras. This strikes me as complexity in spades. But perhaps science students of the future will regard it all as a piece of cake.
Unifications are often thought to be simplifications. The Navier-Stokes equations that unify the action of compressibility and viscosity are a remarkable achievement. Would the Clay Institute consider the equations simple after having declared their solution to be worth a million-dollar prize?
The four equations of Maxwell that unite electricity and magnetism were thought to be the end of physics, with the efforts to follow seen merely as a tedious and routine working out of their implications. I remind myself that Maxwell put forward twenty (scalar) equations; the compact vector form that now allows the equations to be placed conveniently on T-shirts came later from Oliver Heaviside.
I wonder therefore whether the seemingly beautifully compact notations that render things "simple" are little more than rhetorical or semiotic devices that cover up the ugly and the complex. While admitting that "good" notations can embody great conceptual and heuristic strength, I am baffled by the apparent transformation from the complex to the simple. The often repeated paraphrase "The equations are smarter than we are" strikes me as mathematical mysticism. Its fairly recent parallel, "The computers are smarter than we are," strikes me as a very dangerous belief.
Opinions about the role of simplicity in science and mathematics are certainly mixed. Steven Weinberg, in his Towards the Final Laws of Physics, wrote:
"I once heard Dirac say in a lecture, which largely consisted of students, that students of physics shouldn't worry too much about what the equations of physics mean, but only about the beauty of the equations. The faculty members present groaned at the prospect of all our students setting out to imitate Dirac."
Considering beauty in mathematics, Gian-Carlo Rota wrote:
"There are beautiful theorems with ugly proofs, e.g., the prime number theorem which provides an increasingly accurate figure for p(n), the number of primes less than a given number n. Picard's theorem asserting that an entire analytic function takes on all values with two possible exceptions has a most beautiful proof: five lines."
Perhaps---if you suppress the fact that the proof of Picard rests on the previous construction of a modular elliptic function w(z), which takes many pages to do.
In his Religio Medici (1643) Sir Thomas Browne wrote:
"I cannot tell by what logick we call a toad ugly"
but would not stay for an answer.
When the fundamental structures of nature have been modeled in beautiful mathematics, therein lies a profound mystery. Some will merely accept it, while others, including John Adam, will be right there to explain it to us.
Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island, and can be reached at firstname.lastname@example.org.