Fuzzy set theory can be considered as "the art of focusing”. The art of searching the best way of affording new problems. To obtain this goal was at the root of Zadeh’s aims not only in his 1965 seminal paper but also in his “From Circuit Theory to System Theory” of 1962, a very innovative survey of the transition - described in the title - that was happening in those days, A survey which, incidentally, was also focusing something very new:

“There are some who feel that this gap (the one between "animate" systems and "inanimate" systems) reflects the fundamental inadequacy of the conventional mathematics, the mathematics of precisely-defined points, functions, sets, probability measures, etc. for coping with the analysis of biological systems, and that to deal effectively with such systems, which are generally orders of magnitude more complex than manmade systems, we need a radically different kind of mathematics, the mathematics of fuzzy or cloudy quantities which are not describable in terms of probability distributions.”

It is clear that - in those remote years - Zadeh was looking for something very ambitious: a “radically different kind of mathematics”. Now that we all can see more than half a century of important and original achievements we should ask whether this initial ambition, expressed in informal terms, has been fulfilled. Paradoxically, the answer to this question is “no”. The stubborn success we all witness has been obtained by putting into parentheses some crucial foundational questions.

Among the topics that will be briefly surveyed in the talk let’s mention: Trillas’s idea of looking at fuzzy sets as an “experimental science”, the import of “measuring fuzziness”, Zadeh’s “computing with words” in dialogue with Church-Turing thesis, the relationships between fuzzy logic and quantum logic.

Some of the original aims, along these decades, have been considered - implicitly or explicitly - vagaries. Now, after fifty years, it is the moment to look again at the vagaries. And this is a work that can be smartly done by young people whose ideas are less conditioned by the paths traditionally followed.