BOOK REVIEW
Not All Equations Are Equal
Robert Osserman
It Must Be Beautiful: Great Equations of Modern Science. Edited by Graham Farmelo. xvi + 283 pp. Granta Books, 2002. $25.
It Must Be Beautiful is a collection of essays, some of them outstanding, on a variety of topics in 20th-century science. In his foreword to the book, editor Graham Farmelo speaks of "the poetry of science" being "embodied in its great equations." He points out that although it has so far been impossible to define poetry uncontroversially, "No such problems beset mathematicians asked to define the term 'equation.'" The remainder of the book, however, seems designed to demonstrate just how slippery the term can be.
From a purely mathematical point of view, there are at least three distinct types of equations. Perhaps the most common is the one that faces students who are told, "Solve the following equation"—for example, "Find x if
x^{2} – 3x + 2 = 0." Students are thus justifiably confused when they are also presented with equations such as "x2 – 1 = (x – 1)(x + 1)" and are told that in this equation the equal sign has a very different meaning, since here the two sides really are equal, no matter what value we use for x. Therefore this type of equation should really be called an identity. The first equation is what is called a "conditional equation," which holds only for particular values of x. Applied to a conditional equation, the command "Solve the equation: . . ." means "Determine the particular values of x that make the two sides in fact equal." Students meet the third common type of equation in the phrase "Graph the equation: . . ." For example, y = x^{2} – 3x + 2 in essence defines the quantity y in terms of the quantity x by means of the expression on the right.
Careful mathematicians will sometimes distinguish among these distinct types of equations by using a different notation for each. Identities may be written in the form x^{2} – 1 ≡ (x – 1)(x + 1), with the symbol "≡" meaning "identically equal." For the third type of equation, many mathematicians have taken recently to adopting the notation "=:" (which may be read "is defined to be equal to") when they are introducing a new notation for a certain expression. However, in the vast majority of cases one finds the simple equal sign, and the reader is left to decipher what is meant from the context.
Equations from all three categories form the subject matter of the essays in It Must be Beautiful. The first category, conditional equations to be solved, includes some of the most famous in 20th-century physics: the Einstein equation of general relativity, Schrödinger’s wave equation, the Dirac equation and the Yang-Mills equation. Essays on these (by Roger Penrose, Arthur I. Miller, Frank Wilczek and Christine Sutton, respectively) are at the center of the book.
Those by Penrose and Sutton are gems: Both tackle their equations head-on and do a magnificent job of describing the physics behind them. The Einstein equation is notoriously difficult to explain in relatively nontechnical terms, because it relates the curvature of space-time to Newton's gravitational constant and the "mass-energy tensor." Providing some intuitive feeling for the terms themselves, much less the meaning of the equation, is a challenge that Penrose meets brilliantly.
The Yang-Mills equation is far less well known but has received increasing attention from both mathematicians and physicists ever since its introduction in 1953. Sutton tells the story of the physics leading up to it beautifully and describes its consequences well, coming up with particularly felicitous examples to illustrate the concepts introduced along the way.
Miller and Wilczek are far more skittish in dealing with their equations. Miller gives us a good deal of interesting information about Erwin Schrödinger's life, interests and proclivities, as well as a fine account of the bitter controversy between Schrödinger and Werner Heisenberg over their respective approaches to quantum theory. But Miller can’t quite bring himself to state the equation, except in a note, which appears at the end of the book. After the opening section of his essay, Wilczek drops the Dirac equation like a hot potato, promising to explain its terms in an appendix (which requires diligence to locate).
The first two essays in the book are devoted to two equations that are identities: Planck’s E = hf and Einstein's E = mc^{2}. The latter is probably the epitome of the "great equation" in physics. It carries a double whammy, informing us that two almost opposite entities, mass and energy, are related by a simple equation, and that the factor for converting one to the other is numerically enormous—the square of an already large number, the speed of light. It has two further qualities of "greatness": Anyone can understand it, and it has momentous implications, including the fearsome force of nuclear explosions.
In his foreword, Farmelo tells us that a "beautiful" equation has "universality, simplicity, inevitability and an elemental power"—a characterization that would, in my view, exclude most, if not all, of the equations in this book. His definition also leaves out the one characteristic I consider essential—the element of the unexpected. What is so striking about E = mc^{2} is not at all its "inevitability," but perhaps the exact opposite: the almost shocking realization that entities as different as mass and energy are in some sense just different forms of one and the same thing. Purely mathematical equations that I find beautiful have the same element of surprise over the fact that combining certain quantities yields a totally unexpected answer. A famous example is Euler’s equation: e^{iח}=-1, in which the imaginary number "i" is combined with an important constant ("e") and the ratio of a circle’s circumference to its diameter ("ח"), yielding most improbably the simple quantity "–1."
The intuitive feeling that the greatness of an equation reflects the element of surprise receives a kind of confirmation in Shannon’s equation I = –plog_{2}p, which forms the basis of information theory. Igor Aleksander describes the message it conveys as follows:
. . . [T]he amount of information I depends on the surprise that the message holds. This is because the mathematical way of expressing surprise is as a probability p; the less probable an event is, the more surprising it is and the more information it conveys.
Both the above equation and a second equation of Shannon's that introduces the famous "signal-to-noise ratio" fall into our third category—they do not assert the equality between known quantities, nor are they something to "solve." Rather, they are simply definitions of new quantities. The element of surprise here, qualifying these as truly great equations, is first, that one can give a quantitative definition of something as apparently elusive as the information content of a message, and second, that the ramifications of that definition have been so widespread.
Some equations turn out to be useful, even if it would be pushing things a bit to classify them as "great." Consider the Drake equation, which is the subject of an essay by Oliver Morton on the search for extraterrestrial intelligence. Frank Drake said that coming up with the equation "didn’t take any deep intellectual effort or insight." Although I can see that no grand intellectual effort was required, it did take a bit of insight to realize that it might be worthwhile to take a quantity that would appear to be completely beyond any hope of determining (the number N of radio-transmitting civilizations in our galaxy) and write it as the product of seven factors, such as the fraction of stars with planets, the average number of habitable planets around such stars and so on. This simple exercise forced those who believed that the subject was science fiction rather than science and that the number N was obviously zero to examine their assumptions a little more closely to decide which one (or ones) of the separate factors must be zero. As Morton puts it, "Most equations sit at the end of a creative process . . .; the Drake equation, on the other hand, is just a starting point."
With regard to "great equations," the book only goes downhill from here. The essay "Equations of Life," by John Maynard Smith, includes exactly one "equation," which could have been written as N = R, where N is the number of neutral mutations fixed in each generation and R is the neutral mutation rate per generation. The point is that the population size enters in two ways that cancel each other out. This could be classified as a neat observation, but it’s something of a stretch as a "great equation." In fact, the actual subject of this piece—the application of game theory to population biology—has little or nothing to do with that equation. Rather, the focus is on payoff matrices and the way they can lead to strikingly different qualitative and quantitative behavior of populations over time.
Robert May has also contributed an essay on population biology, which concentrates on chaos theory. The mathematical tool at its heart is an iteration, wherein one takes the expression ax(1 – x) and chooses some fixed value for the constant a; one then starts with some value for x, puts it into the expression, and forms an endless loop by feeding the answer back in as a new value for x. It came as a great surprise to mathematicians that this simple expression, now known as the "logistic map," could lead to so complicated a set of behaviors when subjected to a simple iteration. Biologists were perhaps even more surprised to discover these behaviors actually occurring in animal populations (in the fluctuations of grouse populations on moorland, for example).
The final essay in the book, by Aisling Irwin, tells the fascinating story of the gradual discovery of the complex workings of the ozone layer. Here the equal sign does not even make an appearance. Central to the story are the Molina-Rowland equations, which are of the form O_{3} --> O_{2} + O, O_{2} --> O + O, and O + O_{2} --> O_{3}. In other words, these "equations" are really chemical processes described in shorthand, where what is asserted is not the equality of the two sides, but the transformation of certain molecules into other ones. By now, the reader will have accepted the "great equation" format as the gimmick that it is and will be able to appreciate these stories for what they really are: compelling chapters in the advance of science, most of them beautifully told.