High Priest of Fuzziness: Bart Kosko

Bart Kosko: High Priest of Fuzziness

A brash 30-year-old assistant professor ofelectrical engineering systems has become the point man for a new branch ofmathematics thatposes a direct challenge to the 3,000-year-old system ofAristotelian logic underpinning contemporary scientific methodology.

By Sheldon Teitelbaum

His name was Bart Kosko, and he was a tall, fresh-faced kid from Kansas who had left the farm to write symphonies. He was a good enough composer to net a scholarship to the USC School of Music, and to have won a coveted Young Composer's Award. Somewhere along the way, though, after brief and not-so-brief pit stops in philosophy and economics and sojourns in adventure fiction-writing and science-based industry, he ceased contem- plating the music of the spheres and started thinking seriously about the geometry of hypercubes.

After that, well, things got a little fuzzy.

Now a 30-year-old assistant professor of electrical engineering systems in the School of Engineering, Kosko has become the point man for a new branch of mathematics that poses a direct challenge to the 3,000-yearold system of Aristotelian logic underpinning contemporary scientific methodology.

They call it "fuzzy theory" or, as Kosko prefers, "fuzziness." It holds that mathematics as we know it is only a first approximation of a much larger, richer and messier reality. However convenient it may be for mathematicians or computers, "real" reality isn't always black or white, yes or no, one or zero.

Fuzzy theorists prefer to describe situations — a person's tallness, perhaps, or the degree to which a mound might be a mountain — in terms of membership in a set. The Santa Monica Mountains, for instance, might possess a 15 percent degree of membership in the set of the world's highest ranges, while the Himalayas might have a 95 percent degree of membership in that set. In other words, the Himalayas are extraordinarily high, whereas the Santa Monicas are only marginally so.

Kosko in class: Fuzziness holds that mathematics as we know it is only a first approximation of a much larger, richer and messier reality.

Yet if one subscribes to Aristotelian logic — and most Westerners do — a thing either is or isn't. A man is either bald or he's not; a mound is either a hill or a mountain; and if that means a man with 5,000 hairs is bald and one with 5,001 isn't, or that a heap of earth 500 feet high is a mound whereas one 501 feet high is a mountain, that, unfortunately, is the law of the excluded middle, and there's no way around it.

Fuzzy theory, though, says that this law, however serviceable as a mathematical principle, is quite obviously inane in the real world. In fact, Kosko says, the only place it holds at all is in mathematics, where 2 is an even, not an odd, integer. The truth can lie somewhere in the middle, where words like "maybe" and "somewhat" describe reality more accurately than "is" or “isn't,” or "one" and "zero."

Bart Kosko's contribution to this kind of reasoning — which was first articulated by Lotfi Zadeh, a University of California, Berkeley electrical engineer and erstwhile department chairman who expanded upon ideas introduced by Polish mathematician Jan Lukasiewicz during the 1920s — has been to geometrize fuzziness in the same manner that Einstein geometrized gravity.

Kosko pictures the set of all fuzzy sets as a Rubik's cube, or "hypercube." Conventional sets, like the set of Stealth bombers now in existence, are on the corners of the cube. Elements belong to these sets all or none, one or zero. The fuzzier the sets — which means the more ambiguous they are, like the set of somewhat hairy men — the deeper they are in the heart of Kosko's hypercube.

At the midpoint of this fuzzy Rubik's cube, says Kosko, are the phenomena Western thought has labeled as paradoxes. Kosko calls this "the black hole of set theory, where a thing can be its own opposite." Here, for instance, you will find the proverbial cup that is half empty or half full. Also present are the Taoist concept of yin-yang, the liar from Crete who said all Cretans are liars, Bertrand Russell's set of all sets that are not members of themselves, and Russell's barber.

Russell’s barber, explains Kosko, is a bewhiskered man who lives in a town and who shaves a man if and only if he does not shave himself. So who, one might ask, shaves the barber? If he shaves himself, by definition he does not. But if he doesn't, he does.

This is enough to drive most Westerners crazy, and according to the annals of mathematical history, some have succumbed to. despair at ever resolving these enigmas. But, Kosko believes, despair sets in only if one rigidly ascribes to an arbitrary insistence on bivalence. People in the Far East, he says, would not fall into such a trap. In Japan and other societies experienced in Buddhism, the idea of a thing containing its own opposite is not hard to swallow.

“In the end,” he says, “the insistence on either-or interpretations of reality, which Western science relies upon, is merely a cultural preference.”

What is socially significant about fuzziness, he believes, is that it is forcing the world’s first confrontation — at the level of science — between Eastern and Western belief systems. For the first time, there is a mathematical method for expressing the possibility that a thing and its opposite can at once both be and not be, or be partly and partly not — concepts which appear more at home within Taoism than the Western system of logical positivism.

Kosko is convinced that fuzziness will, in fact, force a fundamental reassessment of some of the widespread notions of probability and randomness that Western scientists have relied upon to account for those aspects of reality not easily accommodated by Aristotelian tenets. “This has never before happened,” he says.

B art Kosko’s transition from art to science is an unusual one. He was 14 when he discovered a love for classical music kindled by an abiding curiosity about how Beethoven had achieved his “special effects.” (The choice of words here is telling, because Kosko would ultimately demonstrate a considerable talent for film scoring, and would, in fact, score various junctures in a 12-hour academic videotape series on fuzziness.)

He obtained recordings and the scores of Beethoven's nine symphonies, began getting up an hour earlier than the ungodly hour he was already used to, and pored over them. By the age of 15 he was composing violin and piano sonatas and string quartets, many of which were performed at a local college. Eventually, he wrote an overture to The Count of Monte Cristo for the Northland Symphony Orchestra which, in conjunction with the Young Composer's Award, secured him the USC School of Music scholarship. He enrolled in 1978.

Unfortunately, the match was not one made in heaven. “I wanted to write symphonies and tonal music, which wasn't what some of the avant-garde faculty here had in mind,” he says. “But I passed the exams, so the music faculty let me keep the scholarship for a year. That was quite decent of them. They also suggested I take the graduate film scoring course, which I did. I then finished my first symphony as a protest.”

He switched his major to philosophy and economics, supporting himself with various work-study jobs and by writing commercial fiction for a series of men’s adventure magazines. He was also active, during those years, in libertarian politics, and remains a staunch champion of libertarian ideals, even calling his vision “scientific libertarianism.” However, he continued to be restless in his intellectual pursuits.

“Increasingly,” he recounts, “I had the feeling that the world was not black and white. In philosophy one either studies logic and linguistics or reads other philosophers. In logic, though, everything is either zero or one, false or true. This hadn't changed in millennia, since it first was codified by Aristotle. That is never symptomatic of a healthy theory.

“Formal logic was too dogmatically definitive. Yet the statements of science were always such that the truth could be otherwise — it could be neither zero nor one, it could be .007 or .99. So you ended up with a formal system that didn’t apply to the real world, and that was always breaking down at the edges.”

Kosko switched his allegiance to mathematics, but he could find no solace in the probability theory he studied as a graduate student at UC San Diego. Unfortunately, though useful as arule of thumb, probability does not exist in nature, and woe to the scientist who comes to believe that it does.

Probability, explains Kosko, works with large statistical populations. But it is incapable of dealing with individual situations.

“Tt assumes the world is black or white, but it is never sure which one. There's uncertainty in fuzziness too, but it’s deterministic uncertainty.

“Probabilistic uncertainty turns out to be something strange — the degree to which a set is contained within one of its own subsets. Ifyou flip a coin 100 times and it comes up heads 50 times, we culturally interpret the selective frequency of head flips as a “50 percent probability.” In fact, we know now by theorem that relative frequency is the degree to which all trials are successful, all flips are heads.

“Which is harder to believe in,” Kosko found himself asking, “God or probability?”

Probability, he asserts, is merely a convenient construct. Randomness is fiction. “The ultimate fraud,” he says, “is the scientific atheist who believes in probability.”

Kosko received his master’s degree in mathematics from UCSD in 1983. While he was there, he had received some serious job offers in the signal processing field, and worked as coordinator for artificial intelligence at General Dynamics and, later, for a company called VERAC, where he pursued basic research and design in battle management systems for the strategic and air defense initiatives.

Kosko encountered his first reference to fuzziness in 1983 in a scientific paper. He recalls being struck by revelatory wonder. “This really seemed to tie together many of my concerns,” he said.

“I saw Lotfi Zadeh (the field’s venerable originator and sage) as a 20th-century Zarathustra who came down the mountain and said this is the way things are — here is the big picture. His influence was really seminal.”

Kosko chose electrical engineering as the field in which to pursue his doctorate, seeing it as the area in which most of his interests converged. A year later, at an artificial intelligence conference in Austin, Texas, Kosko encountered Zadeh in the flesh for the first time.

Born in the Soviet Union and educated in Iran and the United States, Zadeh first fuzzified his own thinking in 1964, while attempting to figure out how a computer might be taught to read handwriting.

While staying in a New York hotel, it occurred to him that membership in sets might be a matter of degree. Contemplating letters in this fashion would give a computer considerable leeway in determining the nature of a handwritten letter.

Zadeh might have christened the discipline that grew out of this idea something like “continuous set theory,” becoming a multivariable extension of Lukasiewicz’s earlier continuous logic. He settled on the term “fuzziness,” however, because its inherent accuracy and self-explanatory quality outweighed, he thought, any pejorative connotations the term might have.

In Austin, Kosko walked up to the venerable scientist and breathlessly blurted out that he had actually proven some of his theorems. Zadeh responded graciously, suggesting that the young man send his work along in the mail.

The next day, at an after-conference cocktail party, Kosko overheard one belligerent mathematician, an outspoken probabilist, ridiculing Zadeh and his theories. Most of the crowd seemed disposed toward this point of view, if only because Zadeh himself had never really conceptualized fuzzy theory in distinct mathematical terms that they could quantify. Kosko the graduate student rushed to Zadeh’s defense, and an intense argument broke out. There was more than reputation involved here, as they both knew — this particular argument was about as basic as they get. If Kosko was right, then 3,000 years of science was wrong or, at the least, in need of some major finetuning, which was not what most people wanted to hear — not, anyway, from someone who hadn't even completed his doctorate. After meeting Zadeh in Austin, Kosko complied with his request and sent along his theorems, noting in the cover letter that he had become involved in a public spectacle regarding the efficacy of fuzziness. This resulted in a phone call from Zadeh. A correspondence ensued, and Zadeh eventually agreed to become one of Kosko’s Ph.D. advisers.

Kosko’s own contributions to the field of fuzziness grew out of an experience at another artificial intelligence conference a year later, in 1985. One of the participants mentioned a theorem which, by its very nature, seemed to challenge Zadeh’s notions. Zadeh let the challenge roll off him, but Kosko recalls that a chill ran down his spine; suddenly he found himself experiencing moments of grave doubt about whether fuzziness was all that Zadeh said it was.

“There was a feeling that, maybe, there really is no fool like an old fool. Maybe I had simply been swayed by Zadeh’s personality.

“At that point, I had to know definitively whether or not he was right. I was like a theist who simply had to establish the existence of God. If true, I'd become a priest. If not, I would become the biggest atheist who ever lived.”

What this meant was going back to scratch and building up a mathematical basis for Zadeh’s ideas. Kosko’s current department chairman at USC, Jerry Mendel, says this readiness to go back to first principles, unusual in the world of science, may have assured Kosko’s success in this endeavor.

Many people still aren't buying into Bart Kosko’s notions. Scientists are, after all, people. And some of them think Kosko is still too young and too outspoken to bemaking these kinds of waves. The record, however, shows that Kosko has paid his dues. As a scientist, he is a recognized neural-network and fuzzy theorist who has organized several important international conferences and is about to publish a major textbook. NASA has sought his services — as a consultant on the space shuttle — as did the Strategic Defense Initiative organization while he was still in his mid-20s.

Outside the realm of science, moreover, Kosko is an accomplished symphonist who scores his own videotaped lectures, a published writer and essayist and, lately, a wouldbe screenwriter now collaborating on a sweeping science fiction epic with Bill Gray, who wrote the 1984 film The PhiladelphiaExperiment.

MIT artificial intelligence pioneer Marvin Minsky considers Kosko to be one of the few genuine polymaths in modern science, calling him “a bewildering combination of talents.” Others, like George Klir, editor of the prestigious /nternational Journal of General Systems (which will soon publish the latest in an already significant body of work produced by Kosko), characterizes his mathematical ideas as “quite profound.” Zadeh, his mentor, calls him “one of the most unusual and most highly gifted individuals I have ever met.”

This, certainly, is the view of many Japanese scientists, who hold both Kosko and Zadeh in enormous regard. The Japanese have bought into fuzziness in a big way. They have set up well-funded national centers devoted to the field. Some 60 Japanese companies are, in fact, adapting fuzziness to their computer systems.

Fuzziness appeals to Japanese engineers because it works. Fuzzy chips and software afford a systematic, not-too-complex framework for using approximate reasoning to solve complicated programming problems.

Conventional Al approaches to various tasks are too complex and rigid, attempting to replicate the various heuristics, or rules of thumb, that underlie human reasoning. With the present state of technology, unfortunately, this is often far more trouble than it seems worth. Few experts charged with designing “expert systems” can articulate what makes them an expert.

Fuzzy engineering, on the other hand, has little difficulty accommodating those imprecise quantifiers — values like “most,” “many,” “nearly zero,” “sometimes” — so characteristic of normal human reasoning in real-world situations. Fuzziness, perhaps for the first time, offers a rigorous mathematical language capable of expressing these concepts.

Your traditional air conditioner, to illustrate the benefits of fuzzy applications, is activated when it senses a room is either too hot or too cold. It either blasts at full power or shuts down. A fuzzy air conditioner, in contrast, can identify a range of comfortable temperatures. Instead of shutting down, it tapers off its activities as the area being cooled falls into that range.

Recently, when Japanese manufacturers started coming out with automobile transmission control systems, handheld camcorders and entire factories that engaged in approximate rather than clear-cut reasoning, critics of the field were taken aback.

After all, one of the biggest arguments against artificial intelligence is that it simply hasn't delivered the applications that were promised when these fields opened. Yet in the Japanese city of Sendai, engineers set up a fuzzy subway system that produced more efficient routings, and a smoother ride for less energy, than conventional subways. In fact, from now on all new Japanese subways will incorporate fuzzy technology. Fuzzy chips and software, which follow often vague and consequently more flexible rules of thumb than other computer setups, have put conventional Al approaches away.

“Machine intelligence,” says Kosko, “has so far affected the imagination more than the global economy. Fuzziness affects both.”

The critics weren't giving up without a fight, though. Presented with applications, they demanded proof of the underlying theory. This time, however, Kosko was one step ahead of them. Recently, he worked out the actual theorems which, he says, indisputably establish conventional mathematical approaches to reality as a rough approx-imation, and probability as rather a special case of fuzziness. People may not like the idea of fuzziness, but whether they do or not, the theorems, Kosko believes, are incontrovertible.

Bart Kosko believes that fuzziness is not only going to force a radical reassessment of the way scientists model reality, but will inevitably work fundamental changes in Western culture. Western culture, however, may not survive the experience intact.

“In the short term, fuzziness will give us new machine intelligence applications. The way we systematize knowledge will also change. Fuzzy techniques will permit us to encode, say, the analysis of a Henry Kissinger on something called a fuzzy cognitive map, which we will be able to process numerically, synthesize and adapt with others, and use for purposes now beyond anyone’s ken,

“In the longer term, though, we will witness cultural cHanges. We will no longer need notions of randomness to describe our determin-istic, chaotic universe. Notions of ‘space and time, of matter and antimatter, will change. Our insistence upon maintaining artificial dualities between these will be harder tomaintain.

“Ultimately,” he notes, “we are headed for a clash of Eastern and Western belief systems at the level of science. This has never happened before. Cultural biases accreted for 3,000 years stand to fall. Western beliefs, like Western economies, will be given a run for their money. It will make great 21st-century entertainment.” 0