"How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of reality?" asked Albert Einstein.
If you're like me, you understand readily how one can describe nature's wonders using poetry or music, painting or photography. Wordsworth's "I Wandered Lonely as a Cloud" and Vivaldi's "Four Seasons" richly depict their natural subjects, as do Monet's water lilies and Ansel Adams' photos of Yosemite. But mathematics? How can you describe a tree or cloud, a rippled pond or swirling galaxy using numbers and equations?
Extremely well, as Einstein knew better than most, of course. In fact, most scientists would agree that, when it comes to teasing out the inherent secrets of the universe, nothing visual, verbal, or aural comes close to matching the accuracy and economy, the power and elegance, and the inescapable truth of the mathematical.
How is this so? Well, for the math-challenged, for that person who has avoided anything but the most basic arithmetic since high school, who feels a pit in his stomach when he sees an equation—that is, for myself—I will attempt to explain, with the help of some who do mathematics for a living. If you're math-phobic, too, I think you'll get a painless feel for why even that master of describing nature with words, Thoreau, would hold that "the most distinct and beautiful statements of any truth must take at last the mathematical form."
While many early civilizations, including Islamic, Indian, and Chinese, made important contributions to mathematics, it was the ancient Greeks who invented much of the math we're familiar with. Euclid fathered the geometry we named after him—all those radii and hypotenuses and parallel lines. Archimedes approximated pi. Ptolemy created a precise mathematical model that had all of the heavens wheeling around the Earth.
"With a few symbols on a page, you can describe a wealth of physical phenomena."
The Greeks' discoveries are timeless: Euclid's axioms are as unimpeachable today as when he devised them over 2,000 years ago. And some Greek proto-physicists did use their newfound skills to tackle mysteries of the natural world. With basic trigonometry, for example, the astronomer Eratosthenes estimated the diameter of the Earth with over 99 percent accuracy—in 228 B.C.
But while the Greeks believed that the universe was mathematically designed, they largely applied math only to static objects—measuring angles, calculating volumes of solid objects, and the like—as well as to philosophical purposes. Plato wouldn't let anyone through the front door of his acclaimed Academy who didn't know mathematics. "He is unworthy of the name of man," Plato sniffed, "who is ignorant of the fact that the diagonal of a square is incommensurable with its side." And so it remained for a millennium and a half.
The measure of all things
Galileo changed all that in the early 17th century. Eschewing the Greeks' attempts to explain why a pebble falls when you drop it, Galileo set out to determine how. The "great book" of the universe is written in the language of mathematics, he famously declared, and unless we understand the triangles, circles, and other geometrical figures that form its characters, he wrote, "it is humanly impossible to comprehend a single word of it [and] one wanders in vain through a dark labyrinth." (Wordsworth or Monet might take issue with that statement, but wait.)
Galileo sought characteristics of our world that he could measure—variable aspects like force and weight, time and space, velocity and acceleration. With such measurements, Galileo was able to construct those gems of scientific shorthand—mathematical formulas—which defined phenomena more concisely and more powerfully than had ever been possible before. (His contemporary, the German mathematician Johannes Kepler, did the same for the heavens, crafting mathematical laws that accurately describe the orbits of planets around the sun—and led to the scrapping of Ptolemy's Earth-centric model.)
A tidy sum
A classic example is the formula commonly shown as d = 16t2. (Hang in there, math-phobes. Your queasiness, which I share, should go away when you see how straightforward this is.) What Galileo discovered and ensconced in this simple equation, one of the most consequential in scientific history, is that, when air resistance is left out, the distance in feet, d, that an object falls is equal to 16 times the square of the time in seconds, t. Thus, if you drop a pebble off a cliff, in one second it will fall 16 feet, in two seconds 64 feet, in three seconds 144 feet, and so on.
Galileo's succinct formula neatly expresses the notion of acceleration of objects near the surface of the Earth, but that is just the start of its usefulness. First, just as with any value of t you can calculate d, for any value of d you can figure t. To get to t, simply divide both sides of the formula d = 16t2 by 16, then take the square root of both sides. This leaves a new formula:
t = √ d 16
This compact equation tells you the time needed for your pebble to fall a given distance—any distance. Say your cliff is 150 high. How long would the pebble take to reach the bottom? A quick calculation reveals just over three seconds. A thousand feet high? Just shy of eight seconds.
What else can you do with a pithy formula like d = 16t2? Well, as hinted above, you can make calculations for an infinite number of different values for either d or t. In essence, this means that d = 16t2 contains an infinite amount of information. You can also substitute any object for your pebble—a pea, say, or a boulder—and the formula still holds up perfectly (under the conditions previously mentioned). Could a single poem or painting do as much?
"Mathematics captures patterns that the universe finds pleasant, if you like."
And because the same mathematical law may govern multiple phenomena, a curious scientist can discover relationships between those phenomena that might have otherwise gone undetected. Trigonometric functions, for instance, apply to all wave motions—light, sound, and radio waves as well as waves in water, waves in gas, and many other types of wave motions. The person who "gets" these trig functions and their properties will ipso facto "get" all the phenomena that these functions govern.
A wealth of data
The power of a potent equation extends still further. Take Isaac Newton's universal law of gravitation, which brilliantly combines Galileo's laws of falling bodies with Kepler's laws of planetary motion. Many of us know gravity vaguely as that unseen force that keeps the pebble in your palm or your feet on the ground. Newton described it this way:
F = Gm1m2 r2
I won't go into this formula, but just know that from it you can calculate the gravitational tug between just about any two objects you can think of, from that between your coffee cup and the table it rests on, to that between one galaxy and another. Or, depending on which variables you know, you can nail down everything from the acceleration of any freely falling object near the Earth's surface (32 feet per second during every second of its fall) to the mass of our planet (about 6,000,000,000,000,000,000,000 tons).
"With a few symbols on a page, you can describe a wealth of physical phenomena," says astrophysicist Brian Greene, host of NOVA's series based on his book The Fabric of the Cosmos. "And that is, in some sense, what we mean by elegance—that the messy, complex world around us emanates from this very simple equation that you have written on a piece of paper."
And like Galileo's d = 16t2, Newton's formula is amazingly accurate. In 1997, University of Washington researchers determined that Newton's inverse-square law holds down to a distance of 56,000ths of a millimeter. It may hold further, but that's as precise as researchers have gotten at the moment.
What amazes me most about Galileo and Newton's formulas is their exactitude. In Galileo's, the distance equals exactly the square of the time multiplied by 16; in Newton's, the force of attraction between any two objects is exactly the square of the distance between them. (That's the r2 in his equation.) Such exactness crops up regularly in mathematical descriptions of reality. Einstein found, for instance, that the energy bound up in, say, a pebble equals the pebble's mass times the square of the speed of light, or E = mc2.
Even things we can see and touch in nature flirt with mathematical proportions and patterns. Consider the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… Notice a pattern? After the first, every number is the sum of the previous two. The Fibonacci sequence has many interesting properties. One is that fractions formed by successive Fibonacci numbers—e.g., 3/2 and 5/3 and 8/5—get closer and closer to a particular value, which mathematicians know as the golden number. But what about this: Many plants adhere to Fibonacci numbers. The black-eyed susan has 13 petals. Asters have 21. Many daisies have 34, 55, or 89 petals, while sunflowers usually have 55, 89, or 144.
Is God a mathematician?
The apparent mathematical nature of Nature, from forces to flowers, has left many since the time of the Greeks wondering, as the mathematician Mario Livio does in his book of the same title, "Is God a mathematician?" Does the universe, that is, have an underlying mathematical structure? Many believe it does. "Just as music is auditory patterns that the human mind finds pleasant," says Stanford mathematician Keith Devlin, "mathematics captures patterns that the universe finds pleasant, if you like—patterns that are implicit in the way the universe works."
"Einstein used mathematics to see a piece of the universe that no one had ever seen before."
So did we humans invent mathematics, or was it already out there, limning the cosmos, awaiting the likes of Euclid to reveal it? In his book Mathematics in Western Culture, the mathematician Morris Kline chose to sidestep the philosophical and focus on the scientific: "The plan that mathematics either imposes on nature or reveals in nature replaces disorder with harmonious order. This is the essential contribution of Ptolemy, Copernicus, Newton, and Einstein."
Seeing the invisible
Formulas like Galileo's and Netwon's make the invisible visible. With d = 16t2, we can "see" the motion of falling objects. With Newton's equation on gravity, we can "see" the force that holds the moon in orbit around the Earth. With Einstein's equations, we can "see" atoms. "Einstein is famous for a lot of things, but one thing that is often overlooked is he's the first person who actually said how big an atom is," says Jim Gates, a physicist at the University of Maryland. "Einstein used mathematics to see a piece of the universe that no one had ever seen before."
Today, with advanced technology, we can observe individual atoms, but some natural phenomena defy any description but a mathematical one. "The only thing you can say about the reality of an electron is to cite its mathematical properties," noted the late mathematics writer Martin Gardner. "So there's a sense in which matter has completely dissolved and what is left is just a mathematical structure." Charles Darwin, who admitted to having found mathematics "repugnant" as a student, may have put it best when he wrote, "Mathematics seems to endow one with something like a new sense."
Mathematics also endows one with an ability to predict, as Galileo's and Newton's formulas make clear. Such predictive capability often leads to new discoveries. In the mid-1990s, Kyoto University researchers realized to their surprise that equations originally devised by the mathematical genius Alan Turing predicted that the parallel yellow and purple stripes of the marine angelfish have to move over time. Stable, unmoving patterns didn't jive with the mathematics. To find out if this was true, the researchers photographed angelfish in an aquarium over several months. Sure enough, an angelfish's stripes do migrate across its body over time, and in just the way the equations had indicated. Math had revealed the secret.
"There really is a facing-the-music that math forces, and that's why it's a wonderful language for describing nature," Greene says. "It does make predictions for what should happen, and, when the math is accurately describing reality, those predictions are borne out by observation."
A math for all seasons
So much mathematics exists now—one scholar estimates that a million pages of new mathematical ideas are published each year—that when scientists face problems not solvable with math they know, they can often turn to another variety for help. When Einstein began work on his theory of general relativity, he needed a mathematics that could describe what he was proposing—that space is curved. He found it in the non-Euclidean geometry of 19th-century mathematician Georg F. B. Riemann, which provided just the tool he required: a geometry of curved spaces in any number of dimensions.
Or, if necessary, they invent new math. When the late mathematician Benoit Mandelbrot concluded that standard Euclidean geometry, which is all about smooth shapes, fell short when he tried to mathematically portray "rough" shapes like bushy trees or jagged coastlines, he invented a new mathematics called fractal geometry. "Math is our one and only strategy for understanding the complexity of nature," says Ralph Abraham, a mathematician at the University of California Santa Cruz, in NOVA's Hunting the Hidden Dimension. "Fractal geometry has given us a much larger vocabulary, and with a larger vocabulary we can read more of the book of nature." Galileo would be so proud.
Galileo would also be proud of just how much his successors have achieved with his scientific method. Formulas from his own on falling bodies to Werner Heisenberg's on quantum mechanics have provided us the means to collect and interpret the most valuable knowledge we have ever attained about the workings of nature. Altogether, the most groundbreaking advances of modern science and technology, both theoretical and practical, have come about through the kind of descriptive, quantitative knowledge-gathering that Galileo pioneered and Newton refined.
"Do not worry about your difficulties in mathematics; I can assure you that mine are still greater."
Newton's law of gravity, for instance, has been critical in all our missions into space. "By understanding the mathematics or force of gravity between lots of different bodies, you get complete control and understanding, with very high precision, of exactly the best way to send a space probe to Mars or Jupiter or to put satellites in orbit—all of those things," says Ian Stewart, an emeritus professor of mathematics at the University of Warwick in England. "Without the math, you would not be able to do it. We can't send a thousand satellites up and hope one of them gets into the right place."
Mathematics underlies virtually all of our technology today. James Maxwell's four equations summarizing electromagnetism led directly to radio and all other forms of telecommunication. E = mc2 led directly to nuclear power and nuclear weapons. The equations of quantum mechanics made possible everything from transistors and semiconductors to electron microscopy and magnetic resonance imaging.
Indeed, many of the technologies you and I enjoy every day simply would not work without mathematics. When you do a Google search, you're relying on 19th-century algebra, on which the search engine's algorithms are based. When you watch a movie, you may well be seeing mountains and other natural features that, while appearing as real as rock, arise entirely from mathematical models. When you play your iPod, you're hearing a mathematical recreation of music that is stored digitally; your cell phone does the same in real time.
"When you listen to a mobile phone, you're not actually hearing the voice of the person speaking," Devlin told me. "You're hearing a mathematical recreation of that voice. That voice is reduced to mathematics."
And I'm reduced to conceding that math doesn't scare me so much anymore. How about you? If you still feel queasy, perhaps you can take solace from Einstein himself, who once reassured a junior high school student, "Do not worry about your difficulties in mathematics; I can assure you that mine are still greater."