One night in July, 1964, the logician Lotfi Zadeh found himself alone in his parents’ New York apartment, his dinner plans cancelled. At the time, Zadeh later wrote, he was doing “a lot of thinking about basic issues in systems analysis, especially the issue of unsharpness of class boundaries”—that is, the failure of things in the physical world to conform to classical Boolean logic, the true-or-false, black-or-white, zero-or-one mathematics that underpins much of computer science. “It was at that point that the simple concept of a fuzzy set occurred to me,” Zadeh recalled. “It did not take me long to put my thoughts together and write a paper on the subject.” Published the following summer in the journal Information and Control, the paper began with a brief accounting of what Zadeh considered fuzzy:
For example, the class of animals clearly includes dogs, horses, birds, etc. as its members and clearly excludes such objects as rocks, fluids, plants, etc. However, such objects as starfish, bacteria, etc. have an ambiguous status with respect to the class of animals. The same kind of ambiguity arises in . . . the “class of all real numbers which are much greater than 1,” or “the class of beautiful women” . . . Yet, the fact remains that such imprecisely defined “classes” play an important role in human thinking, particularly in the domains of pattern recognition, communication of information, and abstraction.
Zadeh’s argument resembled an idea that Albert Einstein had expressed four decades earlier, in his book “Geometry and Experience.” “So far as the laws of mathematics refer to reality, they are not certain,” Einstein wrote. “And as so far as they are certain, they do not refer to reality.”
Zadeh, who died earlier this month, at the age of ninety-six, had modest hopes for his paper; he figured that the main reason it had been accepted was because the author was a member of Information and Control’s editorial board. Gradually, though, his innovation found a following, particularly in the East. In the nineteen-eighties, engineers in Sendai, Japan, incorporated fuzzy logic into the design of the city’s new subway, using it to program the system’s famously smooth starts and stops. A catalogue of fuzzy consumer electronics followed—cameras, washers and dryers, vehicle transmissions and anti-skid braking systems, air-conditioners and thermostats, rice cookers, vacuum cleaners, and unmanned helicopters. Still, most of Zadeh’s colleagues in the West continued to express disdain. The electrical engineer Rudolph Kálmán called fuzzy logic “a kind of scientific permissiveness.” The mathematician William Kahan dismissed it as “the cocaine of science.” But Zadeh’s idea persisted. Since 1965, that inaugural paper has accumulated nearly ninety-three thousand academic citations, according to Google Scholar.
Zadeh was born in Baku, Azerbaijan. According to family history, his mother was a Russian Jew and his father was of Turkish origin, with roots in Azerbaijan and Iran. The family left the Soviet Union when Zadeh was ten, moving to Tehran, where he studied at the American College, a missionary school. Most of his teachers were Presbyterians from the Midwest; “at a distance, I fell in love with the United States and American values,” Zadeh wrote. After graduating from the University of Tehran, in 1942, with a degree in electrical engineering, he went on to the Massachusetts Institute of Technology; then to Columbia; then to the Institute for Advanced Study, in Princeton, New Jersey; and, finally, in 1959, to the University of California, Berkeley, where he remained for the rest of his career. By that point, he and his wife, Fay, whom he’d met in Tehran, had two children, Norman and Stella. (Stella, a journalist and talent agent, died in 2006; Fay died earlier this year.) When I spoke with Norman, he said that his father was a fine dancer, a skilled tennis player, and an avid photographer—he snapped portraits of many of the good and the great who came through Berkeley, and President Richard Nixon—but otherwise he was all work, work, work. “Once he had an opinion on something, the probability of persuading him that his opinion was incorrect was pretty small,” Norman told me.
Zadeh’s penchant for nonconformity earned him some ardent fans. In the eighties, Bart Kosko, now an information scientist at the University of Southern California, sought the logician out as one of his Ph.D. advisers. “Intellectually, he was a giant,” Kosko said. “I’ve heard a giant described as someone whose head is in the clouds but whose feet are on the ground, and that was certainly Lotfi.” Fittingly enough, Zadeh’s first and most famous illustration of fuzzy logic involved tall men. “Membership increases smoothly with height, so every man is tall to some degree,” Kosko explained. “If man x is tall to degree seventy per cent, then he is also not tall to degree thirty per cent. To be very tall is to concentrate the membership. Very tall men are tall, but not all tall men are very tall. This leads at once to the result that very not tall men are not very tall.” (Zadeh himself was not the tallest man in the room; at his peak, he reached five feet eleven.)
Practically speaking, there is a very large class of mathematical applications for fuzzy logic—in algebra, game theory, geometry, linear programming, probability, statistics, topology. Kosko, for instance, introduced the idea of fuzzy cognitive maps, an artificial-intelligence tool that researchers are starting to apply in medicine, engineering, defense analysis, and elsewhere. As a fat new book, “Fuzzy Logic and Mathematics,”
explains, Zadeh’s alternative “does not abandon the classical truth values—true and false—but allows for additional ones.” The choice used to be between one, for true, and zero, for false; now all the intervening numbers, a potential infinity, are available, too. “Zadeh showed that it is the spectrum of possibilities between these strict absolutes that is far richer and more interesting—‘degrees of truth’ that model much more realistically the actual situations we face in the ‘real’ world,” Joseph Dauben, a historian of science at the City University of New York and one of the book’s co-authors, told me in an e-mail. “Fuzzy logic, like chaos theory, helps to handle situations that otherwise would be hard to deal with in a rational, sensible way.”
Among pure-blooded logicians, however, there is still uncertainty as to the merits of fuzziness. This spring, I attended a conference at Berkeley celebrating the sixtieth anniversary of the Group in Logic and the Methodology of Science, of which Zadeh was an early member. One of the speakers at the event, the computer scientist and I.B.M. Fellow Ron Fagin, described using fuzzy logic to solve the problem of combining information from two types of databases; the result was a very efficient, and very short (ten-line), algorithm. But at lunch one day, when I asked a handful of attendees what they thought of Zadeh’s approach, the mood turned uneasy. “I’m a logician; I like to sort things,” Jeremy Avigad, a professor of philosophy at Carnegie Mellon, said. His colleague Michael Rathjen, of Leeds University, concluded simply, “Too fuzzy.” (Rathjen ate his sandwich with a fork and knife.)
Zadeh would no doubt have enjoyed taking up the debate. “He was a contrarian’s contrarian,” Kosko told me. In a photo from his student days in Tehran, Zadeh is pictured sitting in his study with a sign above the desk reading, in Russian, “один”—“ALONE.”