Mathematicians From The United States

"Fermat had recourse to the principle of the economy of nature. Heron and Olympiodorus had pointed out in antiquity that, in reflection, light followed the shortest possible path, thus accounting for the equality of angles. During the medieval period Alhazen and Grosseteste had suggested that in refraction some such principle was also operating, but they could not discover the law. Fermat, however, not only knew (through Descartes) the law of refraction, but he also invented a procedure—equivalent to the differential calculus—for maximizing and minimizing a function of a single variable. … Fermat applied his method … and discovered, to his delight, that the result led to precisely the law which Descartes had enunciated. But although the law is the same, it will be noted that the hypothesis contradicts that of Descartes. Fermat assumed that the speed of light in water to be less than that in air; Descartes' explanation implied the opposite."

"Carl B. Boyer … is more or less the Gibbon of math history."

"Robert [Grosseteste] became much interested in science and scientific method … He was conscious of the dual approach by means of induction and deduction (resolution and composition); i.e., from the empirical knowledge one proceeds to probable general principles, and from these as premises one them derives conclusions which constitute verifications or falsifications of the principles. This approach to science was not that far removed from Aristotle ..."

"In ancient classical literature the rainbow sometimes was deified as Iris; at other times it was regarded merely as the route traversed by the messenger of Hera. The conception of the rainbow as a pathway or bridge has been widespread. For some it has been the best of all bridges, built out of three colors; for others the phrase "building on the rainbow" has meant a bootless enterprise. North American Indians were among those who thought of the rainbow as the Pathway of Souls, an interpretation found in many other places. Among the Japanese the rainbow is identified as the "Floating Bridge of Heaven"; and Hawaiian and Polynesian myths allude to the bow as the path to the upper world. In the Austrian Alps the souls of the righteous are said to ascend the bow to heaven; and in New Zealand the dead chieftains are believed to pass along it to reach their new home. In parts of France the rainbow is called the pont du St. Esprit, and in many places it is the bridge of St. Bernard or of St. Martin or of St. Peter. Basque pilgrims knew it as the 'puente de Roma'. Sometimes it is called instead the Croy de St. Denis (or of St. Leonard or of St. Bernard or of St. Martin). In Italy the name arcu de Santa Marina is relatively familiar. Associations of the rainbow and the milky way are frequent. The Arabic name for the milky way is equivalent to Gate of Heaven, and in Russia the analogous role was played by the rainbow. Elsewhere also the bow has been called the Gate of Paradise; and by some the rainbow has been thought to be a ray of light which falls on the earth when Peter opens the heavenly gate. In parts of France the rainbow is known as the porte de St. Jacques, while the milky way is called chemin de St. Jacques. In Swabia and Bavaria saints pass by the rainbow from heaven to earth; while in Polynesia this is the route of the gods themselves. In Eddic literature the bow served as a link between the gods and man — the Bifrost bridge, guarded by Heimdel, over which the gods passed daily. At the time of the Gotterdamerung the sons of Muspell will cross the bridge and then demolish it. Sometimes also in the Eddas the rainbow is interpreted as a necklace worn by Freyja, the "necklace of the Brisings," alluded to in Beowulf; again it is the bow of Thor from which he shoots arrows at evil spirits. Among the Finns it has been an arc which hurls arrows of fire, in Mozambique it is the arm of a conquering god. In the Japanese Ko-Ji-Ki (or Records of Ancient Matters), compiled presumably in 712, the creation of the island of Onogoro is related to the rainbow. Deities, standing upon the "floating bridge of heaven," thrust down a jeweled spear into the brine and stirred with it. When the spear was withdrawn, the brine that dripped down from the end was piled up in the form of the island. In myth and legend the rainbow has been regarded variously as a harbinger of misfortune and as a sign of good luck. Some have held it to be a bad sign if the feet of the bow rest on water, whereas a rainbow arching from dry land to dry land is a good augury. Dreambooks held that when one dreams of seeing a rainbow, he will give or receive a gift according as the bow is seen in the west or the east. The Crown-prince Frederick August took it as a good omen when, upon his receiving the kingdom form Napoleon in 1806, a rainbow appeared; but others interpreted it as boding ill, a view confirmed by the war and destruction of Saxony which ensued. By many, a rainbow appearing at the birth of a child is taken to be a favorable sign; but in Slavonic accounts a glance from the fay who sits at the foot of the rainbow, combing herself, brings death."

"Ptolemy left in his Optics, the earliest surviving table of angles of refraction from air to water. … This table, quoted and requoted until modern times, has been admired … A closer glance at it, however, suggests that there was less experimentation involved in it than originally was thought, for the values of the angles of refraction form an arithmetic progression of second order … As in other portions of Greek Science, confidence in mathematics was here greater than that in the evidence of the senses, although the value corresponding to 60° agrees remarkably well with experience."

"Robert Grosseteste … was born at the decisive moment when Greek and Arabic science became accessible in Latin versions."

"The Introductio does not boast an impressive number of editions, yet its influence was pervasive. In originality and in the richness of its scope it ranks among the greatest of textbooks; but it is outstanding also for clarity of exposition. Published two hundred and two years ago, it nevertheless possesses a remarkable modernity of terminology and notation, as well as of viewpoint. Imitation is indeed the sincerest form of flattery."

"But what, after all, are the integers? Everyone thinks that he or she knows, for example, what the number three is — until he or she tries to define or explain it."

"It should be mentioned also that the Hindus, unlike the Greeks, regarded irrational roots of numbers as numbers. This was of enormous help in algebra, and Indian mathematicians have been much praised for taking this step; but one must remember that the Hindu contribution in this case was the result of logical innocence rather than of mathematical insight. We have seen the lack of nice distinction on the part of Hindu mathematicians between exact and inexact results, and it was only natural that they should not have taken seriously the difference between commensurable and incommensurable magnitudes. For them there was no impediment to the acceptance of irrational numbers, and later generations followed their lead uncritically until in the nineteenth century mathematicians established the real number system on a sound basis."

"Much of T. W. Anderson's work in statistical theory has had an important influence on econometrics. But it is his work on structural estimation that has had the greatest impact on subsequent developments in the subject."

"A Hebrew belief asserted that if Yahweh lays aside his bow and hangs it in the clouds, this is a sign that his anger has subsided. Other peoples have had similar ideas, based upon the tradition that an archer carries his bow with the ends pointing downward when he wishes to indicate his peaceful intentions."

"In mathematical statistics, as well as econometrics, Ted Anderson is a scholar of immense stature. The scope and diversity of his research in statistical theory is almost a phenomenon in itself. Sometimes it seems that wherever one turns the subject, Ted's mark and influence is already firmly established. His books on multivariate analysis and time series rapidly became accepted as major treatises and are now integral parts of the bookshelves of every statistician."

"A time series is a sequence of observations, usually ordered in time, although in some cases the ordering may be according to another dimension. The feature of time series analysis which distinguishes it from other statistical analysis is the explicit recognition of the importance of the order in which the observations are made. While in many problems the observations are statistically independent, in time series successive observations may be dependent, and the dependence may depend on the positions in the sequence. The nature of a series and the structure of its generating process also may involve in other ways the sequence in which the observations are taken."

"[At Princeton Samuel Wilks...] was the major interest for me... I did have a contact with Oscar Morgenstern... Von Neumann was developing theory of games at the time... but I must say, I really didn't understand it very well when he lectured it."

"Early in 1943 I got full time war research work... My first project was to evaluate long term weather forecast. Longe range, in that time, meant three to four days."

"... speaking of Doug, he's the unsung hero of Unix. He was manager of the group that produced it and a huge creative force in the group, but he's almost unknown in the Unix community. He invented a couple of things you might have heard of: pipes and — get this — macros. Well, someone had to do it and that someone was Doug. As Ken once said when we were talking one day in the Unix room, "There's no one smarter than Doug.""

"McIlroy ... is the smartest of all of us and the least remembered or written down of all of us [programmers at Bell Labs' Computing Techniques Research Department]."

"Doug McIlroy's influence on the development of both C and C++ cannot be overestimated. I cannot remember a single critical design decision in C++ that I have not discussed at length with Doug."

"I don't know the counts of Unix and Linux servers. I do know that my heart sinks whenever I look under the hood in Linux. It is has been so overfed by loving hands. Over 240 system calls! Gigabytes of source! A C compiler with a 250-page user manual (not counting the language definition)! A simple page turner, 'less,' has over 40 options and 60 commands! It's proof that open-source can breed monsters just like the commercial pros. Miraculously, though, this monster works."

"Word and Excel and PowerPoint and other Microsoft programs have intimate — one might say promiscuous — knowledge of each others' internals. In Unix, one tries to design programs to operate not specifically with each other, but with programs as yet unthought of."

"Everything was small... and my heart sinks for Linux when I see the size of it. [...] The manual page, which really used to be a manual page, is now a small volume, with a thousand options... We used to sit around in the Unix Room saying, 'What can we throw out? Why is there this option?' It's often because there is some deficiency in the basic design — you didn't really hit the right design point. Instead of adding an option, think about what was forcing you to add that option."

"Ritchie and Thompson made an amazing team; and they played Unix and C like a fine instrument. They sometimes divided up work almost on a subroutine-by-subroutine basis with such rapport that it almost seemed like the work of a single person. In fact, as Dennis has recounted, they once got their signals crossed and both wrote the same subroutine. The two versions did not merely compute the same result, they did it with identical source code! Their output was prodigious. Once I counted how much production code they had written in the preceding year − 100,000 lines! Prodigious didn’t mean slapdash. Ken and Dennis have unerring design sense. They write code that works, code that can be read, code that can evolve."

"This is the Unix philosophy: Write programs that do one thing and do it well. Write programs to work together. Write programs to handle text streams, because that is a universal interface."

"The notion of "intricate and beautiful complexities" is almost an oxymoron. Unix programmers vie with each other for "simple and beautiful" honors — a point that's implicit in these rules, but is well worth making overt."

"Heinz performs the magic trick of convincing us that the familiar objects of our existence can be seen to be nothing more than tokens for the behaviors of the organism that apparently create stable forms. These stabilities persist, for that organism, as an observing system. This is not to deny an underlying reality that is the source of objects, but rather to emphasize the role of process, and the role of the organism in the production of a living map, a map that is so sensitive that map and territory are conjoined."

"Cybernetics is the study of systems and processes that interact with themselves and produce themselves from themselves."

"When James Tobin, a Nobel Prize winner in economics, was on the US Council of Economic Advisors some years ago, I asked him if he used complicated macroeconometric models in his work for the Council. He responded that he didn’t because he and others could not understand the workings and output of such models and thus did not have much confidence in them. When I asked what he actually did use to analyse problems, he remarked that he used a simple multiplier–accelerator model on the back of an envelope and even though he didn’t have too much confidence in it, he said at least he understood what he was doing."

"I want to tell you a story about Tobin. There was a conference at the Minnesota Fed in honor of Walter Heller, and all the old people, the heroes of my youth in economics, were there. It was a really nice occasion. Gardner Ackley was giving a talk; it was very negative about the direction of modern macro. saying that there was too much mathematics and that too many assumptions were unrealistic. He also was very critical of rational expectations and of all sorts of modern theorizing. It occurred after two days of really negative things being said about rational expectations. Nobody on our side said anything, in deference to Heller, I suppose. Then Tobin stood up and said that listening to Ackley's talk reminded him of going to the AEA meetings, after the war in 1946, at which the most-prominent old members of the profession were criticizing the young Keynesians for being unrealistic and too mathematical. Tobin said "We've had our chance, we haven't solved all the problems. There are a lot of problems left, there are unsatisfactory things about our models, and it's time to give these guys a chance." It was very gracious."

"Tobin was a person who really impressed me, because he had a passion for social justice and for public policy (...) for Tobin it was always something that was about making the lives of people better."

"I suspect that many of the world's financial lords are somewhat embarrassed to tell Japan repeatedly at G-7 meetings and elsewhere to adopt a Keynesian solution. Within Europe, central banks and governments think Keynesian theories and policies are absolutely wrong. Despite the remarkable success of pragmatic policies in the United States, true believers in the Invisible Hand reject Keynesian diagnoses and prescriptions. Many observers of Japan have found it intellectually comforting to blame the slump on the plight of the banks, flooded with bad loans dated from the land and equity bubbles and their collapse. They hope that a governmentmanaged and -subsidized rectification of bank balance sheets will trigger overall economic recovery. I think this is a false hope. The bank problem is only a small part of the macroeconomic disaster. It has to be resolved, of course, but resolution that is no substitute for the needed fiscal and monetary stimuli."

"Back in 1982, a brief but brusque exchange … took place between James Tobin … and Robert Nozick … Tobin exclaimed at Nozick: ‘There is nothing more dangerous than a philosopher who’s learned a little bit of economics.’ To which Nozick immediately responded: ‘Unless it’s an economist who hasn’t learned any philosophy.’"

"The other economic camp made for quite a different story. James Tobin (an east-coast Ivy League policy advisor) had already won the Nobel Prize when I spoke with him. A true gentleman, he spoke softly about his life and his Keynesian approach to economics. With due respect, I worried after a time that the interview sounded so automatic, so “done” before, that it would add little to the book. Then I brought up Lucas’s criticism. Tobin began to speak much louder and faster (on transcribing the tape I actually had to adjust the volume). He remained reasonable and gentlemanly but his voice betrayed his indignation toward Lucas and his camp, about how they were misleading sensible Keynesian economic thought."

"The important Keynesian insight is that a high propensity to save will not generate high national saving unless it goes into investment, into accumulation of real capital. The "paradox of thrift" makes this point in an extreme way. In certain circumstances, when there is no demand for investment around, the economy can be no better off, or even worse off, if a thrifty public cuts consumption."

"Keynes did not challenge the efficacy of price adjustment mechanisms in clearing particular markets in the Marshallian partial equilibrium theory on which he had been reared. He did challenge the mindless application of those mechanisms to economy-wide markets. Founding what came to be known as macroeconomics, he was modeling a whole economy as a closed system. He knew he could not use the Marshallian assumption that the clearing of one market could be safely described on the assumption that the rest of the economy was unaffected."

"I probably always say the same things; I hope people don't remember. One of the same things I say is that Japanese macroeconomic policy is perversely and inexcusably incompetent, and I surely would say that again. It's true-as Paul Krugman, a fellow participant in this program, has been saying and as I have said here in previous years-that Japan has reinvented the Keynesian liquidity trap. It can now reappear in classrooms where it had been long ignored or at best barely mentioned as a curiosum of the Great Depression."

"When I tried to sort out the pernicious disagreements between new classical and new Keynesian economists, I conducted a series of conversations with the protagonists (Klamer 1983). The personal differences were revealing. The viva cious Robert Solow (with a taste for the quick quip), the serious Robert Lucas (never less than self-composed), the chatty Franco Modigliani (not shy of self promotion), and the unassuming James Tobin (wanting an interview at least as long as Lucas’s) quickly taught me how trenchant the rhetorical differences were."

"The central message is still that, as Keynes argued, fiscal policy is the answer to liquidity traps, financial or political. The arguments against fiscal policy in Japan, so far as I understand them are intellectually fallacious; they would receive failing grades in an undergraduate macro exam."

"I had, to be sure, been drawn into economics when the General Theory was an exciting revelation for students hungry for explanation and remedy of the Great Depression. At the same time, I was uncomfortable with several aspects of Keynes’ theory, and I sought to improve what would now be called the microfoundations of his macroeconomic relations."

"I shall argue that Keynesian macroeconomics neither asserts nor requires nominal wage and/or price rigidity. It does assert and require that markets not be instantaneously and continuously cleared by prices."

"Tobin asked me to come and talk at Yale in 1977, and I was there for a couple of days. He was really nice in a personal way. He made it clear that he respected my work as a professional, so I didn't feel my professional standing was at risk. When I got there he gave me a nice introduction. It wasn't a question of whether I was a competent economist or not; there was nothing personal involved. These guys, however, had lots of disagreements and criticisms. I'd been thinking along these lines for many years, so I thought I did a pretty good job of taking care of the questions, but I don't know how it looked from the other side. It left me with a feeling of being way ahead of the game. It was an exciting feeling."

"Ball, Mankiw, Romer and others style themselves as New Keynesians. Their program is to develop improved microeconomic foundations for imperfectly flexible prices. In the process, they hope to illuminate the paradox that individually rational or near-rational behavior can result in significant collective market failures. These are certainly laudable objectives. In the end, I suspect, the program will not change the essential substance of Keynesian macroeconomics. But it will make Keynes more palatable to theorists."

"A long decade ago economic growth was the reigning fashion of political economy. It was simultaneously the hottest subject of economic theory and research, a slogan eagerly claimed by politicians of all stripes, and a serious objective of the policies of governments. The climate of opinion has changed dramatically. Disillusioned critics indict both economic science and economic policy for blind obeisance to aggregate material "progress," and for neglect of its costly side effects. Growth, it is charged, distorts national priorities, worsens the distribution of income, and irreparably damages the environment. Paul Erlich speaks for a multitude when he says, "We must acquire a life style which has as its goal maximum freedom and happiness for the individual, not a maximum Gross National Product.""

"Economics has always flourished and acquired energy from controversies generated by practical policy questions of the day. That was true in the times of Smith, and Ricardo, and Keynes, and it is true today. These periods of division and revolution and counterrevolution are generally followed by periods of synthesis and consolidation from which the science emerges stronger. i am optimistic that this will happen again, and that the best of the insights of the new clasicals will be absorbed in a mainstream, in which the essential insights of Keynesian economists also survive."

"Monetarism—both of the older Friedman version stressing adherence to money stock targets and of the newer rational expectations variety—has been badly discredited. The stage has been set for recovery in the popularity of Keynesian diagnoses and remedies. I do not mean to imply, of course, that there is some Keynesian truth, vintage 1936 or 1961, to which economists and policymakers will or should now return, ignoring the lessons of economic events and of developments in economics itself over these last turbulent fifteen years. I do mean that in the new intellectual synthesis which I hope and expect will emerge to replace the divisive controversies and chaotic debates on macroeconomic policies, Keynesian ideas will have a prominent place."

"There is no reason to think that the impact [of monetary policy] will be captured in any single [variable]... , whether it is a monetary stock or a market interest rate."

"[C]onsider an economy with only one private sector and only two assets: money issued by the government to finance its budget deficits, and homogeneous physical capital. Let p be the price of currently produced... consumer goods and capital goods. ...[A]llow the value of existing capital goods ...to diverge from their current reproduction cost—let qp be the market price of existing capital goods."

"The rate of investment – the speed at which investors wish to increase the capital stock – should be related, if to anything, to q, the value of capital relative to its replacement cost."

"Keynesian economics at a minimum provides a license for welfare state measures and other government efforts towards redistribution of wealth. The license is the faith that macroeconomic stabilization and prosperity are compatible with a wide range of social policies, that modern capitalism and democracy are robust enough to prosper and progress while being humane and equitable. That faith conflicts with the visions of extreme Right and Left, which agree that extremes of wealth and poverty, of security and insecurity, are indispensable to the functioning of capitalism. Keynesian policies helped to confound those dismal prophecies in the past; I think they will do so again."

"According to [the general equilibrium approach to monetary theory], the principal way in which financial policies and events affect aggregate demand is by changing the valuations of physical assets relative to their replacement costs."

"The key behavioral assumption... spending... and portfolio decisions are independent... [A]ccumulation of wealth... [is] separable from... its allocation. As savers, people... add to their wealth; as portfolio managers, they... distribute among... assets and debts the net worth... The propensity to consume may depend upon interest rates, but... not... directly on the... mix of asset supplies or... rates... these... are growing."