Correction Appended

BACK during China's Shang dynasty, around 3,500 years ago, sages foretold the future by casting oracle bones -- the clairvoyant equivalent of crap shooting.

Things have changed somewhat since then, even though predictions are as much in demand as ever, and even though we still send plenty of business to astrologers, tarot-card readers, numerologists, phrenologists, necromancers and psychics of all stripes. Aside from the traditional soothsayers, we sometimes also heed forecasts based on observations, scientific synthesis and reasoning.

Scientific sages depend not on tea leaves or the positions of planets but on tools like mathematical modeling, statistical analysis, complexity theory, celestial mechanics, geology, economics and epidemiology. Needless to say, scientists discount the naive paradigms of fortune tellers, ancient and modern.

Alas, rational approaches to prediction also fail all too often, and it may be that there are some phenomena for which predictions will remain forever out of reach.

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As the stock market lurches between the tugs of bulls and bears, financial analysts armed with statistics and the latest marketing theories throw up their hands in despair when asked to guess what will happen from one day to the next.

Picnic planners know better than to rely on short-term weather forecasts, and meteorologists offer little hope that truly accurate weather predictions for specific places and times will ever be possible.

Relief agencies get little useful help from experts as they try to brace for disasters that might, with equal likelihood, occur tomorrow or a thousand years from now. Understanding the mechanisms of events like earthquakes and volcanoes offers little help in predicting precisely when they will occur.

Geologists have a pretty good idea that earthquakes are caused by the movement of tectonic plates, yet the art of earthquake forecasting remains notoriously imprecise.

The largest earthquake in four years -- an 8.2-magnitude tremor centered under the ocean between Australia and Antarctica -- caught seismologists completely off guard when it shook the sea floor last March 25, and geologists are still puzzled by the quake; it did not occur at the junction of tectonic plates where most quakes occur, but struck within a single plate.

Future seismic surprises, including those causing catastrophic loss of life, seem inevitable.

The tsunami that roared over Papua New Guinea on July 17, killing more than 2,000 people, caught everyone by surprise. Experts excused their failure to warn the population on the ground that a giant tsunami wave materializes only when a scarcely noticeable wave spawned by an earthquake or some other event far out at sea reaches shallow coastal water. By then it is almost on top of some hapless coastline, leaving no time to prepare.

There have been recent reminders that even in principle, some things are impossible to predict.

Among the puzzles that have perplexed mathematicians and physicists at least since the time of Isaac Newton in the 17th century is the ''N-body problem'' (sometimes called the ''Many-body problem'').

One of Newton's monumental discoveries was that any two objects attract each other with a force proportional to their masses and inversely proportional to the square of their distance apart. But when three or more objects -- the Sun, the Earth and the Moon, for example -- are interacting gravitationally, exact solutions of the equations describing their motions generally remain beyond reach.

Fortunately, since Newton's day, some very good approximate solutions of N-body equations have been devised, and such solutions have allowed space scientists to send vehicles to the distant reaches of the solar system with astounding precision.

This month, Dr. Gregory R. Buck of Saint Anselm College, in Manchester, N.H., disclosed a new class of approximate N-body solutions based on the analogy of a closed loop of beads, in which the beads, evenly spaced, chase each other around the curves and tangles of the loop. The system may help to work out the interactions of particles within a plasma, Dr. Buck suggested.

But none of the approximations now known exactly solve Newton's equations, and this means that the motions of asteroids and comets can be predicted only up to a certain point in the distant future. Beyond that, forecasting an impact by one of these objects on the earth may be intrinsically impossible. A hit or miss may depend so sensitively on the minuscule ''initial conditions'' of all the objects involved that precise calculation becomes impossible. This is a property of a large class of systems scientists describe as chaotic.

Long-term foreknowledge of the hit or miss -- a question of life or death for millions -- is ruled out by the chaos inherent in the N-body system.

Scientists have yet to come to terms with chaos in all its manifestations. Dr. Steven Weinberg, winner of a Nobel prize in physics, once said he considered an understanding of chaotic turbulence in fluids as the single most intractable problem in physics. Until scientists reach a deeper understanding of turbulence, many physicists believe, the dynamics of climate change, the behavior of galaxies and many other phenomena can never be fully penetrated.

Chaos can also prevent reliable predictions of group decisions, including those that investors make. Group decisions by investors can cause chaotic price fluctuations of commodities that are intrinsically unpredictable, it seems.

In a report published Aug. 24 in the journal Physical Review Letters, Dr. David A. Meyer of the University of California at San Diego and Dr. Thad A. Brown of the University of Missouri presented formal proof that collective decisions can be chaotic, even when the views of all participants are known and a standard voting rule is strictly applied.

When a group of decision-makers must choose between three or more options by comparing two of them at a time, the collective outcome often depends on the order in which the choices are presented. The outcome can cycle chaotically, the mathematicians found. Even nonhuman decision-makers -- the computers that buy and sell commodities according to programmed rules, for example -- are subject to chaotic uncertainty, a situation in which prediction becomes impossible.

Frustrated by the complex and chaotic behavior of the real world, some theorists have invented forecasting techniques based on little more than pure mathematics.

In the 1970's, Rene Thom, a French mathematician, developed an approach called Catastrophe Theory, which, for a time, enjoyed a considerable vogue among physicists, biologists and even sociologists. The theory is an application of topology, a field of mathematics that deals with the shapes of surfaces.

Dr. Thom's theory holds that many sequences of events can be represented as smooth trajectories along a saddle-shaped surface, at one point of which an abrupt discontinuity, or ''catastrophe cusp,'' shunts them off to one side of the saddle or the other.

Dr. Thom and his followers proposed that mathematical models based on catastrophe cusps could be used to predict the reproduction of bacteria, the behavior of the stock market, heart attacks, biological evolution and the outbreak of war, among many other things. To some, Dr. Thom's theory seemed to offer an explanation of almost everything, but many others condemned it as useless. Today, Catastrophe Theory is all but forgotten.

Predicting life spans in the absence of detailed knowledge has long interested scientists as well as insurance statisticians. A forecast of life expectancy based on the average age at death of a person's four grandparents is a simple example of statistical forecasting. But a much more daring approach was devised a few years ago by Dr. J. Richard Gott 3d, a professor of astrophysics at Princeton University.

Dr. Gott's scheme is based on the ''Copernican principle,'' which assumes that the odds are overwhelmingly against any particular place or time being ''special.'' From this, Dr. Gott reasoned that the mere knowledge of how long something (or someone) has been around is sufficient to estimate how much longer it could last. Based on this system, and the assumption that Homo sapiens appeared on earth about 200,000 years ago, Dr. Gott calculated that intelligent human beings are 95-percent certain to survive a minimum of 5,128 years more, and a maximum of 7.8 million years more.

There are those who contend that predictions like these are so vague that they are scarcely more useful than the prophecies of the Delphic oracle in ancient Greece, which was consulted by Socrates, Oedipus and other luminaries of the day. The oracle (operated by a concealed priest or priestess) was so ambiguous it could nearly never be proved wrong.

Scientists will never be able to answer all our questions about future events or to satisfy a deep-seated human yearning to foresee what's coming at us. Some scientific efforts at prediction will always be defeated by the nature of Nature.

Mystic oracles have never shed light on future events either, but even 1,398 years after Socrates' suicide, legions of people continue to visit palmists, astrologists and psychics. It's human to prefer something to nothing at all.

Correction: September 23, 1998, Wednesday An essay in Science Times yesterday about predictions misstated the time elapsed since the suicide of Socrates, who relied on oracles. It was 2,397 years ago, not 1,398.

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