Unless the laws of celestial mechanics are repealed (or have been misapplied), this Saturday, on the very date 49 years ago that the scientists of the Manhattan Project produced the largest manmade sound, Jupiter will be the stage for what some astronomers are billing as the loudest noise in the solar system, except for the constant, silent roaring of the Sun.
For almost a week, about 20 pieces of the Shoemaker-Levy 9 comet, some perhaps two or three miles in diameter, are expected to hit the far, hidden side of the planet with an impact equivalent to many nuclear bombs. Jupiter is considered big and mushy enough to absorb the attack with aplomb. Closer to home, the concern is not whether the cataclysm might damage the Earth but whether it will be detectable at all.
Separated by half a billion miles of empty space, we Earthlings won't hear a thing. If scientists didn't know to watch for the explosions' visible effects -- subtle ripples in Jupiter's stormy atmosphere, the slightest sign of a reflective flash in some of its many moons -- the mighty cosmic crashes might have gone forever unmarked, like the tree that falls in the forest with no one to hear the sound.
Working scientists tend to avoid philosophical discussions, and few are likely to be heard speculating on the epistemological implications of measuring light and calling it sound. But a few of their colleagues are beginning to question the bravado with which science piles assumption on top of assumption, climbing toward the heavens on great theoretical towers. Along the way, they are taking up some of the questions scientists have considered intrusive when posed by philosophers.
How secure is the foundation of scientific knowledge? Are there limits to what we and our cranial computers can understand? In some ways this might seem part of the preoccupation with limits that has arisen in the humanities. But the scientists insist on an important distinction: They are approaching these questions scientifically and mathematically. The result is the early stirring of what might be called a science of science.
Celestial mechanics itself is certainly secure. But science seeks to understand not just the big bangs on Jupiter but the Big Bang said to have given rise to the unfolding of all space and time. Or, turning its sights in the other direction, toward the interior of atoms, it tries to construct a theory of everything, in which all matter and all energy can be explained by a crystalline array of equations.
But as science climbs higher and higher on ladders of abstraction, mathematical Towers of Babel, will it eventually hit a ceiling? In May, at a workshop called "Limits to Scientific Knowledge," at the Santa Fe Institute in New Mexico, the mathematician John Casti put the question like this: "Is the real world too complex for us?" Are there important problems that lie forever beyond our reach? Otto Rossler, a physicist visiting from the University of Tubingen, called the event "a postmodern" scientific meeting. Science was turning its sights back on itself -- trying to plumb its own depths.
The Familiar and the Strange
So far scientists have had little reason to worry about limits. Supplementing our eyes with lenses and electronic detectors and our brains with mathematics and computers, we have succeeded in taking the familiar and using it to explain the very strange.
Newton, the legend goes, saw that the laws describing a falling apple could be used to predict the motions of the planets. And so the science of celestial mechanics was born, allowing people to predict things like comets hitting Jupiter. So successful were the predictions of nearby events that when faraway objects like galaxies were discovered rotating so fast that they should have spun apart eons ago, no one doubted the validity of Newton's laws. In a heroic act of scientific imagination, astronomers inferred instead the existence of some kind of undetectable stuff called dark matter whose gravity holds the galaxies together. To shore up the Big Bang theory, some cosmologists propose that as much as 99 percent of the universe is made of this mysterious cosmic glue.
The bravura that allows these great leaps of abstraction comes from the feeling that we can stand outside the universe and see it whole. But even as we strain for this God's-eye view, we know deep down that we are inevitably part of what we are trying to comprehend.
In 1931, the Austrian mathematician Kurt Godel captured this dilemma in one of his famous incompleteness theorems. Once a logical system becomes as complex as arithmetic, he showed, it cannot prove its own consistency. Doing so requires a more complex system. But then to prove the consistency of that system one needs a yet more powerful system, and so on ad infinitum.
Godelian Knots
One of the questions the science of science is considering is whether humanity, in its quest to build consistent logical explanations of the universe, is stuck on the Godelian treadmill, confronting a universe inevitably more complex than our brains.
Jim Crutchfield, a Berkeley computer scientist who studies chaotic systems -- those that are hypersensitive to the most infinitesimal perturbations -- has estimated that the gravitational pull of an electron, randomly shifting position at the edge of the Milky Way, can change the outcome of a billiard game on Earth. In such a precarious world, the very act of constructing a theory of the universe might significantly change the universe itself. James Hartle, a cosmologist at the University of California at Santa Barbara, has written that the notion of separate laws that exist independent of the lawmakers might have to be jettisoned as so much "excess baggage." We can't stand separate from creation and view it as though it were one of our computer simulations.
Science has been rooted in the unspoken assumption that the mind and knowledge -- especially mathematics -- are ethereal essences somehow separate from the universe. In the interest of demystification, some scientists suggest that we think of brains instead as computational machinery and scientific laws as programs -- algorithms, mathematicians call them -- that describe the phenomena we observe. Ptolemy's system, in which the Sun, stars and planets circle the Earth in complex spiralling epicycles, does a fine job of predicting astronomical events. But Kepler's laws, with the planets swinging around the Sun in simple ellipses, are so much more elegant and compact that they have the ring of truth.
"Entities should not be unnecessarily multiplied," said the medieval philosopher William of Ockham, and the quest for the simplest, most elegant explanation has become science's guiding light. But is Ockham's razor, as it's called, a universal truth or, as Joseph Traub, a Columbia University computer scientist, suggested at the Santa Fe conference, a prejudice born from our limited mental powers?
In any case, the search for ultimately elegant laws may be doomed from the start. Another participant at the meeting, Gregory Chaitin, a mathematician for the I.B.M. Research Division, has proved that it is generally impossible to know whether an algorithm (read "law") is the most concise description. We must remain forever tantalized by the possibility that more elegant laws lie barely beyond our reach. Even worse, if we are presented with data that appears lawless and random, we cannot know if subtle patterns lurk within, order that we haven't been clever enough to ferret out.
Plato thought of numbers, concepts and laws as ideals existing in a nonphysical realm. But the work of Dr. Chaitin's colleague at I.B.M., Rolf Landauer, suggests that the laws science discovers are themselves subject to limitations. Information is physical -- whether it consists of magnetic spots on a disk drive or patterns of neurons in a brain -- and so it must obey the laws of physics. Again, science seems constrained by the impossibility of separating itself from the very world it strives to understand.
In trying to construct a science of science, people like Dr. Chaitin and Dr. Landauer are questioning some of the deepest assumptions of their craft. Since Newton, scientific laws have been expressed in the form of differential equations, which have exact solutions, and with the so-called real numbers, which can be expressed as infinitely long decimal expansions. Pi equals 3.14159 . . . In practice, science inevitably falls short of this ideal of infinite precision. In quantum physics, the simplest atom -- hydrogen, with one proton and one electron -- can be described precisely. But the equation for the helium atom, with its additional proton -- is intractable. We must make do with good approximations. Estimates of the size of the shards of the Shoemaker-Levy comet vary so widely that some scientists predict there will be no measureable impact on Jupiter at all.
Science has long operated on the assumption that space is continuous, with infinitely many points between two marks on a line. Mathematicians have calculated pi beyond a billion decimal places. But 61 decimal places are enough to describe a circle girding the visible universe with a deviation of less than a single Planck length -- a unit 10<2><0> (1 followed by 20 zeroes) times smaller than a proton. This seems as close to perfectly circular as a real circle can be. Do the rest of the decimal places have any meaning?
The mathematician Herman Weyl once said that the belief in an infinite continuum of numbers "taxes the strength of our faith hardly less than the doctrines of the early Fathers of the Church or the Scholastic philosophers of the Middle Ages."
Few scientists are ready to abandon differential equations and real numbers for the more realistic mathematics Dr. Chaitin is proposing. But in seeking a foundation for science, everything is up for grabs, including the universality of mathematics.
For centuries philosophers have debated whether mathematics is invented or discovered. Taking a middle ground, the 19th-century mathematician Leopold Kronecker declared, "God made the integers; all else is the work of man."
Einstein, it seems, went even further. Even the integers, he wrote, are "obviously an invention of the human mind, a self-created tool which simplifies the ordering of certain sensory experiences."
News of those distant Jovian explosions will come to us in a shower of numbers, interpreted -- perhaps invented -- by human minds. GRAPHIC: Drawing LANGUAGE: ENGLISH