Mathematics goes soft around the edges
Cambridge, Mass. — These computer-driven, quantified, categorically stratified, actuarially, hypothetically, statistically tabulated times notwithstanding, numbers just don't tell the whole story anymore.
So a new approach begins to catch the imagination of those who whisper in decisionmakers' ears: fuzzy- mindedness.
It begins in the undersize and disheveled offices of professors of applied mathematics, and, as wandering pollen wafts through esoteric symposiums and scholarly journals -- such as the bimonthly Fuzzy sets and systems, published in Amsterdam -- and into think tanks and onto the charts of consultants. From there it spreads gradually into the world of merger, marketing, and government regulation.
It eventually becomes part of the picture of how the world works that the ordinary person carries in the back of his head. And perhaps never particularly thinks about.
The hard and fast is making room for the soft and fuzzy. Some scientists and theorists acknowledge that, in charting the tangled and uncertain webworks of real life, science can be something of an art.
The world just won't fit into a mathematical equation.Economic forecasts, given precisely, are sometimes wildly wrong; business decisions made by MBAs with heads for figures don't quite hit the mark. Similarly, no high-powered computer that can blink in pages of arithmetic more or less instantaneously can decipher a handwritten word. Where people are concerned, things go blurry around the edges.
So now there is a burgeoning interest in pulling into accont the vague, intuitive, and impressionistic factors that have been ignored in the modern quest for figures.
On one front, businessmen and their consultants begin to consider corporate culture, esprit de corps, and people-centered values more seriously in their decisons. On another, academics are at work refining the formal theory of "fuzzy sets."
When is a banana, mathematicians turn to one another and ask, no longer ripe? A car full? An expense worth it? When does a boy become a man?
How attractive is a building? And they set their pencils to scratching at their pads and make their programmable calculators hum.
Questions like these are traditionally not the stuff of mathematicians and calculators. Exactly what, after all, is meant by ripe, or attractive? But some fuzzy mathematicians are attempting to ponder the imponderable.
Most simply, they are trying to reconcile numbers and words -- numbers representing the exact and literal world of the sciences, words representing the subtle and ambiguous humanities. Science and art.
The fuzzy school of thought traces its history back to 1965, when Prof. Lofti A. Zadeh of the University of California at Berkeley wrote a paper entitled "fuzzy sets." He had noticed a mismatch between applied mathematics and the real world. "I gradually came to the conclusion," he says, "that classical mathematics was too precise to describe systems where a human element was involved."
"Nature," Professor Zadeh explains in a favorite phrase, "writes with a spray can rather than a ball-point pen."
Now there are at least three scholarly journals on fuzzy- set theory published in the Netherlands, France, and China, respectively. The Japanese have fuzzy-set work groups that meet monthly and, according to Zadeh, use fuzzy mathematics extensively in fields like robotics. Fuzzy-set conferences are held every year or two -- the most recent one was in Peking. Fuzzy thinking seems to be spreading.
It becomes important in areas like federal regulation, where experts are expected to take all sorts of fuzzy variables and come up with clear-cut numbers as to what constitutes "dangerous" or "dirty." Or in business, where a collection of working people try to behave according to financial analysis charts -- or try not to.
A certain company, for example, has been looking for another firm to buy for years. According to Joseph Fiksel, an operations research specialist with the Arthur D. Little consulting firm here (ADL), this company knows all its financial goals -- knows what it wants in number terms -- but nothing has clicked. So ADL took a group of the company's managers through a strictly subjective exercise probing for what it wanted, beginning with the most basic categories, such as transportation of foodstuffs. Common interests were found.
The gist of what makes Professor Zadeh's logic fuzzy -- and the bridge over the age-old gap between the hard and the soft, the Aristotelian and t he Platonic -- is the idea of "grades of membership." Fuzzy mathematicians take words (they call them "linguistic variables") such as "dangerous emission levels" or "visual privacy" and figure out the degree to which they apply to a given situation.
Some men, for instance, are clearly members of the set of tall men. Some are clearly not. But it isn't at all clear whether other men are tall or not. Not because their height is uncertain, but because "tall" is a fuzzy term.
Classical logic says yes or no, in out, tall or not tall. But by fuzzy logic a man may have an 80 perceent membership in the set of tall men, not because he is only 80 percent of a certain height, but because he is perrcent into the gray haze that surrounds the term "tall."
Once Zadeh had conceived this idea of graded membership, he says, fuzzy mathematics grew from it smoothly. Now there is a fuzzy algebra and a fuzzy calculus. (In other languages, he says, "fuzzy" doesn't carry the stigma it does in English.) Discussion has spread, converts have been won, and resistance aroused.
Mathematics begins to take on the ambiguity of language. Ideally, two things follow. Predictions, models, and estimates no longer show unrealistic precision. And the numbered, structured view of the scientist figures in -- through language -- qualities that defy the accuracy of numbers.
It has yet to make its way into public acceptance. Opposition to the fuzzy approach among academics, Zadeh says, is natural enough, since it marks a retreat from the whole direction of modern math and science toward precision. And the public, uncomfortable with uncertainty, often prefers experts to speaks with clear-cut authority.
"In its broadest context," ADL's Joseph Fiksel says, "it represents a turnaround from a highly mechanistic world view we inherited from the industrial revolution." Fiksel, with an academic background in applied mathematics, began taking fuzzy-set theory seriously three years ago after hearing Dr. Zadeh deliver a paper at a Mexico City conference. Since, he has written on the subject himself and given seminars for colleagues at ADL.
Accidents like the one at Three Mile Island two years ago show the weakness of the classical approach, Dr. Fiksel says. There, engineers had rigorously measured the risks, set up "fault trees," calculated every possible combination of factors.
This careful precision, Fiksel believes, leads to a false sense of security. In this case, the unexpected happened. Human error wasn't sufficiently figured in. And this, he adds, is typical of such analyses everywhere.
"Imponderables enter in and fall between the cracks," he explains. "Through hindsight we see that the original analysis was inadequate." The answer lies back with the human element.
Just as the human eye can immediately read a message scrawled sloppily on a note pad, making a dazzling set of associations and inferences from context that the most sophisticated computer scanner is helpless to emulate, so people who understand other people know best how human error at Three Mile Island can be avoided next time.
Keep the structure of the scientific approach, Fiksel advises, but tie it to fuzzy-type, subjective judgment as well. "People familiar with a particular realm can give you an intuitive understanding of how all these forces come together and what is likely to happen."
In other words, when a businessman waves aside the computer data and accountants' statistics and asserts, "I know my company," he is probably right. His intuitive grasp of his business is probably made up of a "constellation of insights" too subtle and full of complex priorities to be expressed in mathematical symbols.
These insights also manage to compare what are often, in mathematical terms, incomparable. Weighing in life styles and the legislative environment with financial factors for, say, charting the future of the automobile business often requires a mix of words and figures, since some of the social and political elements are difficult to quantify.
This intuitive element can sew the numbers and equations together and fill in the cracks, allowing a grasp of the whole blurry picture and pointing up what probability and statistics fail to predict.
"If you look at history, yoy get these sort-of-surprise happenings, these quantum leaps," says Fiksel, that throw the numbered and graphed view on its ear. He counts the formation of OPEC or the assassination of a president among such historical surprises.
Fuzzy-set theorists like to describe it as a measure of possibility, or plausibility, in Fiksel's term. This is distinct from probability. Chance and possibility are different kinds of uncertainty, according to the fuzzy theorists.
While a statistician might ask the likelihood of a coup d'etat in a certain country based on the occurences of coups in comparable situations, a fuzzy theorist might gather a variety of opinions and viewpoints and predict something the statistics cannot show.
A key difference lies in the acknowledgment by fuzzy mathematics that society is not a series of random occurrences following the laws of chance. Rather, they maintain, it is the sometimes surprising outcome of improvising and purposeful, albeit a little fuzzy, people trying to beat the odds.