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What do you get when you add a couple of math professors, a Roman Catholic nun, and a computer? Not the start of a bad joke, but a major breakthrough in mathematics — and a pretty one, too.

By Mark F. Bernstein

MATHEMATICIANS HAVE BEEN writing proofs since Euclid first took up geometry 2,300 years ago, but a corner of the field that has long been regarded as one of its highest intellectual endeavors is now obsolete. Thanks in large measure to the work of Dr. Herbert S. Wilf, the Thomas A. Scott Professor of Mathematics, and Dr. Doran Zeilberger of Temple University, at least one type of proof is now the province of the computer. Adapting a technique discovered 50 years ago by a Roman Catholic nun, they have helped demonstrate that even some of the most intractable mathematical problems can not only be simplified, but made elegant and even beautiful, as well. 
   Wilf and Zeilberger have shown that computers can prove a certain class of equations, known as combinatorial identities, instantly and infallibly. They first published that discovery eight years ago in a paper for The Journal of the American Mathematical Society. This past January, the American Mathematical Society acknowledged the importance of their work by awarding Wilf and Zeilberger its Leroy P. Steele Prize for Seminal Contributions to Research, one of the highest honors a mathematician can receive. 
   What exactly are combinatorial identities? Dissecting mathematical terminology is like walking through a dark cave: It is much less frightening if one stays close to the wall and proceeds in small steps. 
Combinatorics is simply a mathematician’s fancy term for the art of counting or arranging things. If you have 30 couples at a dinner party and want to seat them at tables of four so that no husband sits next to his own wife, determining the number of different ways this can be done — and finding a simple formula for doing so — is an exercise in combinatorics. 
   An identity, the other part of that imposing-looking term, is just another way of saying that what is to the left of an equals sign is the same as what is to the right; in other words, that A equals B. All of us are familiar with identities, though we may not know it. Einstein’s famous theory of relativity — e=mc2 — and the Pythagorean Theorem — a2+b2=c2 — are two famous identities. So is 2+2=4. 
   To prove a combinatorial identity, one must find, or attempt to find, that the often numbingly complex set of equations to the left of an equals sign (the “A” in our A=B equation) is in fact identical to a much simpler equation to the right (the “B” part). It is the art of proving that something ungainly can be reduced to something simple, the distillation of something monstrous into something elegant. The standard analogy is that combinatorial identities fold together like an origami swan, although given the size of some of the equations Wilf and Zeilberger now can simplify, a more fitting comparison might be to an inflatable life raft. 
   Proving combinatorial identities has long been an intellectual challenge — as well as a source of intense pleasure — for mathematicians, but the exercise could be full of uncertainty. Frequently an equation could not be simplified, though it was impossible for the frustrated mathematician to tell whether this was because no simplification existed or because he or she just was not clever enough to find it. Thanks in part to Wilf, the computer can now find those elusive combinatorial identities or, which is just as helpful, state with certainty that no simplified identity exists. Small wonder that some have called Wilf’s and Zeilberger’s creation an “automatic proofing machine.” 
   Very well, a philistine might say. A fine little mathematical trick. But of what use is this breakthrough in the real world? The short answer is that it has many practical applications, chief of which is perhaps enabling computer programmers to determine the number of steps — and hence the amount of memory — required to perform a given task. But the question of practicality makes Wilf bristle, though in a cheerful, Wilfian way. “Nobody ever subjected Picasso to that test,” he points out. “They’d just hang his pictures on the wall, and they’d smile.” 
   To Wilf, the distillation is its own reward. “Mathematics,” he says, “is a deeply satisfying aesthetic experience. The human race is deeply in need of aesthetically satisfying experiences, and we’re happy to provide them. That’s a very practical, real-world application — to make you smile and think.” 
   Mathematics — good for the soul? Smiling and thinking as balanced parts of the same equation? To the great innumerate mass of us, that would indeed be an identity worth proving.

WILF WAS FASCINATED with math problems from an early age. As a child, he says he can recall reading math books in his room at night after he was supposed to be asleep. His first and perhaps strongest lesson in the distillation of complexity came from his father. It is a lesson Wilf recalls vividly to this day. 
   Alexander Wilf, his son admits, was not a man to do things by halves. A successful Philadelphia rug and appliance merchant, he had his own moment of simplification shortly after World War II when he attended an address given by a member of the Irgun, the right-wing Zionist military organization. At the end of the talk, Alexander Wilf approached the speaker and gave him a check for a few hundred dollars. 
   “But this guy,” Wilf recalls, “was a very savvy fundraiser.” He refused to accept the check, telling Alexander Wilf that if that was all he could afford, he obviously needed the money more than the Irgun did. His son still retells the story with a mixture of amusement and amazement. “That was a seminal moment in my father’s life.” 
   Stung, Wilf’s father pondered what the man had said and decided that what was most important to him was to devote himself to the creation of a Jewish state. With that, he sold his house in Wynnefield along with his interest in the family business and moved to New York, where he became executive director of the American League for a Free Palestine. Young Herb was sent to live with relatives while continuing his education at Central High School, feeling, as he says, “deserted” by his family. 
   Nevertheless, he proudly recounts his father’s work. “My father was a very brilliant guy,” Wilf insists. He had “a very strong worldview and awesome power to verbalize what he thought. Also, an awesome power to get things done in the real world.” The American League for a Free Palestine staged fundraisers for the Irgun at Madison Square Garden, performing a show, A Flag is Born, written by the noted writer Ben Hecht. It used the proceeds to smuggle refugees into Israel along what came to be known as a “European underground railway.” An article in The New York Times in January 1947 lists Alexander Wilf as one of the owners of a mysterious ship, the Abril, which was believed to be ferrying refugees through the British blockade. Occasionally, it appears, Wilf also attempted to smuggle arms. The following year, another ship in which he had invested much of his own money, the Altalena, was bombed off the coast of Tel Aviv while carrying refugees and munitions. 
   Alexander Wilf’s passion for Israel did not overshadow his interest in his own son’s future. “My father said, ‘Why don’t you go to MIT and be a scientist, because that’s what you’re good at,” Wilf recalls. “I had so much faith in him that I applied just to MIT — nothing else — and I got in! I thought it was that way for everybody.” 
   Shortly before he left for Cambridge, though, the Jewish state was founded and Alexander Wilf decided to move the family again, this time to Israel, where he planned to start an opposition newspaper. He pressed his son to join them. 
   “I can’t quite figure myself out at the point,” Wilf says now, “but I was very, very resistant. I was very strongly on a professional track. I really wanted to be a scientist … and I was just as stubborn as he was.” Within a few years, political machinations in Israel had doomed his father’s newspaper, Wilf says, and the family returned to the United States. 
   Wilf, in the meantime, had gone on to Columbia University to get his Ph.D. — but his education, he adds, came in the real world. Married and with a young family to support, while in graduate school he worked at various full-time jobs, helping design jet engines and some of the first nuclear power plants. 
   It was during these years that he first discovered computers. His introduction came not at MIT, where some of the earliest work was being done (“I must have missed that,” Wilf says), but at IBM, where he worked one summer while an undergraduate. With only a few weeks’ training on a mainframe IBM 701, Wilf wrote a program used by Esso (now Exxon) to control the quality of refined gasoline. 
   Wilf took his first teaching position at the University of Illinois in 1959. Three years later, he returned to Philadelphia to join Penn’s math department, where he has worked ever since.

IN 1946, THE YEAR the ENIAC computer was built at Penn and Alexander Wilf joined the Irgun, a Roman Catholic nun had an epiphany of her own that was to have an important influence on Wilf’s mathematical career. Sister Mary Celine Fasenmyer was raised in Central Pennsylvania and graduated from tiny Mercyhurst College after taking her vows. Because she was, as she later modestly put it, “always good in math,” her order permitted her to pursue her Ph.D. at the University of Michigan during World War II under the supervision of the distinguished mathematician, Earl Rainville. 
   In her dissertation, Sister Celine showed that certain types of mathematical patterns known as recurrence relations could be satisfied in a mechanical, or algorithmic, way. Rainville recognized, as Wilf puts it, “that there was some very pretty mathematics going on,” and presented his student’s ideas to the world in two chapters of his book, Special Functions, published in 1960. 
   There Sister Celine’s breakthrough remained for almost two decades, until, in 1978, Doron Zeilberger recognized that her technique could also be used to prove combinatorial identities. Wilf recalls first reading about Zeilberger’s discovery in his office one quiet afternoon after grades had been submitted. “I remember feeling that I was about to connect to a parallel universe that had always existed but which until then had remained very well hidden, and I was about to find out what sort of creatures lived there.” 
   As revolutionary as Zeilberger’s application of Sister Celine’s thesis may have been, it still had its rough edges. Wilf and Zeilberger had known each other casually since Zeilberger had sent Wilf a fan letter for another combinatorial paper Wilf had written, thanking him for restoring his faith in the beauty of mathematics. Wilf managed both to simplify Zeilberger’s new technique and to devise a way to make it produce at least one, and sometimes more, entirely new and unanticipated identities in the process of proving the first. Their integrated discoveries became known as “WZ theory.” 
   Zeilberger credits Wilf with making his idea aesthetically appealing. “Before Herb, it was correct, but it definitely was not pretty. His contribution made it beautiful.” And this beauty is more than just skin deep. “It’s not cosmetics,” Zeilberger adds. “It leads to new breakthroughs. If it’s not pretty, it’s obscure.” 
   Far from remaining obscure, their discoveries were set forth in the paper for which the two have been awarded this year’s Steele Prize. They have since expanded upon it in a book published last year with Slovenian mathematician Marko Petkovsek, which they aptly titled A=B. Wilf and Zeilberger’s paper met with enthusiastic reviews in the field. The eminent computer-scientist and mathematician Donald Knuth raved about it. “I fell in love with these procedures as soon as I learned them because they worked for me immediately,” he wrote in the foreword to A=B. “The success rate was astonishing.” So astonishing that Knuth abandoned two projects on which he had been working, regarding them as now obsolete. 
   In the midst of this success, however, Wilf did not forget Sister Celine. An enthusiastic amateur pilot, he flew to Erie where she lived in a Catholic retirement home and invited her to attend an upcoming mathematical conference in Boca Raton. Hearing no reply in the following weeks, Wilf flew down to Florida — only to discover that Sister Celine had received a travel grant from the diocese and was in attendance. 
   When he introduced her from the audience, the 87-year-old nun slowly rose to her feet. She said she had only two remarks to make. First, she wanted to thank Professor Wilf for the invitation. And second, she said, casting a level gaze at the assemblage of distinguished mathematicians, “I want you all to know — I really did that work.” 
   “There wasn’t a dry eye in the house,” Wilf says.

AFTER ALMOST 40 years in the classroom, it isn’t surprising that Wilf holds strong views about education. Prominent among them is his belief that academe does not suffer from an excess of good teachers. 
   “A lot of attention is paid in this country to the programmatic aspects of education; that is to say, to curricula,” he says. “Shall we teach this or that or the other thing, and if so in what grade and by what textbook? I think that’s mostly irrelevant. Teaching is mostly a chemical activity that goes on between Human A and Human B. Forget the curriculum.” 
   Wilf’s faith in the chemistry of learning leads him to scorn standardized tests. “I’m so glad our scores are so low on those tests” compared to students in other countries, he says, because the societies that excel in them “do nothing but take tests. The cost to society of regimenting kids so they all listen to their teacher and go home and do homework for four hours is tremendous. I do not want this society to take that course.” 
   Instead, Wilf has faith that curiosity will eventually prevail. “I want my kids to be free and untrammeled to screw up their mathematics and flunk all those standardized tests. All the best mathematics research is coming out of this country, where the kids run wild — so somewhere along the line we must be doing something right. I don’t care if we’re 98th, with only Lithuania and Fiji behind. What do I care? That’s measuring something in eleventh grade. But life is a long game.” 
   That is not to say that Wilf tolerates lazy thinking or lazy teaching. Quite the contrary, he believes students must learn the apparent paradox that simplification is impossible without mastery, and that elegance — like luck — is the residue of design. 
   “The biggest single problem we have to block understanding in mathematics is the fact that instructors don’t push students hard enough to force them to verbalize it,” he insists. Yes, mathematics is a universal language, “but you’ve got to speak it. It’s a formal language. You can’t use informal conversation. It won’t work.” 
   He has returned to the subject several times over the last several years. In an article titled, “Epsilon Sandwiches,” he once wrote that “what is at the highest premium is the ability of students to wrap complete English sentences around their mathematical thoughts.” In his junior math course, he insists, “I don’t care about ‘lively’ and I don’t care about ‘motivated.’ ‘Clear’ is important, but ‘complete and correct and readable’ is really what it’s all about.” 
   For his own part, this professor — the author of five books and founder or editor of three academic journals — believes that the classroom and the blackboard make equal demands on a mathematician’s time. Research and teaching each enrich the other, he says. “If you’re just teaching without doing research, what are you teaching? You’re not teaching stuff that’s in your own head, you’re teaching something somebody else did 20 years ago in his research. Research means asking questions. If you don’t do research, so far as I’m concerned, you’re not asking questions.” 
   Wilf seems to have struck a stable balance. In 1972, he won Penn’s Lindback Award for excellence in undergraduate teaching and in 1996 received the Haimo Award of the Mathematical Association of America for Distinguished Teaching of College or University Mathematics.

BY REMOVING SOME aspects of proving identities from the human intellect and turning them over to computers, Wilf has been called “the Grinch who stole mathematics.” Zeilberger, for one, is quick to defend him. 
   “There is a silly rivalry between those who like computers and those who like humans,” he explains. “Herb likes humans.” 
   Nevertheless, Wilf accepts the appelation with a mixture of pride and regret. He particularly regrets that many of the famous combinatorial identity problems posed in the back pages of the American Mathematics Monthly have been rendered obsolete. “It was like the Sunday New York Timescrossword puzzle,” Wilf says. “How would you feel if they automated it? ‘Well, it’s good for the human intellect, but what do I do with my Sunday mornings?'” 
   With computers taking over new fields of math, what, in fact, is left for humans to do? The realm of inspiration, Wilf says, has not been foreclosed, just pushed further along. He gives an example as old as the Greeks: proving that the sum of the interior angles of any triangle equals 180 degrees. It may have been Euclid who first reasoned that this could best be done by drawing a line parallel to the base of the triangle, then showing that the three angles thus formed mirrored those of the triangle. A computer, Wilf says, could run the proof, but it could not conceive the “brilliant idea” of that simple parallel line. 
   “The human part is going to be very hard to extinguish,” he adds with the hint of a smile. “But we’re working on it.” 

Mark F. Bernstein is a freelance writer in Philadelphia. He last wrote for the Gazette in December on a Penn-Princeton football game played at the Academy of Music

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