| | Ivars Peterson's MathTrek |
September 15, 1997
I first visited the CN Tower in 1976, soon after it was completed. Recently, I returned to the tower's observation deck, welcoming the chance to experience the breathtaking view once again. For my son Eric, it was a great thrill, especially sitting on a glass floor suspended above the retractable, clamshell roof of the SkyDome 1,300 feet below.
I couldn't quite bring myself to step onto the glass. At the same time, I couldn't help wondering at the tremendous engineering effort that had gone into constructing the tower and ensuring that it remained stable and safe, year after year. Here was human knowledge and insight applied in a daring, spectacular way -- a bold statement of mastery and a visible tribute to predictability.
Of course, not all engineering efforts are equally successful. Bridges can collapse, planes crash, and software fail. There are limits to what can be calculated and foretold. And there are often unintended consequences.
I have always been fascinated by the limits of human knowledge. From physical theory and experiment, we can derive laws and constants to describe the world in which we live. Mathematical logic furnishes clear answers to precisely defined questions. Yet in all these efforts to grasp the meaning of what we see around us, we constantly face the random and the unknowable. As in the Heisenberg uncertainty principle of quantum mechanics, we can sometimes define with remarkable precision the extent and shape of our ignorance. But nothing rids us of the element of chance.
In our lives, we deliberately carve out pockets of reason and logic; we endeavor to circumscribe and control chaos and chance. Out of these hard-won, though limited certainties, we build our remarkable technologies, and in our everyday lives we find ways to hack through and negotiate the jungles of uncertainty that constantly press in upon us. So we persist in our efforts to learn, calculate, organize, classify, and predict.
Science involves finding formulas -- the physical laws that we believe underlie the machinations of nature. Mathematics tells us that, given a formula, it's relatively easy to compute answers. The task is far more difficult, however, when we are given the answers and we must work out the formulas.
We collect data -- strings of numbers -- and from patterns among the numbers in those strings we make stabs at the formulas. As the strings get longer and the numbers more precise, the formulas get better, and they can work amazingly well.
Yet we have no guarantee that we have the right formulas, that we are applying them appropriately, or even that the formulas exist at all. We could be trapped in a local, accidental pattern in an otherwise random existence. All we can do is keep testing our hypotheses and revel in the curious freedom that chance and inevitable uncertainty provide from the straitjacket of determinism.
Those are some of the thoughts that underlie my newest book, The Jungles of Randomness: A Mathematical Safari. The book represents a distillation of more than 15 years of reporting and writing for Science News. I believe that mathematics plays a crucial role in helping us to make sense of the natural world. In my book, I use games and puzzles, fractals and chaos, along with viruses, fireflies, drums, dice, and amusement park rides to illuminate the need to sort through the various meanings of randomness and to recognize the limits of certainty. I delve into graph theory, random-number generators, Ramsey numbers, random walks, resampling statistics, probability theory, error-correcting codes, and a host of other mathematical topics.
Humans are predisposed to seeing order and meaning in a world that too often presents random choices and chaotic evidence. Random events sometimes get linked into a meaningful, though ultimately bogus, scenario. In a review of the recent movie Conspiracy Theory, one critic disparaged the credibility of the film's convoluted plot and commented on the unlikelihood of coordinated, clandestine activity on the vast scale implied by the narrative. Yet the movie was involving and enjoyable, the critic admitted. Would you go see a movie titled Chaos Theory instead?
In a recent commentary on the peculiar attraction of conspiracy films, another critic noted, ". . .the conspiracy theory gives the event meaning that it would otherwise lack. The phenomenon is familiar from the culture of JFK assassination buffs, who cannot abide the squalid possibility that a grim little nobody with stained teeth and rancid breath reached out to twist the shape of history. If that were true, the news is very bad: It means there is nothingness in the universe and that random winds wreck lives and nations on no principle save whimsy."
At the same time, the mathematics of Ramsey theory demonstrates that patterns can, and indeed must, arise out of pure randomness and pure chance. Highly regular patterns exist in sufficiently large sets of randomly selected objects, whether they are gatherings of people, piles of pebbles, stars in the night sky, or sequences of numbers generated by the throw of a die.
On a clear, moonless night, we see thousands of stars scattered across the sky. With so many stars visible, it's not particularly difficult to pick out groups that appear in a certain pattern: four stars that very nearly form a straight line or a square, six stars that define a cross, seven stars in the shape of a dipper. The human imagination fills in the rest, connecting dots to create a menagerie of celestial creatures that inhabit the sky.
The same sort of connecting of dots can occur in other contexts. In his book The Bible Code, Michael Drosnin claims there is a hidden code in the Bible that foretells the future. The strings of Hebrew symbols, arrayed in crossword fashion, describing those events can be found among the 304,805 characters of the original version of the first five books of the Bible, he contends.
The trouble is that given so many characters, which can be reorganized into blocks in a large number of ways and interpreted either as letters or numbers, all sorts of patterns can appear. The human mind inevitably finds something that it identifies as meaningful among so many possibilities, especially when links are loosely construed and spelling mistakes are allowed. Kennedy is there near the word Dallas; Shakespeare keeps company with Hamlet.
In a stinging commentary, mathematician Shlomo Sternberg of Harvard University dismisses the entire matter as a hoax. He points out that the decoding depends on the letter-for-letter accuracy of the current electronic version of the Bible, when historical evidence indicates discrepancies among different editions of the text. Moreover, hidden messages can be produced in any sufficiently long text, he argues.
Mathematicians Brendan McKay of the Australian National University and Dror Bar-Natan of Hebrew University are also extremely skeptical. "We believe there are serious flaws in the mathematics, and we have serious concerns about other aspects," McKay notes. He goes on to show how it's possible to apply the same techniques to the text of the Bible to find Bill Gates, along with nearby references to MS-DOS, virtual reality, and software!
There are times, however, when identifying a pattern and establishing a causal link are crucial to our lives, even when the evidence may be scanty. We notice all sorts of coincidences. Some result from chance events that turn out to be far more probable than many people imagine. Others have hidden causes, so they don't really count as coincidences. When researchers find odd clusters of certain diseases, birth defects, or cancers in their data, how can we know when these events represent the luck of the draw or whether they reflect some underlying cause?
The immense difficulty of establishing a causal relationship between a type of childhood leukemia and water polluted with traces of the chemical trichloroethylene underlies the poignant, true-life story told in Jonathan Harr's compelling legal thriller, A Civil Action. Faced with daunting scientific, statistical, and legal intricacies, the lead lawyer for the plaintiffs has to convince a jury that industrial contaminants had leaked into well water in concentrations large enough to affect the health of residents using that water.
In The Jungles of Randomness, I conclude: "With our mathematical and statistical machetes, we can hack out extensive networks of trails and clearings that provide for most of our day-to-day needs and make sense of some fraction of human experience. The vast jungle, however, remains close at hand, never to be taken for granted, never to reveal all its secrets, and always teasing the inquiring mind."
Copyright © 1997 by Ivars Peterson.
Drosnin, Michael. 1997. The Bible Code. New York: Simon & Schuster.
Duncan, Ronald, and Miranda Weston-Smith. 1977. The Encyclopedia of Ignorance: Everything You Ever Wanted to Know About the Unknown. Oxford, England: Pergamon Press.
Harr, Jonathan. 1995. A Civil Action. New York: Vintage.
Jackson, Allyn. 1997. Book review of The Bible Code. Notices of the American Mathematical Society 44(September):935-938
Peterson, Ivars. 1997. The Jungles of Randomness: A Mathematical Safari. New York: Wiley.
______. 1996. Fatal Defect: Chasing Killer Computer Bugs. New York: Vintage.
Rothstein, Edward. 1997. Is destiny just a divine word game? New York Times (Aug. 12): B1.
Schickel, Richard. 1997. Full-service paranoia. Time (Aug. 18):62.
Sternberg, Shlomo. 1997. Comments on The Bible Code. Notices of the American Mathematical Society 44(September):938-939 (also available at http://www.ams.org/index/committee/publications/notices/199708/comm-sternberg.html).
Tenner, Edward. 1996. Why Things Bite Back: Technology and the Revenge of Unintended Consequences. New York: Knopf.