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April 10, 2026
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"One should clearly distinguish between two aspects of the estimation problem:"
"::(ii) Conditions which guarantee a priori the existence, physical realizability, and stability of the optimal filter."
"Among Kalman's early work was the development of what is now called the Kalman filter for detection of signals in noise. This revolutionized the field of estimation, by providing a recursive approach to the filtering problem. Before the advent of the Kalman filter, most mathematical work was based on Norbert Wiener's ideas, but the 'Wiener filtering' had proved difficult to apply. Kalman's approach, based on the use of state space techniques and a recursive least-squares algorithm, opened up many new theoretical and practical possibilities. The impact of Kalman filtering on all areas of applied mathematics, engineering, and sciences has been tremendous. It is impossible to even begin to enumerate its practical applications. Just as examples of their diversity, one may mention the guidance of the Apollo spacecraft and of commercial airplanes, uses in seismic data processing, nuclear power plant instrumentation, and demographic models, as well as applications in econometrics."
"Let me say quite categorically that there is no such thing as a fuzzy concept... We do talk about fuzzy things but they are not scientific concepts. Some people in the past have discovered certain interesting things, formulated their findings in a non-fuzzy way, and therefore we have progressed in science."
"I would like to comment briefly on Professor Zadeh's presentation. His proposals could be severely, ferociously, even brutally criticized from a technical point of view. This would be out of place here. But a blunt question remains: Is professor Zadeh presenting important ideas or is he indulging in wishful thinking? No doubt Professor Zadeh's enthusiasm for fuzziness has been reinforced by the prevailing climate in the U.S.-one of unprecedented permissiveness. 'Fuzzification, is a kind of scientific permissiveness; it tends to result in socially appealing slogans unaccompanied by the discipline of hard scientific work and patient observation."
"I have been aware from the outset (end of January 1959, the birthdate of the second paper in the citation) that the deep analysis of something which is now called were of major importance. But even with this immodesty I did not quite anticipate all the reactions to this work. Up to now there have been some 1000 related publications, at least two Citation Classics, etc. There is something to be explained. To look for an explanation, let me suggest a historical analogy, at the risk of further immodesty. I am thinking of Newton, and specifically his most spectacular achievement, the law of Gravitation. Newton received very ample "recognition" (as it is called today) for this work. it astounded - really floored - all his contemporaries. But I am quite sure, having studied the matter and having added something to it, that nobody then (1700) really understood what Newton's contribution was. Indeed, it seemed an absolute miracle to his contemporaries that someone, an Englishman, actually a human being, in some magic and un-understandable way, could harness mathematics, an impractical and eternal something, and so use mathematics as to discover with it something fundamental about the universe."
"A of the Riccati type is derived for the covariance matrix of the optimal filtering error. The solution of this 'variance equation' completely specifies the optimal filter for either finite or infinite smoothing intervals and stationary or non-stationary statistics. The variance equation is closely related to the Hamiltonian (canonical) differential equations of the calculus of variations. Analytic solutions are available in some cases. The significance of the variance equation is illustrated by examples which duplicate, simplify, or extend earlier results in this field. The duality principle relating stochastic estimation and deterministic control problems plays an important role in the proof of theoretical results. In several examples, the estimation problem and its dual are discussed side-by-side. Properties of the variance equation are of great interest in the theory of s. Some aspects of this are considered briefly."
"At present, a nonspecialist might well regard the Wiener-Kolmogorov theory of filtering and prediction [1, 2] as "classical' — in short, a field where the techniques are well established and only minor improvements and generalizations can be expected. That this is not really so can be seen convincingly from recent results of Shinbrot [3], Stceg [4], Pugachev [5, 6], and Parzen [7]. Using a variety of methods, these investigators have solved some long-stauding problems in nonstationary filtering and prediction theory. We present here a unified account of our own independent researches during the past two years (which overlap with much of the work [3-71 just mentioned), as well as numerous new results. We, too, use time-domain methods, and obtain major improvements and generalizations of the conventional Wiener theory. In particular, our methods apply without modification to multivariate problems."
"As an indispensable constituent of AI, fuzzy logic is a superset of conventional (Boolean) logic that has been extended to handle the concept of partial truth, where the truth value can range between completely true and completely false. As the creator of a new field of mathematics—fuzzy set theory and fuzzy logic—Lotfi Zadeh’s intellectual contributions are myriad. He is also known for his research in system theory, information processing, AI, expert systems, natural language understanding, and the theory of evidence. His current research is focused on fuzzy logic, computing with words, and soft computing, which is a coalition of fuzzy logic, neurocomputing, evolutionary computing, probabilistic computing, and parts of machine learning"
"Conceptual graphs (CGs) (Sowa 1976; 1984) and fuzzy logic (Zadeh 1965; 1975a) are two logical formalisms that emphasize the target of natural language, each of which is focused on one of the two mentioned desired features of a logic for handling natural language. Conceptual graphs, based on semantic networks and Peirce's existential graphs, combine the visual advantage of graphical languages and the expressive power of logic."
"Well, I knew it was going to be important. That much I knew. In fact, I had thought about sealing it in a dated envelope with my predictions and then opening it 20-30 years later to see if my intuitions were right. I realized this paper marked a new direction. I used to think about it this way-that one day Fuzzy Logic would turn out to be one of the most important things to come out of our Electrical Engineering Computer Systems Division at Berkeley. I never dreamed it would become a worldwide phenomenon. My expectations were much more modest."
"In many, many fields. I expected people in the social sciences-economics, psychology, philosophy, linguistics, politics, sociology, religion and numerous other areas to pick up on it. It's been somewhat of a mystery to me why even to this day, so few social scientists have discovered how useful it could be. Instead, Fuzzy Logic was first embraced by engineers and used in industrial process controls and in "smart" consumer products such as hand-held camcorders that cancel out jittering and microwaves that cook your food perfectly at the touch of a single button. I didn't expect it to play out this way back in 1965."
"The term fuzzy logic is used in this paper to describe an imprecise logical system, FL, in which the truth-values are fuzzy subsets of the unit interval with linguistic labels such as true, false, not true, very true, quite true, not very true and not very fake, etc.... As a consequence, the truth tables and the rules of inference in fuzzy logic are (i) inexact and (ii) dependent on the meaning associated with the primary truth-value true as well as the modifiers very quite."
"To what degree is something true or false?"
"I can't say that anything has been "exciting". Rather, I would choose the word "interesting". Not too long ago, the Chinese University of Hong Kong conducted a survey to determine which consumer products were using Fuzzy Logic. The result was a thick report, some 150-200 pages long-washing machines, camcorders, microwave ovens, etc. What interested me wasn't the particular applications so much as the breadth of applications-so many products were incorporating Fuzzy Logic."
"A linguistic variable is a variable whose values are words or sentences in a natural or synthetic language."
"Essentially, a fuzzy algorithm is an ordered sequence of instructions (like a computer program) in which some of the instructions may contain labels or fuzzy sets, e.g.: Reduce x slightly if y is very large Increase x very slightly if y is not very large and not very small If x is small then stop; otherwise increase x by 2."
"[T]he successes of modern control theory in the design of highly accurate space navigation systems have stimulated its use in the theoretical analyses of economic and biological systems. Similarly, the effectiveness of computer simulation techniques in the macroscopic analyses of physical systems has brought into vogue the use of computer-based econometric models for purposes of forecasting, economic planning, arid management."
"A linguistic variable is defined as a variable whose values are sentences in a natural or artificial language."
"The question really isn't whether I'm American, Russian, Iranian, Azerbaijani, or anything else. I've been shaped by all these people and cultures and I feel quite comfortable among all of them."
"The advent of the Computer age has stimulated a rapid expansion in the use of quantitative techniques for the analysis of economic, urban, social, biological and other types of systems in which it is the animate rather than in dominant role. At present, most of the techniques employed for the analysis of humanistic, i.e., human centred systems are adaptations of the methods that have been developed over a long period of time for dealing with mechanistic systems, i.e., physical systems governed in the main by-the laws of mechanics, electromagnetism, and thermodynamics. The remarkable successes of these methods in unraveling the secrets of nature and enabling us to build better and better machines have inspired a widely held belief that the same or similar techniques can be applied with comparable effectiveness to the analysis of humanistic systems."
"A frequent source of misunderstanding has to do with the interpretation of fuzzy logic. The problem is that the term fuzzy logic has two different meanings. More specifically, in a narrow sense, fuzzy logic, FLn, is a logical system which may be viewed as an extension and generalization of classical multivalued logics. But in a wider sense, fuzzy logic, FLw is almost synonymous with the theory of fuzzy sets. In this context, what is important to recognize is that: (a) FLw is much broader than FLn and subsumes FLn as one of its branches; (b) the agenda of FLn is very different from the agendas of classical multivalued logics; and (c) at this juncture, the term fuzzy logic is usually used in its wide rather than narrow sense, effectively equating fuzzy logic with FLw"
"More often than not, the classes of objects encountered in the real physical world do not have precisely defined criteria of membership. 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 case of a number such as 10 in relation to the “class” of all real numbers which are much greater than 1."
"It was a biologist — Ludwig von Bertalanffy — who long ago perceived the essential unity of system concepts and techniques in the various fields of science and who in writings and lectures sought to attain recognition for “general systems theory” as a distinct scientific discipline. It is pertinent to note, however, that the work of Bertalannfy and his school, being motivated primarily by problems arising in the study of biological systems, is much more empirical and qualitative in spirit than the work of those system theorists who received their training in exact sciences. In fact, there is a fairly wide gap between what might be regarded as “animate” system theorists and “inanimate” system theorists at the present time, and it is not at all certain that this gap will be narrowed, much less closed, in the near future. There are some who feel this gap 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, we need a radically different kind of mathematics, the mathematics of fuzzy or cloudy quantities which are not describable in terms of probability distributions. Indeed the need for such mathematics is becoming increasingly apparent even in the realms of inanimate systems"
"In general, complexity and precision bear an inverse relation to one another in the sense that, as the complexity of a problem increases, the possibility of analysing it in precise terms diminishes. Thus 'fuzzy thinking' may not be deplorable, after all, if it makes possible the solution of problems which are much too complex for precise analysis."
"A fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one. The notions of inclusion, union, intersection, complement, relation, convexity, etc., are extended to such sets, and various properties of these notions in the context of fuzzy sets are established. In particular, a separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint."
"[ Fuzzy logic is ] a logic whose distinguishing features are (i) fuzzy truth-values expressed in linguistic terms, e.g., true, very true, more or less true, or somewhat true, false, nor very true and not very false, etc2.; (2) imprecise truth tables; and (3) rules of inference whose validity is relative to a context rather than exact."
"Engineers were the only members of the community "who understand the needs of the nation, desires of the workmen, and the power of the productive forces""
"For more than twenty years Mr. Gantt has been closely interested in advanced work in the field of labor management. For more than ten years his name has been identified with certain methods which, nevertheless, are yet but partially and imperfectly understood by many, and because of this incomplete understanding are sometimes supposed to be summed up in the "Bonus System" of wage payment."
"It is becoming perfectly clear that the principles underlying industrial and military efficiency are the same and that a nation, to be efficient in a military sense, must first be efficient industrially"
"There is another and higher leadership, that of the intellect, by which the methods and thoughts of one man may affect the whole civilized world. Industrial leaders who have most prominently attracted our attention in the past are those who have, by their inventions or their direction of activities, accumulated large fortunes ; but none of these are as great as the man who by the force of his intellect leads people throughout the civilized world to benefit themselves and others. Such a man was the late Frederick Winslow Taylor who, in his determination to eliminate error and to base our industrial relations on fact, set an example which will have an effect all over the world"
"Taylor’s friend Henry Gantt explains to his fellow engineers in the middle of the First World War that they must "develop a task system on the basis of democracy that will yield as good, or better, results than those now in operation under autocracy""
"The achievement of Gantt offers a means of measuring the human or social efficiency of industry... Gantt's method has made it possible to ascertain the cause of the diseased industry just as blood analysis established the cause of malaria. While the latter made the completion of the Panama Canal possible, the former will transform industry from servitude into creative service and its pensioners into respectable members of the community... Unlike statistical diagrams, curve records, and similar static forms of presenting facts of the past (Gantt) charts... are kinetic, moving, and project through time the integral elements of service rendered in the past toward the goal in the future."
"The mechanical engineer today is carrying forward, under the direction of science, the work that was begun by the mechanic who first learned to chip flint or make a fire ; and it is he alone that can lead the mechanic of today to a better understanding of his problems, and the capitalist to a better appreciation of their solution."
"Scientific investigation is rapidly putting at our disposal vast amounts of knowledge concerning materials and forces, which it is the business of the engineer to utilize for the benefit of the community. Well-designed plants and efficiency labor-saving devices, to be seen on every hand, bear testimony that he is doing at least a portion of his work well. When, however, it comes to the operation of these plants and the utilization of these labor-saving devices, the lack of co-operation between employer and employee, and the inefficient utilization of labor, very much impair their efficiency. The increase of this efficiency is essentially the problem of the manager, and the amount to which it can be increased by proper study is, in most cases, so great as to be almost incredible."
"Finance and industry must be socialized somehow. If we refuse to do it from the bottom we shall have to do it from the top, and doing it from the top means the emergence of many Prussias — with wars upon wars."
"Mr. Gantt concentrated his attention on the development of a method of charting which would show a comparison between performance and promises... he had used a chart on which the work for machines was "laid out" according to the time required to do it. The Gantt Progress Chart, as developed from this early form, was found to help in the making of definite plans and to be highly effective in getting those plans executed. The rate at which the work goes forward is continuously compared with the advance of time, which induces action to accelerate or retard that rate. These charts are not static records of the past - they deal with the present and future and their only connection with the past is with respect to its effect upon the future."
"The aim of our efficiency has not been to produce goods, but to harvest dollars... The production of goods was always secondary to the securing of dollars."
"Whatever we do must be in accord with human nature. We cannot drive people; we must direct their development... The general policy of the past has been to drive, but the era of force must give way to that of knowledge, and the policy of the future will be to teach and to lead, to the advantage of all concerned"
"For continuous flow production such as this I know of nothing better for recording output and comparing performance with capacity or what ought to be produced, than the straight line charts developed by Mr. H. L. Gantt, which show required and actual production in terms of both quantity and time. Their use, however, is not limited to the class of work just described."
"The greatest problem before engineers and managers today is the economical utilization of labor. The limiting of output by the workman, and the limiting by the employer of the amount a workman is allowed to earn, are both factors which militate against that harmonious co-operation of employer and employee which is essential to their highest common good."
"When Taylor began his efforts at the Midvale Steel Company in the 1880s, several members of the American Society of Mechanical Engineers were likewise interested in labor management. Industrial capitalism was running up against renewed resistance from the growing ranks of labor, still committed to a sense of work integrity and craftsmanship. Task management, or scientific management as it came to be called, began to take shape in the eighties as the way to break the worker's threatening resistance. The heart of this approach is the systematic reduction of work into discrete, routinized tasks, totally separated from any policy decisions about the job. ... For capitalism to be firmly in control, it must monopolize information and techniques as surely as it controls the rest of the means of production. The worker must be permitted only to perform certain specific narrow tasks as planned by management."
"Despite the pseudo-scientific apologies for the Taylorist approach, the public readily developed a very negative view of it. As the Taylor Society admitted with surprising candor, scientific management was widely seen as "the degradation of workmen into obedient oxen under the direction of a small body of experts—into men debarred from creative participation in their work.""
"With the triumph of scientific management, unions would have nothing left to do, and they would have been cleansed of their most evil feature: the restriction of output. To underscore this idea, Taylor fashioned the myth that 'there has never been a strike of men working under scientific management', trying to give it credibility by constant repetition. In similar fashion he incessantly linked his proposals to shorter hours of work, without bothering to produce evidence of "Taylorized" firms that reduced working hours, and he revised his famous tale of Schmidt carrying pig iron at Bethlehem Steel at least three times, obscuring some aspects of his study and stressing others, so that each successive version made Schmidt's exertions more impressive, more voluntary and more rewarding to him than the last. Unlike Harrington Emerson, Taylor was not a charlatan, but his ideological message required the suppression of all evidence of worker's dissent, of coercion, or of any human motives or aspirations other than those his vision of progress could encompass."
"Frederick W. Taylor was the first man in recorded history who deemed work deserving of systematic observation and study. On Taylor's 'scientific management' rests, above all, the tremendous surge of affluence in the last seventy-five years which has lifted the working masses in the developed countries well above any level recorded before, even for the well-to-do. Taylor, though the Isaac Newton (or perhaps the Archimedes) of the science of work, laid only first foundations, however. Not much has been added to them since – even though he has been dead all of sixty years."
"Commons provided the cardinal reason for the unions' absence of hostility to Taylorism: "... the unions have generally come to the point of confining their attention to wages — that is, to distribution — leaving to employers the question of production." ... The "management's rights" clause found in every U.S. union contract ... vests the sole right to set work methods, job design, assignments, etc. with management. ... In fact, well before the War the idea began spreading that unionization, with its standard "management's rights" clause contracts, was the best approach for fitting the Taylorist yoke on the workers. The efficacy of this "trojan horse" tactic of union mediation led Thompson to prescribe industrial unionism over the AFL's craft unionism as the best way the secure the Taylor system in industry."
"We never had any use for Taylor or any of the efficiency or scientific management crowd. They never realized that human toil was the last thing in the world you had to be efficient about; the only way to be really efficient is to eliminate it entirely, and this would have been heresy to any of the Taylor, Gant, Barth, Cook efficiency crowd."
"Among the names of those who have led the great advance of the industrial arts during the past thirty years, that of Frederick Winslow Taylor will hold an increasingly high place. Others have led in electrical development, in the steel industry, in industrial chemistry, in railroad equipment, in the textile arts, and in many other fields, but he has been the creator of a new science, which underlies and will benefit all of these others by greatly increasing their efficiency and augmenting their productivity. In addition, he has literally forged a new tool for the metal trades, which has doubled, or even trebled, the productive capacity of nearly all metal-cutting machines. Either achievement would entitle him to high rank among the notable men of his day; — the two combined give him an assured place among the world's leaders in the industrial arts."
"The near future will show us the service which has been rendered to the mechanical arts by this generous publication of researches pursued with such uncommon perseverance. But even now we can admire without reserve the scientific method which has controlled this whole work. It is an example unique in the history of the mechanic arts. We have all admired the researches of Sir Lothian Bell, on blast furnaces, and those of Sir on the regenerative furnace; but notwithstanding the high scientific value of the work of these two great engineers, on reading their papers neither of them leaves an impression on the mind which can be compared with that of Mr. Taylor's paper. It is a model which every young engineer will have to study."