THERE is no Nobel prize in metaphysics. But if there were, it would surely go to the thinker who could solve the stubborn mystery of why mathematics so often mirrors the workings of the physical world. Rocks arc through the sky in parabolic trajectories, planets swing in ellipses around the sun. And inside atoms, tiny particles move according to the mathematics of quantum mechanics.

But not every equation is so well behaved. Sometimes they predict absurdities like subatomic particles with negative masses or even negative probabilities of existing. What could it mean for something to weigh less than nothing or to have less than a zero chance of being real? ''Physics in the never-never land,'' the theorist Marvin Goldberger once called these mathematical chimera.

Last week, two Dutch physicists, Gerardus 't Hooft and his former teacher, Martinus (Tini) Veltman, belatedly won a Nobel Prize for showing, in the early 1970's, that a theory connecting two of the four forces of nature belonged not in the land of Peter Pan but among the handful of mathematical inventions that speak truths about the world.

Having parsed reality into four forces -- gravity, electromagnetism and the weak and strong nuclear forces -- physicists have been trying ever since to fit the pieces back together. They dream of ''unifying'' these separate phenomena, showing they are different manifestations of a single superforce that existed in the first moments after the Big Bang.

By the late 1960's, physicists had taken an important step toward unification with a theory blending electromagnetism with the weak force. But it was not clear whether this ''electroweak'' theory was sound. Like a buggy piece of software, it might churn out quantities that were infinite, as when you divide a number by zero. If so, the theory would be in jeopardy.

Decades earlier, spurious infinities had threatened to wreck the crown jewel of early 20th-century physics: quantum electrodynamics. The theory, which explains electromagnetism, is an example of a more general notion called quantum field theory, in which forces are carried by ephemeral particles, called quanta, that pop in and out of existence too quickly to be observed. For electromagnetism these messengers (or ''virtual particles'') are called photons. They shuttle back and forth between two electrons, causing them to repel each another.

But a single electron would also interact with itself, like a snake swallowing its own tail. The result, the equations predicted, was an endlessly looping process that would bestow the particle with infinite mass and charge. This was so obviously false that many physicists came to doubt the whole idea of quantum field theory. There were other crazy infinities as well. Maybe it was time to uproot the entire foundation of particle physics and begin anew.

It came as a relief when, in the late 1940's, Richard Feynman, Julian Schwinger and Sin-itiro Tomonaga independently discovered a way to ''renormalize'' the equations -- manipulating them so the infinities cancel one another out like terms in an algebraic equation. The result was measurements so precise that, as Feynman later put it, it was like gauging the distance from Los Angeles to New York within the width of a human hair.

Still, not everyone was impressed with this shell game. ''Sensible mathematics involves neglecting a quantity when it turns out to be small,'' the great theorist Paul Dirac complained, ''not neglecting it because it is infinitely great and you do not want it!'' Even after he shared a Nobel Prize for the work, Feynman dismissed it as ''hocus pocus,'' insisting that he had been honored for sweeping a problem under the rug.

In fact, by the time electromagnetism and the weak force were on the verge of being unified, quantum field theory had fallen again into disrepute. At the University of Utrecht, Dr. Veltman, who had been laboring over the infinities for years, found an ally in Gerard 't hooft.

The two could hardly have been more different, said Peter van Nieuwenhuizen, who was with them at the time. With a thick black beard and shining eyes, Dr. Veltman was large, imposing, bluntly honest. ''He looked like a character from a Dutch painting,'' said Dr. van Nieuwenhuizen, who is now director of the C. N. Yang Institute for Theoretical Physics at the State University of New York at Stony Brook. Dr. 't hooft was small, soft-spoken, diplomatic. From the day he arrived, it was clear he was a star.

The obsessive Dr. Veltman once said he would gladly give up a finger to make the infinities go away. With an unusual ability to see an old problem in a new light, Dr. 't hooft broke the ice jam. ''He could solve problems that seemed unsolvable in a way so elegant that you could only sit in absolute stupefaction,'' Dr. van Nieuwenhuizen said.

Using Dr. Veltman's earlier work as a platform, the younger scientist found the key to renormalizing electroweak theory, reconciling the infinities. He also deepened the understanding of how, in the early moments of creation, the two forces might have come unraveled. The agent for this ''spontaneous symmetry breaking'' was a particle, still undiscovered, called the Higgs.

The symmetry between teacher and student was also destined to break. Dr. Veltman thought the Higgs particle was baloney. And over the years, he came to feel that his student's contributions were being celebrated more than his own. Gradually they drifted apart. But the ideas that emerged from their uneasy collaboration changed physics.

The two scientists accomplished far more than simply debugging electroweak theory. Their work applied to electromagnetism and the strong force as well, bringing quantum field theory to a new height of respectability. Today's Standard Model, which describes the workings of all the forces but gravity, is cast in this very form.

Physicists still argue about whether renormalization is a mere trick or a clue, still barely understood, to some deep truth of nature. Most leave such questions to the metaphysicians, content to get on with their work.

Photos: The Dutch physicists Gerardus 't Hooft (top) and Martinus (Tini) Veltman shared a Nobel Prize for connecting quantum dots. (Associated Press); (Reuters)