Mathematicians From The United States

"The opinion is widely prevalent that even if the subjects are totally forgotten, a valuable mental discipline is acquired by the efforts made to master them. While the Conference admits that, considered in itself this discipline has a certain value, it feels that such a discipline is greatly inferior to that which may be gained by a different class of exercises, and bears the same relation to a really improving discipline that lifting exercises in an ill-ventilated room bear to games in the open air. The movements of a race horse afford a better model of improving exercise than those of the ox in a tread-mill."

"Our so-called "Arabic" notation owes its excellence to the application of the principle of local value and the use of a symbol for zero. It is now conclusively established that the principle of local value was used by the ns much earlier than by the Hindus and that the Maya of Central America used the principle and symbols for zero in a well-developed numeral system of their own. The notation of Babylonia used the scale of 60, that of the Maya, the scale 20 (except in one step). It follows, therefore, that the present controversy on the origin of our numerals does not involve the question of the first use of local value and symbols for zero; it concerns itself only with the time and place of the first application of local value to the decimal scale and with the origin of the forms or shapes of our ten numerals. ... Hurt by the alleged arrogance of certain Greek scholars, Sebokht praises the science of the Hindus and speaks of "their valuable methods of computation. . . . I wish only to say that this computation is done by means of nine signs." Unfortunately, he leaves it to us to guess whether or not he used the zero. The passage, written about 662 A.D., is the earliest reference that has been found outside of India to our numerals. ...The form of the symbols with the zero, used in India, differed so widely from the old forms without the zero used there, that the former seem to have had an independent origin and to have been imported into India. ...The following are outstanding facts: 1. The earliest reliable record of the use of our numerals with zero is an inscription of 867 A.D. in India. 2. The validity of the testimony of early Arabic writers ascribing to India the numerals with zero is shaken, but not destroyed. 3. There is not a scintilla of evidence in the form of old manuscripts or numeral inscriptions to support the Greek origin of our numerals. 4. At present the hypothesis of the Hindu origin of our numerals stands without serious rival. But this hypothesis is by no means firmly established."

"Professor Sylvester's first high class at the new university Johns Hopkins consisted of only one student, G. B. Halsted, who had persisted in urging Sylvester to lecture on the modern algebra. The attempt to lecture on this subject led him into new investigations in quantics."

"My quotations from Newton suggest the motive which induced him to take a stand against the use of hypotheses, namely, the danger of becoming involved in disagreeable controversies. ...Newton could no more dispense with hypotheses in his own cogitations than an eagle can dispense with flight. Nor did Newton succeed in avoiding controversy."

"As regards algebra, the early Arabs failed to adopt either the Diophantine or the Hindu notations. An examination of [the algebra of Al-Khwarizmi] shows that the exposition was altogether rhetorical, i.e., devoid of all symbolism."

"The history of mathematics may be instructive as well as agreeable ; it may not only remind us of what we have, but may also teach us to increase our store. Says De Morgan, "The early history of the mind of men with regards to mathematics leads us to point out our own errors; and in this respect it is well to pay attention to the history of mathematics." It warns us against hasty conclusions; it points out the importance of a good notation upon the progress of the science; it discourages excessive specialization on the part of the investigator, by showing how apparently distinct branches have been found to possess unexpected connecting links; it saves the student from wasting time and energy upon problems which were, perhaps, solved long since; it discourages him from attacking an unsolved problem by the same method which has led other mathematicians to failure; it teaches that fortifications can be taken by other ways than by direct attack, that when repulsed from a direct assault it is well to reconnoitre and occupy the surrounding ground and to discover the secret paths by which the apparently unconquerable position can be taken."

"The history of mathematics is important also as a valuable contribution to the history of civilization. Human progress is closely identified with scientific thought. Mathematical and physical researches are a reliable record of intellectual progress."

"A. Wayne Wymore founded the first academic department of Systems Engineering in the world at the University of Arizona in 1960. He pioneered Mathematical-based Systems Engineering and later led Model-based Systems Engineering. He was an early and ardent supporter of the fomenting of INCOSE. He has led self-evaluation of Systems Engineering education, and continues to be one of the most prominent theoreticians of the Systems Engineering community. In addition to his teaching, writing, and consulting, he has participated in pro bono projects to bring a Systems Engineering approach to social service organizations."

"System design can be requirements based, function based, or model based. Model based system engineering and design has an advantage of executable models that improve efficiency and rigor. One of the earliest developments of this technique was Wymore’s (1993) book entitled Model-based System Engineering, although the phrase “model-based system design” was in the title and topics of Rosenblit’s (1985) PhD dissertation. Model-based systems engineering depends on having and using well-structured models that are appropriate for the given problem domain."

"Wayne’s book, A Mathematical Theory of Systems Engineering: The Elements, John Wiley, New York [1967] appeared in an era that spawned the concept of a mathematical system as a generalization of the control systems of engineers and the automata theories of computer scientists. Wymore employed the unusual term “Tricotyledon Theory of System Design” to portray the tri-partite nature of his theory. Tricotyledon, or three leaved seed, pictures a leaf for the class of models of the behavior of the system being designed, a second leaf for the technologies that are available to implement the system, and a leaf for the intersection of the two previous classes, namely, the technologies that can implement the models according to specified evaluation criteria. In subsequent work over three decades, he deepened the theory and applied it to numerous system engineering problems. The theory became the basis for his book, Model-based System Engineering, CRC Press, Inc. Boca Raton, FL, USA © 1993 and helped to spawn the trend toward use of systems modeling tools for systems design. However, Wayne himself did not emphasize the use of modeling and simulation as essential tools in systems engineering practice."

"[In the year 1957] I have just returned from an exciting meeting of the American Society for Engineering Education where I heard a paper on the new discipline of systems engineering. It is no longer sufficient for engineers merely to design boxes such as computers with the expectation that they would become components of larger, more complex systems. That is wasteful because frequently the box component is a bad fit in the system and has to be redesigned or worse, can lead to system failure. We must learn how to design large-scale, complex systems from the top down so that the specification for each component is derivable from the requirements for the overall system. We must also take a much larger view of systems. We must design the man-machine interfaces and even the system-society interfaces. Systems engineers must be trained for the design of large-scale, complex, man-machine-social systems."

"In the folklore of systems engineering, there are sayings, “If it isn't testable, it isn't a requirement,” and, “If it is not quantifiable, it is not testable.” I had always tended to subscribe to the particle of wisdom in these sayings. One of the many lessons that I learned from the CATIE experience, however, was that systems engineering had to be able to deal directly and purposively with what are usually considered to be strictly qualitative (unquantifiable) aspect of systems, such as quality of life, for example. The agricultural practices of these small farmers in Central America is such a large part of their lives that to change any part of their agricultural practices might be seen as diminishing their quality of life - even with respect to the color of the beans that the new practices might require. But even if we cannot deal directly with quality of life or user friendliness or enmity/sympathy, we can identify quantifiable figures of merit relating to these qualitative aspects and if we find enough figures of merit and combine them suitably, we might eventually obtain agreement that we have captured the essence of the qualitative aspect, at least as far as this particular set of stakeholders is concerned."

"During this period I was able to carry out only one project of real interest to me. Pure Oil was a fully integrated oil company in the sense that they engaged in exploration for oil, construction of oil wells, production of crude oil, transportation to refineries, distribution of refinery products to company owned storage facilities and to gas stations. We developed a linear programming model of the whole network hoping to discover the optimum allocation to and from connected nodes in order to meet required deliveries at minimum cost. Then we collected all the data needed by our model and ran the model. The computations took several days and the results were disappointing: The optimum allocations were not significantly different from their current practices. Is it possible that we had discovered a system of human beings in which everyone knew the payoff functions and constraints, and, over time, had evolved behavior patterns that enabled them to achieve near optimum performance? Or was it possible that our model was not detailed enough as it was based so closely on current practice that no new behavior could emerge? Or was our data incorrect? This was disappointing but great experience."

"General systems theory is considered as a formal theory (Mesarovic, Wymore), a methodology (Ashby, Klir), a way of thinking (Bertalanffy, Churchman), a way of looking at the world (Weinberg), a search for an optimal simplification (Ashby, Weinberg), didactic method (Boulding, Klir, Weinberg), metalanguage (Logren), and profession (Klir)."

"Over 80 million people have participated in Cub Scout Pinewood Derbies. Pinewood is a case study of the design of a Cub Scout Pinewood Derby for one particular scout pack. The system helps manage the entire race from initial entry through final results. Many alternatives for race format, scoring, and judging are presented."

"Wayne Wymore is now well established as an important leader in systems engineering and a founder of a highly original "school of thought" in the area of systems design. His contribution to this area, which will be the subject of a special issue of this journal in the near future, is best exposed in a trilogy consisting of this book and its two predecessors [Wymore, 1967, 1976]. Wymore's approach to systems design is characterized by mathematical rigor, comprehensiveness, and broad applicability."

"After earning the PhD degree and acquiring some relatively extensive experience in digital computers… It was time to leave the University. The result of an extensive search for the right job was a family move to Arlington Heights, Illinois, where it was a short commute to the Research Laboratories of the Pure Oil Company at Crystal Lake. I was given the title of Mathematical and Computer Consultant. The Labs were set in a beautiful campus, the professional personnel were eager to learn what I had to teach and to include me in many interesting projects where my knowledge and skills could be put to good use. I was encouraged to initiate my own program of research. I went to work with enthusiasm. The corporate headquarters of Pure Oil were located in down town Chicago. Pure Oil had been trying to install an IBM 705 computer system for all their accounting needs including calculation of all data necessary for the management of exploration, drilling, refining and distribution of oil products and even royalties to shareholders in oil wells. Typical for those early days, the programming team was in deep difficulties and needed help; they lacked adequate resources and suitable training. The Executive Vice President of Pure Oil, when he heard that there was a computer expert already on the payroll at the Crystal Lake lab, ended our family blissful dream and I was reassigned to the down town office."

"Every author has several motivations for writing, and authors of technical books always have, as one motivation, the personal need to understand; that is, they write because they want to learn, or to understand a phenomenon, or to think through a set of ideas."

"If all the theories pertinent to systems engineering could be discussed within a common framework by means of a standard set of nomenclature and definitions, many separate courses might not be required."

"The problem of the design of a system must be stated strictly in terms of its requirements, not in terms of a solution or a class of solutions."

"Only if mathematical rigor is adhered to, can systems problems be dealt with effectively, and so it is that the systems engineer must, at least, develop an appreciation for mathematical rigor if not also considerable mathematical competence."

"[The process of system design is]... consisting of the development of a sequence of mathematical models of systems, each one more detailed than the last."

"All that can be done pedagogically is to show the student how some phenomena have been modeled, let him model some phenomena under supervision, and then hope he will be successful on his own—or know enough to secure assistance."

"[The word system is often defined in a way] that seems the most appropriate for the purpose of any given discussion."

"In the minds of many writers systems engineering is synonomous with component selection and interface design; that is, the systems engineer does not design hardware but decides what types of existing hardware shall be coupled and how they shall be coupled. Complete agreement that this function is the essence of systems engineering will not be found here, for, besides the very important function of systems engineering in systems analysis, there is the role played by systems engineering in providing boundary conditions for hardware design."

"[Arrow's] experience as an Air Force weather forecaster during the Second World War "added the news that the natural world was also unpredictable." ... One incident that occurred while Arrow was forecasting the weather illustrates both uncertainty and the human unwillingness to accept it. Some officers had been assigned the task of forecasting the weather a month ahead, but Arrow and his statisticians found that their long-range forecasts were no better than numbers pulled out of a hat. The forecasters agreed and asked their superiors to be relieved of this duty. The reply was: "The Commanding General is well aware that the forecasts are no good. However, he needs them for planning purposes.""

"It is not easy to see him. Not because he’s short. Rather because the piles of books and papers scattered around his admittedly relatively narrow office in the Landau Building are so high. ‘They’ve moved us here and now I’m trying to look through all my stuff and see what I can possibly throw out’, says Kenneth J. Arrow, rising swiftly from behind one of the book piles, with an apologetic gesture. In spite of his 86 years, he seems physically quite fit and, no wonder, he prides himself in cycling to the office every day. The room is nice, bright, modern and new, but there is no way Arrow could possibly fit everything into the book shelves along the walls. ‘Never mind’, he says, laughing. He has a great warm open smile, and is eager to talk – which he then does at an amazing speed. Politely, he liberates a chair for me, opens a drawer next to himself for me to place the microphone in since there is no more space on the desk, of course – and off we go."

"And I am glad that you talked to Ken Arrow. But Nobel laureates, who have wide responsibilities and much on their mind, are not necessarily on top of what has been going on in research outside their usual field. I happen to know of one laureate who, circa 1991, was quite unaware that anyone had thought about increasing returns in either growth or trade."

"But the old dilemma between efficiency and fairness was about to be shattered by a young New Yorker called Kenneth Arrow, who knew all about unfairness after watching helplessly as a teenager while his father lost his successful business and all his savings in the Great Depression. The desire for social justice stayed with Arrow, but intellectually he couldn’t just ignore the question of efficiency. The young economist set his logical mind to wrestling with the tension between the unerring efficiency of the free market and the imperative that some kind of fairness should prevail. His solution was brilliant, twisting the traditional thinking about competitive markets and efficiency on its head. He proved that not only are all perfect markets efficient, all efficient outcomes can be achieved using a competitive market, by adjusting the starting position."

"I’m an optimist, so I am hopeful. I don’t think Samuelsonian economics can be the long-run equilibrium of such an important field such as economics. I think that because of the two secret sins of Samuelsonian economics, the qualitative theorems and significance testing in the absence of the loss function, we can’t make scientific progress. We can do a lot of other stuff. We can take a look at tables and see how large schooling is in the national economy; that kind of thing is science. So I’m sure we can make some progress, off the center of the scientific stage of so-called mainstream Samuelsonian economics. An interesting concrete example of this was a presentation Ken Arrow made when he came to Iowa, where he was trying to make the obvious point that we shouldn’t spend all of our time in undergraduate economics preparing people for graduate school. This is a point that I have made over and over again, that we should be preparing students for life, for business and law school. Arrow said, “Look, the main argument for this is not very complicated: as we all know, half of 1 percent of our students, if that, go on to graduate school in economics, end of argument.” The argument is not special to Arrow, but the fact that it came from Arrow was very interesting. He didn’t come and say there is an existence theorem proof that there doesn’t exist some social welfare function such that blah blah blah. He didn’t do that. He said, “Look, here’s the number that shows obviously that the policy of making undergraduate programs into junior graduate programs is a mistake.”"

"Arrow’s Impossibility Theorem is quite surprising. It shows that three very plausible and desirable features of a social decision mechanism are inconsistent with democracy: there is no “perfect” way to make social decisions. There is no perfect way to “aggregate” individual preferences to make one social preference. If we want to find a way to aggregate individual preferences to form social preferences, we will have to give up one of the properties of a social decision mechanism described in Arrow’s theorem."

"Alchian: Perhaps it might have been more appropriate for the Nobel Prize to have gone to you and Hicks together, and [Kenneth] Arrow and [[Gunnar Myrdal|[Gunnar] Myrdal]] together. Hayek: Oh, surely. [laughter]"

"Then during the early '50s Kenneth Arrow published his book Social Choice and Individual Values (1951). That stirred up a great deal of interest, both in political science and economics. My general reaction was that the people who criticized Arrow, and Arrow himself, really didn't quite get the message in the sense that the concentration was on the fact that majority rule would not give you a political equilibrium, that you get this political cycle and so forth. My criticism basically was, if that's the way the preferences are, that's what you want to have. A democracy should not mean one majority simply ruling. It ought to be a rotation, if that's the way the preferences are. I was kind of an anti-majoritarian then and now. So my critique of Arrow, which not many people paid much attention to, got me further into thinking about these things."

"In 1972 American economist Kenneth Arrow, jointly with Sir John Hicks, was awarded the Nobel Prize in economics for “pioneering contributions to general equilibrium theory and welfare theory.” Arrow is probably best known for his Ph.D. dissertation (on which his book Social Choice and Individual Values is based), in which he proved his famous “impossibility theorem.” He showed that under certain assumptions about people’s preferences between options, it is always impossible to find a voting rule under which one option emerges as the most preferred."

"In technical economics, I see a peak with Paul Samuelson and Kenneth Arrow and some of the core developments in game theory. Since then there are fewer iconic figures being generated in this area of research, even though there are plenty of accomplished papers being published."

"Once individuals are acting to carry out a norm and cannot be fully checked, their natural norm of self-interest will come into play. The rules actually acted upon will therefore move away from any norm that might be acceptable on the basis of general principles."

"We don’t have a laissez-faire system. The intervention of the federal government, as measured by expenditures, is growing. It is not a private system at all. Roughly 50 percent of health costs are paid for by the government, and state governments are spending more and more on health. It’s crowding out education. State budget-support for education, especially higher education, is crowded out by two things: health and prisons. Nobody is prepared for the idea of a laissez-faire system, and we never really had one."

"During this period I also came-across Kenneth Arrow's now famous Social Choice and Individual Values. Although I hardly did full justice to that pioneering work, it emboldened me to take a stab at a much more formal presentation than I had encountered in political theory. The form of my two chapters certainly owes something to my having ploughed my way through that book. For better or worse, Arrow's book must also have influenced my decision to present parts of the argument in a formal notational system (though only in footnotes and appendices)."

"There’s a famous Churchill quote: “democracy is the worst form of government, except for all the others.” That applies to regulation as well. In the late 1800s, we had a natural monopoly: railroads. The government created a regulatory enterprise – the Interstate Commerce Commission – and of course it was captured. But it still made a difference. I think we just have to accept that capture does occur, but it’s limited. The Federal Trade Commission, for example, is a pretty active body. Monopolies have been broken up. The AT&T telephone monopoly was broken up in 1982 – Stigler was still writing about regulatory capture then. AT&T was a classic monopoly, but a pretty benevolent one. It delivered good service – rates were too high, but not by that much. It didn’t necessarily pass on value to the consumers, but Bell Labs were a source of great innovative function. Nevertheless, it was broken up by antitrust measures, brought on by government. So there is regulatory capture, but it is by no means complete. Regulations do play a role. One example of non-regulatory capture fighting against effective regulation was during the run-up to the 2008 crash, when several officials argued for CDOs to be regulated, which means they would have had to meet certain requirements of transparency. This was not accepted. I’m not saying the crash could have been avoided if that happened, but that would have made a big difference."

"We don't have much time left. We are moving towards temperature increases of around two degrees Celsius, which is going to have consequences in the tropics, and we will lose things like glaciers. That's not a theory; it's happening right now. It's not a prediction; it's happening right now. But you just sightsee near those glaciers. But the glaciers are a big source of water. And on the questions of water, in California we store our water in a snowpack. When that's gone, the rain will be the same but it won't accumulate. With warming temperatures the snowpack will not work. It might be possible to substitute with dams, but that's complicated. This is conjoined with a big energy problem and I think that we really have to encourage development in this area. Just waiting for technological improvement won't work. We need to encourage it."

"I then follow up with four major aspects of economic research in the last 60 years, the period of my scholarly activity. One, econometric methodology and practice, is of such fundamental importance that it cannotgo unnoticed, although I played no role in it. With the other three, general equilibrium, dynamic processes, and uncertainty and information, I was more intimately involved."

"There are information asymmetries in this story. Health insurance is limping along. It's limited in scope, and then you other consequences. Insurance companies have high premiums to protect themselves. The ones who come to the insurance company are sicker and the people have to pay more. You have adverse selection. You have moral hazard. And the doctor does what's on the safe side -- defensive medicine -- without regard to cost. These are fundamental conditions that make health insurance difficult. You have some things that help. Some doctors understand that they shouldn't abuse the system. But you still see problems in the way doctors behave towards patients. They goof off. Sometimes it's too much work. Some things are difficult and risky to diagnose."

"Clearly, information is of the utmost importance in making economic decisions, especially when it comes to securities or other assets whose value depends on events not yet known. Now the distribution of information cannot simply be taken as given. On the contrary, information can be acquired at some cost."

"The Austrian a priori dogmatism (von Mises, especially; Hayek, to a lesser degree)."

"As we see it, a choice is not something you do today in isolation; it is part of a plan that includes some choices to be made in the future."

"I was influenced very early on, my first time at the—I spent a year at the Center at Palo Alto, the Center for Advanced Study there, and I became a good friend of Kenneth Arrow. I think he was not at the Center that year but he lived in Palo Alto. [...] Marvelous man. Both as a person and as a scholar. And I became as I say greatly influenced by the way in which he dealt with phenomena. [...] So I was influenced by that as a model, a way of thinking more abstractly, perhaps, than customary, about democratic theory. Making clear the premises, the epistomological assumptions and matters of that kind, and I think that sort of set the stage. And then once you get in of course, into that field, which was not highly— I don’t know how to put this properly—as a formal field of political science was not highly developed at the time, once you get into it you quickly become aware of how rich the potential subject matter is. One of the enormous changes, perhaps anticipating your question, one of the changes in the world is the extraordinary increase in the number of countries that, by the standards that we use today, can be called democratic—always, I repeat this and repeat this, but, always keeping in mind the difference between the ideal and the threshold at which we now accept a country as democratic, or a polyarchy as I would say. And the enormous increase in the number of those available for study—when I was a graduate student, there were maybe half a dozen countries that you could study: France and Britain and, I’m not quite sure of Canada at that time . . . and then the expansion created out there a field . . . that was both a challenge and an opportunity."

"I don't know which way human psychology will go wrong. We have a world in which there is uncertainty. There are new technologies. And one of the problems is that we do things today with a thought to the future. Any purchase is one for the future. If you buy a refrigerator you are making a commitment to the future, so that you have food to eat for the next ten years. That's a simple theory, one of many simple theories -- theories that people agree on and that don't, in a fundamental way, change. But every once in a while there is unemployment, and wages will drop, and there will be several different interpretations."

"There are many unknowns in the creation and use of knowledge as a factor of production. Still, two main lessons stand out: * Every country or firm must have education and training in technology and science, even if the research is not on par with that being conducted elsewhere. Knowledge cannot be absorbed unless some knowledge is already possessed. * Countries and firms must be open to new ideas, have multiple sources of new ideas, and see that ideas are diffused. This point strongly argues for freedom of entry, even when it seems to forgo economies of scale."

"Various medical groups are continually putting out advice as to health-inducing behavior (diet, exercise, and so forth). Individuals who follow this advice will reduce their demand for medical services. An extreme version is the campaign of the American Dental Association for fluoridation of water, a measure designed to reduce cavities and so the demand for dental services. There is also a logical problem. How does the relatively small group of physicians acquire power? In a democratic system, political power derives from numbers or from wealth. While physicians are certainly relatively well off, they do not have the concentration of wealth of many industries. Their power, it seems to me, is derived from moral authority. It is derived from the same professional respect that we are trying to explain. Indeed, when the American Medical Association came into conflict with either the retired or with the business interests that bought medical plans, the weakness of its power base quickly became apparent."

"Krugman's whole attack is directed at a statement made neither by Arthur nor by Cassidy. Krugman has not read Cassidy's piece with any care nor has he bothered to review what Arthur has in fact said."