A mathematician, the Hungarian lover of numbers Paul Erdös once said, is a device for converting coffee into theorems. Here, then, are a few glimpses into the Truth Factory. The Mathematical Sciences Research Institute, sustained mostly by the National Science Foundation, sits on a hill above the University of California at Berkeley, where it attracts people from around the world for stints of up to a year to lose themselves in subjects like algebraic geometry or special holonomy.

Consider it an embassy of another world, a Platonic realm of clarity and beauty, of forms and relations, where the answers to questions not yet asked already exist.

Higher mathematics—as opposed to what we do every April 15—has been relevant ever since Archimedes leaped out of his bath shouting "Eureka!" more than 2,000 years ago. Nobody knows when some abstruse bit of math will float off a blackboard at a place like this and become—often decades later—a key tool in cryptography, biology, physics or economics (as in "A Beautiful Mind").

Take string theory, a mathematically labyrinthine effort to construct a so-called theory of everything out of the notion that the fundamental elements of nature are tiny strings flopping and wriggling in an 11-dimensional space-time. It has been called a piece of 21st-century physics that fell by accident into the 20th century.

In their quest to negotiate this labyrinth, string theorists have made a hot topic of something called Riemann surfaces, invented by the German mathematician Georg Friedrich Bernhard Riemann 150 years ago, but they have also helped blaze new fields of mathematics.

"Since our theories are so far ahead of experimental capabilities, we are forced to use mathematics as our eyes," Dr. Brian Greene, a Columbia University string theorist, said recently. "That's why we follow it where it takes us even if we can't see where we're going."

So in some ways the men and women seen here scrutinizing marks on their blackboards collectively represent a kind of particle accelerator of the mind.

But the "unreasonable effectiveness" of mathematics in explaining the world, as the physicist Eugene Wigner once put it, is a minor motivation at best for those immersed in the field. Most mathematicians say they are in it for the math itself, for the delirious quest for patterns, the thrill of the detective chase and the lure of beautiful answers.

"Math is sense," said Dr. Robert Osserman, a Stanford professor and deputy director of the institute, quoting from the play "Copenhagen." "That's what sense is."