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Friday, August 15, 2014

As Math Grows More Complex, Will Computers Reign?

Shalosh B. Ekhad, the co-author of several papers in respected mathematics journals, has been known to prove with a single, succinct utterance theorems and identities that previously required pages of mathematical reasoning. Last year, when asked to evaluate a formula for the number of integer triangles with a given perimeter, Ekhad performed 37 calculations in less than a second and delivered the verdict: “True.”

This simple computation, written with math software called Maple, verifies a formula for the number of integer triangles with a given perimeter. (Illustration: Simons Science News)

Shalosh B. Ekhad is a computer. Or, rather, it is any of a rotating cast of computers used by the mathematician Doron Zeilberger, from the Dell in his New Jersey office to a supercomputer whose services he occasionally enlists in Austria. The name — Hebrew for “three B one” — refers to the AT&T 3B1, Ekhad’s earliest incarnation.

“The soul is the software,” said Zeilberger, who writes his own code using a popular math programming tool called Maple.

Doron Zeilberger, a mathematician at Rutgers University, believes computers are overtaking humans in their ability to discover new mathematics. (Photo: Tamar Zeilberger)
 A mustachioed, 62-year-old professor at Rutgers University, Zeilberger anchors one end of a spectrum of opinions about the role of computers in mathematics. He has been listing Ekhad as a co-author on papers since the late 1980s “to make a statement that computers should get credit where credit is due.” For decades, he has railed against “human-centric bigotry” by mathematicians: a preference for pencil-and-paper proofs that Zeilberger claims has stymied progress in the field. “For good reason,” he said. “People feel they will be out of business.”

Anyone who relies on calculators or spreadsheets might be surprised to learn that mathematicians have not universally embraced computers. To many in the field, programming a machine to prove a triangle identity — or to solve problems that have yet to be cracked by hand — moves the goalposts of a beloved 3,000-year-old game. Deducing new truths about the mathematical universe has almost always required intuition, creativity and strokes of genius, not plugging-and-chugging. In fact, the need to avoid nasty calculations (for lack of a computer) has often driven discovery, leading mathematicians to find elegant symbolic techniques like calculus. To some, the process of unearthing the unexpected, winding paths of proofs, and discovering new mathematical objects along the way, is not a means to an end that a computer can replace, but the end itself.

In other words, proofs, where computers are playing an increasingly prominent role, are not always the end goal of mathematics. “Many mathematicians think they are building theories with the ultimate goal of understanding the mathematical universe,” said Minhyong Kim, a professor of mathematics at Oxford University and Pohang University of Science and Technology in South Korea. Mathematicians try to come up with conceptual frameworks that define new objects and state new conjectures as well as proving old ones. Even when a new theory yields an important proof, many mathematicians “feel it’s actually the theory that is more intriguing than the proof itself,” Kim said.
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Related links 
Doron Zeilberger's homepage 
Doron Zeilberger (From Wikipedia, the free encyclopedia) 
Proof confirmed of 400-year-old fruit-stacking problem

Source: Quanta Magazine