August 4, 2003

Math prof seeks to reconnect mathematics to its intellectual roots

By Tim Stephens

Anthony Tromba has loved mathematics since he was a child, and it bothers him that the field seems to be losing its appeal to students.

Mathematics professor Anthony Tromba's new edition of Vector Calculus, below, seeks to put mathematics in a historical context.

Tromba, professor and chair of mathematics at UCSC, said that when he was an undergraduate in the 1960s there were nearly six times as many mathematics majors in the United States as there are today. Although there are probably many reasons for this trend, one factor may be the way mathematics tends to be taught, he said.

"People don't teach mathematics as an intellectual discipline, but more as a collection of facts or theorems, without putting it in the context of why one would be interested in it in the first place," Tromba said.

The connection of mathematics to the humanities and to its own roots as a discipline has been severed, he said. Tromba has tried to counter this trend in the new edition of his widely used textbook, Vector Calculus, coauthored with Jerrold Marsden of the California Institute of Technology. The book has been a standard university textbook for more than 27 years and has been translated into six languages. Nevertheless, Tromba said he felt it was time for a change. The revised 5th edition, published in July by W. H. Freeman, features a striking portrait of Isaac Newton on the cover and a whole new approach in the text.

"We radically changed it. We tried to put the mathematics in historical context and illuminate the thought processes that lead to mathematical results," Tromba said.

Tromba's 1996 book The Parsimonious Universe, coauthored with Stefan Hildebrandt, took a similar approach, while aiming for a more general readership. Written in a lively and accessible style, the book explores mathematical insights into the variety of shapes and forms found in the natural world.

Tromba enjoys contemplating the insights of the great mathematicians of the past--intellectual giants like Newton, Leibniz, Huygens, and Galileo. Perhaps the most breathtaking insight of all, he said, was the conclusion of the ancient Greeks that the world could be understood through mathematics.

"This was a very profound insight, because we really don't understand the universe. The universe is mind-boggling--it is beyond human comprehension--and yet we can deal with it on a theoretical basis using mathematical models," Tromba said.

The power of such models is apparent from the many instances in which formal mathematical theories have predicted natural phenomena that were not discovered or confirmed until many years later. Maxwell's equations, for example, predicted radio waves, and Einstein's field equations predicted that gravity would bend light and that the universe is expanding.

To students, mathematics can often seem like little more than a set of arcane rules for manipulating abstract symbols. According to Tromba, however, there is a mystical aspect of mathematics that is not often communicated to students.

"When you discover mathematical structures that you believe correspond to the world around you, you feel you are seeing something mystical, something profound," he said. "You are communicating with the universe, seeing beautiful and deep structures and patterns that no one without your training can see."
Tromba noted that mathematicians do not invent mathematics, they discover it.
"The mathematics is there, it's leading you, and you are discovering it," he said.

"Mathematics is a profound language, an awesomely beautiful language. For some, like Leibniz, it is the language of God. I'm not religious, but I do believe that the universe is organized mathematically."

Tromba has been honored recently for both his teaching and his contributions to mathematics. He received an Excellence in Teaching Award for 2001-02 from UCSC's Academic Senate, and in May 2003, Tromba's colleagues held a symposium in honor of his 60th birthday (which is in August). Dubbed "Tony Fest," the symposium brought together many of the mathematicians Tromba has worked with during his career for two days of lectures and social activities. Though initially reluctant to be the object of such attention, Tromba said he was very moved by the event.

"One of the great things in my life has been to be able to interact with brilliant mathematicians, to work with some of them, and to understand their genius," he said. "I don't consider myself one of them, but to understand and to be blown away by such awesome genius--it's like coming close to Mozart or Bach."

Although Tromba speaks eloquently about the intellectual thrills of mathematics, he acknowledges that such thrills are really only accessible to mathematicians.
"With art and music, people can feel the greatness of the work without actually understanding its structure and complexity. You don't have to be a musician to appreciate Mozart's Requiem, but mathematics doesn't afford you the same opportunity--it's a very small club," he said.

Tromba is doing his best to encourage more students to join that club.

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