March 26, 2002
The Most Seductive Equation in Science: Beauty Equals Truth
By DENNIS OVERBYE
n the fall of 1915, Albert Einstein, living amid bachelor clutter on coffee, tobacco and loneliness in Berlin, was close to scrawling the final touches to a new theory of gravity that he had pursued through mathematical and logical labyrinths for nearly a decade. But first he had to see what his theory had to say about the planet Mercury, whose puzzling orbit around the Sun defied the Newtonian correctness that had long ruled the cosmos and science. The result was a kind of cosmic “boing” that changed his life.
Einstein’s general theory of relativity, as it was known, described gravity as warped space-time. It had no fudge factors — no dials to twiddle. When the calculation nailed Mercury’s orbit Einstein had heart palpitations. Something inside him snapped, he later reported, and whatever doubt he had harbored about his theory was transformed into what a friend called “savage certainty.” He later told a student that it would have been “too bad for God,” if the theory had been subsequently disproved.
The experience went a long way toward convincing Einstein that mathematics could be a telegraph line to God, and he spent most of the rest of his life in an increasingly abstract and ultimately fruitless pursuit of a unified theory of physics.
Rare indeed is the scientist who has not at one point or other been seduced by the beauty of his own equations and dumbfounded by what the physicist Dr. Eugene Wigner of Princeton once called the “unreasonable effectiveness of mathematics” in describing the world.
The endless fall of the moon, the fairy glow of a rainbow, the crush of a nuclear shock wave are all explicable by scratches on a piece of paper, that is to say, equations. Every time an airplane safely touches down on time, a computer boots up, or a cake comes out right, the miracle is recreated. “The most incomprehensible thing about the universe is that it is comprehensible,” Einstein said.
Math is the language of physics, but is it the language of God?
Mathematicians often say that they feel as if their theorems and laws have an objective reality, like Plato’s perfect realm of ideas, which they do not create or construct as much as simply discover. But the equating of math with reality, others say, consigns vast arenas of experience to the darkness. There are no mathematical explanations yet for life, love or consciousness.
“As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality,” said Einstein.
He maintained that it should be possible to explain scientific principles in words to a child, but his followers often argue that words alone cannot convey the glories of physics, that there is a beauty apparent only to the mathematically adept.
That inhuman beauty has long been a lodestone for physicists, says Dr. Graham Farmelo, a physicist at the Science Museum in London and an editor of “It Must Be Beautiful: Great Equations of Modern Science.”
“You can write it on the palm of your hand and it shapes the universe,” Dr. Farmelo said of Einstein’s gravitational equation, the one that produced heart palpitations. He compared the feeling of understanding such an equation to the emotions you experience “when you take possession of a great painting or a poem.”
In the hopes of getting the rest of us to take possession some of our intellectual heritage, Dr. Farmelo recruited scientists, historians and science writers to write about the life and times of 11 of the most powerful or notorious equations of 20th century science.
The book is partly a meditation on mathematical beauty, possibly a difficult concept for many Americans right now as they confront their tax forms. But as Dr. Farmelo noted in an interview, even the most recalcitrant of us have had glimpses of mathematical grace when, say, our checkbooks balanced.
Imagine that your withholdings always turned out to be exactly equal to the tax you wind up owing. Or that your car’s odometer turned over to all zeros every year on your birthday no matter how far you thought you had driven. Such occurrences would be evidence of patterns in your financial affairs or driving habits that might be helpful in preparing tax returns or scheduling car maintenance.
The pattern most highly prized in recent modern physics has been symmetry. Just as faces and snowflakes are prettier for their symmetrical patterns, so physical laws are considered more beautiful if they keep the same form when we change things by, for example, moving to the other side of the universe, making the clocks run backward, or spinning the lab around on a carousel.
A good equation, Dr. Farmelo said, should be an economical compression of truth without a symbol out of place. He looks for attributes like universality, simplicity, inevitability, an elemental power and “granitic logic” of the relationships portrayed by those symbols.
There is, for example, Einstein’s E=mc², which Dr. Peter Galison, a Harvard historian and physicist, describes in the book as “a metonymic of technical knowledge writ large,” adding, “Our ambitions for science, our dreams of understanding and our nightmares of destruction find themselves packed into a few scribbles of the pen.”
When it comes to the quest for beauty in physics, even Einstein was a piker compared with the British theorist Paul Dirac, who once said “it is more important to have beauty in one’s equations than to have them fit experiment.”
An essay by Dr. Frank Wilczek, a physics professor at the Massachusetts Institute of Technology, recounts how the 25-year-old Dirac published an equation in 1928 purporting to describe the behavior of the electron, the most basic and lightest known elementary particle at the time. Dirac had arrived at his formula by “playing around” in search of “pretty mathematics,” as he once put it. Dirac’s equation successfully combined the precepts of Einstein’s relativity with those of quantum mechanics, the radical rules that prevail on very small scales, and it has been a cornerstone of physics ever since.
But there was a problem. The equation had two solutions, one representing the electron, another representing its opposite, a particle with negative energy and positive charge, that had never been seen or suspected before.
Dirac eventually concluded that the electron (and it would turn out every other elementary particle) had a twin, an antiparticle. In Dirac’s original interpretation, if the electron was a hill, a blob, in space, its antiparticle, the positron, was a hole — together they added to zero, and they could be created or destroyed in matching pairs. Such acts of creation and annihilation are now the main business of particle accelerators and high-energy physics. His equation had given the world its first glimpse of antimatter, which makes up, at least in principle, half the universe.
The first antimatter particle to be observed, the antiproton, was found in 1932, and Dirac won the Nobel Prize the next year. His feat is always dragged forth as Exhibit A in the argument to show that mathematics really does seem to have something to do with reality.
“In modern physics, and perhaps in the whole of intellectual history, no episode better illustrates the profoundly creative nature of mathematical reasoning than the history of the Dirac equation,” Dr. Wilczek wrote.
In hindsight, Dr. Wilczek writes, what Dirac was trying to do was mathematically impossible. But, like the bumblebee who doesn’t know he can’t fly, through a series of inconsistent assumptions, Dirac tapped into a secret of the universe.
Dirac had started out thinking of electrons and their opposites, the “holes,” as fundamental entities to be explained, but the fact that they could be created and destroyed meant that they were really evanescent particles that could be switched on and off like a flashlight, explains Dr. Wilczek.
What remains as the true subject of Dirac’s equation and as the main reality of particle physics, he says, are fields, in this case the electron field, which permeate space. Electrons and their opposites are only fleeting manifestations of this field, like snowflakes in a storm.
As it happens, however, this quantum field theory, as it is known, must jump through the same mathematical hoops as Dirac’s electron, and so his equation survives, one of the cathedrals of science. “When an equation is as successful as Dirac’s, it is never simply a mistake,” Dr. Steven Weinberg, a 1979 Nobel laureate in physics from the University of Texas, writes in an afterword to Dr. Farmelo’s book.
Indeed, as Dr. Weinberg has pointed out in an earlier book, the mistake is often in not placing enough faith in our equations. In the late 1940’s, a group of theorists at George Washington University led by Dr. George Gamow calculated that the birth of the universe in a Big Bang would have left space full of fiery radiation, but they failed to take the result seriously enough to mount a search for the radiation. Another group later discovered it accidentally in 1965 and won a Nobel Prize.
Analyzing this lapse his 1977 book, “The First Three Minutes,” Dr. Weinberg wrote: “This is often the way it is in physics. Our mistake is not that we take our theories too seriously, but that we do not take them seriously enough. It is always hard to realize that these numbers and equations we play with at our desks have something to do with the real world.”