Mathematicians From The United States

"[F]or monetary analysis... assign... to each asset a rate of return r_i (i=1,2,... n) and... [imagine] each sector j (j=1,2,... m) to have a net demand for each asset, f_{ij}, which is a function of the vector r_i and possibly of other variables... [I]n practice, many of the cells are empty..."

"Each sector is... constrained by its own net worth... The individual economic unit can neither change the legacy of the past nor... affect by... portfolio choices... market valuations of his assets."

"[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 general accounting framework for... the capital account... Rows represent assets or debts. ...Columns represent sectors of the economy ...Entries in cells ...can be postive, negative, or zero. A negative entry means... the sector... is a debtor in the... asset indicated by the row. ...The sum across the row is the net exogenous supply of the asset to the economy ...For stocks of goods, this ...is the economy's inheritance from the past. For internally generated financial assets the net... supply... is zero. If from the sums in the final column the... government's holdings of an asset are subtracted (or its debt added), the net holdings of the private economy result. The sum of a column represents the net worth of a sector. The sum of the final column is the national wealth. ...[P]rivate wealth differs from this total by the amount of the government's net worth. If government is a net debtor... if its stocks of goods are ignored, then private wealth exceeds national wealth."

"For me, growing up in the 1930s, the two motivations powerfully reinforced each other. The miserable failures of capitalist economies in the Great Depression were root causes of worldwide social and political disasters. The crisis triggered a fertile period of scientific ferment and revolution in economic theory."

"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."

"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."

"In economic surveys of households, many variables have the following characteristics: The variable has a lower, or upper, limit and takes on the limiting value for a substantial number of respondents. For the remaining respondents, the variable takes on a wide range of values above, or below, the limit."

"I... set forth and illustrate a general framework for monetary analysis. It is not... new... but... shared... in spirit with many monetary economists. My purpose... is exposition and recapitulation."

"[M]onetary assets fall into place as a part, but not the whole, of the menu of s; and the ing system is one sector, but not the only one, whose behavior must be specified."

"A forthcoming book by Harry Markowitz, Techniques of Portfolio Selection, will treat the general problem of finding dominant sets and computing the corresponding opportunity locus, for sets of securities all of which involve risk. Markowitz's main interest is prescription of rules of rational behaviour for investors; the main concern of this paper is the implications for economic theory, mainly , that can be derived from assuming that investors do in fact follow such rules."

"Treatment of the capital account separately from the production and income account of the economy is... a first step, a simplification... justified by convenience... The strategy is to regard income account variables as tentatively exogenous... and to find equilibrium in the markets for stocks of assets conditional upon assumed... outputs, incomes, and other flows."

"In a complete equilibrium... financial inputs to the real side must reproduce the assumed values of the real inputs to the financial side."

"A[n]... example of this strategy is the "LM curve." Macroeconomics... immortalized Hick's decomposition of the Keynesian system into sub-models. One... tells what asset stock equilibrium corresponds to... tentative... aggregate real income and the commodity price level. ..."[T]he" interest rate equates... demand and supply of money and clears... markets for other assets."

"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."

"Linear programming was developed as a discipline in the 1940's, motivated initially by the need to solve complex planning problems in wartime operations. Its development accelerated rapidly in the postwar period as many industries found valuable uses for linear programming. The founders of the subject are generally regarded as George B. Dantzig, who devised the simplex method in 1947, and John von Neumann, who established the theory of duality that same year. The Nobel prize in econonmics was awarded in 1975 to the mathematician Leonid Kantorovich (USSR) and the economist Tjalling Koopmans (USA) for their contributions to the theory of optimal allocation of resources, in which linear programming played a key role. Many industries use linear programming as a standard tool, e.g. to allocate a finite set of resources in an optimal way."

"Koopmans complained that macroeconomic models weren't satisfactory because they didn't handle randomness. He talked about building models in continuous time, which is something we're trying to do now."

"Koopmans was also like Kantorovich in generalizing his approach from one sector of the economy to the economy as a whole. Koopmans showed the conditions required for economy-wide efficiency in allocating resources. He also, again like Kantorovich, used his activity analysis techniques to derive efficient criteria for allocating between the present and the future."

"The solution of important problems may be delayed because the requisite tools are not perceived. Or the availability of certain tools may lead to an awareness of problems, important or not, that can be solved with their help."

"According to a frequently cited definition, economics is the study of “best use of scarce resources.” The definition is incomplete. “Second best” use of resources, and outright wasteful uses, have equal claim to attention. They are the other side of the coin."

"… a rather different class of applications of the idea of best allocation of scarce resources... usually referred to as the theory of optimal economic growth. In most studies of this kind made in the countries with market economies there is not an identifiable client to whom the findings are submitted as policy recommendations. Nor is there an obvious choice of objective function, such as cost minimization or profit maximization in the studies addressed to individual enterprises. The field has more of a speculative character. The models studied usually contain only a few highly aggregated variables. One considers alternative objective functions that incorporate or emphasize various strands of ethical, political, or social thought. These objectives are then tried out to see what future paths of the economy they imply under equally simplified assumptions of technology or resource availability."

"Decisions and plans made by others... [can be judged to be] quantitatively at least as important as the primary uncertainty arising from random acts of nature and unpredictable changes in consumers' preferences."

"We look upon economic theory as a sequence of conceptual models that seek to express in simplified form different aspects of an always more complicated reality."

"Without recognising indivisibilities — in human person, in residences, plants, equipment, and in transportation — location patterns, down to those of the smallest village, cannot be understood"

"The principal customers aimed for are other economists or members of other professions, who are somewhat closer to the making of policy recommendations... The question of the clientèle is even more baffling when the problem concerns growth paths for time spans covering several generations. What can at best be recommended in that case is the signal the present generation gives, the tradition it seeks to strengthen or establish, for succeeding generations to take off from."

"[This work urges] a clearer separation, in the construction of economic knowledge, between reasoning and recognition of facts, for the better protection of both."

"One can in particular interpret the proposition as a statement conditions under which the simplicity of incentive structure and the economies of information handling characteristic of a competitive market organization can be secured without loss of efficiency of allocation... The price system carries to each producer, resource holder, or consumer a summary of information about the production possibilities, resource availabilities and preferences of all other decision makers. Under the conditions postulated, this summary is all that is needed to keep all decision makers reconciled with a Pareto optimal state once it has been established."

"It is worth pointing out that in this particular study our authors have abandoned demand and supply functions as a tool for analysis, even as applied to individuals... [The problem] has been reformulated as one of proving that a number of maximizations of individual goals under interdependent restraints can be simultaneously carried out."

"The early thirties brought what liberal economists called the Great Depression and Marxist economists described as the great crisis of capitalism. It dawned on me that the economic world order was unreliable, unstable, and, most of all, iniquitous. I sought intellectual contacts and friendship with a group of socialist students and also with a small handful of communist-oriented students and unemployed workers."

"While it was long possible and sometimes tempting for physicists to deny the usefulness of the molecular hypothesis, we economists have the good luck of being some of the ‘molecules’ of economic life ourselves, and of having the possibility through human contacts to study the behavior of other ‘molecules’."

"Econometrics may be defined as the quantitative analysis of actual economic phenomena based on the concurrent development of theory and observation, related by appropriate methods of inference."

"If evolutionary selection is the basis for a belief in profit maximization, 'then we should postulate that basis itself and not the profit maximization which it implies in certain circumstances."

"Optimizing responses of economic agents are simultaneously feasible only if the proper prices are already known to them. But these prices must somehow themselves be the result of the same responses."

"One is led to conclude that economics as a scientific discipline is still somewhat hanging in the air."

"An object in OOA represents a single typical but unspecified instance of something in the real world - any airplane, I don't care which one, as long as it is typical. The object-oriented analyst distinguishes this concept from that of a specified instance: Airplane number N2713A, Air Force One, or The Spirit of St. Louis, for example."

"While a small domain (consisting of fifty or fewer objects) can generally be analyzed as a unit, large domains must be partitioned to make the analysis a manageable task. To make such a partitioning, we take advantage of the fact that objects on an information model tend to fall into clusters: groups of objects that are interconnected with one another by many relationships. By contrast, relatively few relationships connect objects in different clusters. When partitioning a domain, we divide the information model so that the clusters remain intact... Each section of the information model then becomes a separate subsystem. Note that when the information model is partitioned into subsystems, each object is assigned to exactly one subsystem."

"The key books about object-oriented graphical modeling languages appeared between 1988 and 1992. Leading figures included Grady Booch [Booch,OOAD]; Peter Coad [Coad, OOA], [Coad, OOD]; Ivar Jacobson (Objectory) [Jacobson, OOSE]; Jim Odell [Odell]; Jim Rumbaugh (OMT) [Rumbaugh, insights], [Rumbaugh, OMT]; Sally Shlaer and Steve Mellor [Shlaer and Mellor, data], [Shlaer and Mellor, states] ; and Rebecca Wirfs-Brock (Responsibility Driven Design) [Wirfs-Brock]."

"Professor Florian Cajori died August 14, 1934. In May of the following year I was invited by the University of California Press to edit this work. ...this is a revision of Motte's translation of the Principia. From many conversations with Professor Cajori, I know that he had long cherished the idea of revising Newton's immortal work by rendering certain parts into modern phraseology (e.g., to change the reading of "reciprocally in the subduplicate ratio of " to "inversely as the square root of") and to append historical and critical notes which would provide instruction to some readers and interest to all. This is his last work; one of the most fitting to crown a life devoted to investigation and to the history of the sciences in his chosen field."

"As a mathematician Dean Cajori has achieved a name which very few in this world can equal, a name which is respected all over the globe. His text books and his writings have been published all over the world. We are proud of all the achievements of our "Caj", of course, but we are especially proud of what he has done for us here, and it is for this reason that we shall always hold him in our memory. As a friend and as an instructor he has been more to us than we can ever measure, and we shall always look back upon the days when we had "Caj"."

"They [the students] looked upon "Caj" as one of their best friends in all the College, and thought of him as the one responsible for their later success. He has had a personal appeal to a great many students... It was the appeal of the one with a human interest in what somebody else is doing, the appeal of the true friend and the hearty well-wisher. It was the appeal of "Caj"."

"Fermat died with the belief that he had found a long-sought-for law of prime numbers in the formula 2^{2^n} + 1 = a prime, but he admitted that he was unable to prove it rigorously. The law is not true, as was pointed out by Euler in the example 2^{2^5} + 1 = 4,294,967,297 = 6,700,417 times 641. The American lightning calculator Zerah Colburn, when a boy, readily found the factors but was unable to explain the method by which he made his marvellous mental computation."

"In 1735 the solving of an astronomical problem, proposed by the Academy, for which several eminent mathematicians had demanded several months' time, was achieved in three days by Euler with aid of improved methods of his own... With still superior methods this same problem was solved by the illustrious Gauss in one hour."

"Most of his [Euler's] memoirs are contained in the transactions of the Academy of Sciences at St. Petersburg, and in those of the Academy at Berlin. From 1728 to 1783 a large portion of the Petropolitan transactions were filled by his writings. He had engaged to furnish the Petersburg Academy with memoirs in sufficient number to enrich its acts for twenty years a promise more than fulfilled, for down to 1818 [Euler died in 1793] the volumes usually contained one or more papers of his. It has been said that an edition of Euler's complete works would fill 16,000 quarto pages."

"The miraculous powers of modern calculation are due to three inventions : the Arabic Notation, Decimal Fractions and Logarithms."

"It is a remarkable fact in the history of geometry, that the Elements of Euclid, written two thousand years ago, are still regarded by many as the best introduction to the mathematical sciences."

"Comparatively few of the propositions and proofs in the Elements are his [Euclid's] own discoveries. In fact, the proof of the "Theorem of Pythagoras" is the only one directly ascribed to him."

"J. J. Sylvester was an enthusiastic supporter of reform [in the teaching of geometry]. The difference in attitude on this question between the two foremost British mathematicians, J. J. Sylvester, the algebraist, and Arthur Cayley, the algebraist and geometer, was grotesque. Sylvester wished to bury Euclid "deeper than e'er plummet sounded" out of the schoolboy's reach; Cayley, an ardent admirer of Euclid, desired the retention of Simson's Euclid. When reminded that this treatise was a mixture of Euclid and Simson, Cayley suggested striking out Simson's additions and keeping strictly to the original treatise."

"The Elements has been considered as offering models of scrupulously rigorous demonstrations. It is certainly true that in point of rigour it compares favourably with its modern rivals; but when examined in the light of strict mathematical logic, it has been pronounced by C.S. Peirce to be "riddled with fallacies." The results are correct only because the writer's experience keeps him on his guard."

"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."