IEEE Systems, Man and Cybernetics Magazine - July 2018 - 50

Remembering Lotfi Zadeh (continued from page 49)
radical for you!" Sergei did as he was told, and the picture of

After what was certainly a pyrrhic victory, I duly forgot

the four turned out very nice and was published in a paper

the question that, nevertheless, remained hidden in my

by Sergei as well as in a book in homage to Prof. Zadeh [S3].

subconscious up until many years later, when I returned to it

It is just a shame you can't make out the red in the black

while writing a paper, along with a number of colleagues, on

and white photograph.

computation with antonyms [S7]. It was after I had computed

Not only was Prof. Zadeh very kind with everyone, and

the antonym of "close to four" by using a symmetry in [0, 10]

particularly with young researchers with whom he always

decomposed into several parts, each one corresponding to

engaged in conversation and advised in kind, simple

the subintervals in which maround 4 is monotonic, constructing

terms, but he always gave his oral presentations without

it like a + b - x but with a and b being the extremes of the

any attempt to show one iota of personal superiority over

corresponding subinterval, and obtaining a function easily

the audience. Only once did I ever see him lose his temper

recognizable as a membership function representing "far

in public; it was at a seminar in Berkeley, back in the mid-

from four." For me, sooner or later, discussions with Prof. Zadeh

1980s, when a young researcher presented his ideas in

always turned out to be fruitful.

an excessively shocking and bombastic style. He quoted

Throughout Prof. Zadeh's life, the efforts he made in

Henri Poincaré, but wrote on the blackboard "Pointcarré."

continuously encouraging people to work on fuzzy logic

Prof. Zadeh suddenly leapt up from his seat toward the

were remarkable. Even if he almost always published papers

blackboard and, while rubbing out "Pointcarré" and writing

with just his own signature (he liked to work alone), he

"Poincaré," shouted earnestly at the speaker, "Be careful with

never refused to discuss his results with anybody; his lectures

spellings!" The speaker blushed, finished off quickly, and was

were typically rounded off with, "Please, don't hesitate to

never seen or heard by us again.

contradict me!" A highly frugal person, he never ate too

I did not always fully agree with Prof. Zadeh during our

much, smoked, or drank alcohol. His health was punished

conversations, but I always came out enriched by the

by the many years of making trips around the world to

discussion; he inspired me to keep turning things over and

deliver addresses at a multitude of conferences in different

over in my mind. One example of such an occasion is the

countries. For instance, after the serious heart attack he

following story.

suffered at the end of 2009, he told me that he had flown

Given a predicate P and its membership function mP , to

over 100,000 mi that year.

represent the membership function of its antonym aP,

Prof. Zadeh enjoyed contact with people of different

Zadeh maintained that this should be done by taking

cultures and maintained a healthy respect for them all. He

maP (x) = mP (a + b - x); that is, by considering the symmetry

was not a nationalistic person; rather, he preferred multi-

in [a, b], and defining maP = mP ^s ^x hh,

cultural growth for people. The academic and industrial

s (x) = a + b - x

for all x. Agreeing on the use of a symmetry to reflect the

culture of fuzzy logic spread all over the world.

opposition between P and aP, I tried to convince him that the

AS: Prof. Zadeh was an engineer who loved mathematics.

symmetry should depend on P; that it cannot be the same

He believed that mathematics provided an accurate

for all P in [a, b]; he refused to accept it, and all the examples

analysis of both natural and anthropocentric problems. In

he certainly managed to give me some good antonyms. It

fact, and as he explained, before 1964, he asked prominent

took me a while to realize that if the graph of mP is either

mathematicians Herbert Robbins and Richard Bellman [S4]

decreasing or nondecreasing, as was the case in all the

to develop a mathematical theory of sets with unsharp

examples given by Prof. Zadeh, just the symmetry a + b - x

boundaries, a request which, owing to the numerous

is fine, but that the problem arises when this graphic is not

commitments that the two mathematicians had back then,

monotonic [S9]. Knowing full well that to convince Prof. Zadeh

was not met. In 1964, on his own, Prof. Zadeh came up with

I would have to provide him with a concrete counterexample,

the notion of fuzzy sets specified through a membership

I decided to postpone the discussion. The next day and,

function, which he published in 1965 [S11] and is currently

finally, the predicate "close to four" in [0, 10] whose antonym

one of the most-cited papers from the 20th century.

should be "far from four," gave me the solution, since with

Prof. Zadeh's main concern was with precisiation

the symmetry 10 - x, we obtain a membership function

and formalization. Precisiation is to change vagueness,

corresponding to the predicate "close to six." I immediately

a linguistic-philosophical concept, into fuzziness, a

returned to broach the subject with Prof. Zadeh and, after

measurable or scientific one [S9]. Formalization is to provide

joking with me by saying,"Well, but 'close to six' already implies

mathematical models to understand problems from science

'far from four,'" he accepted that the symmetry should be

or humanities; as they usually show borderline cases, Prof.

taken accordingly with the characteristics of P, that it should
be sP but not only s.

50

IEEE SyStEmS, man, & CybErnEtICS magazInE July 2 0 1 8

(continued on page 51)



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