Harvard University Alumni

"While spectacularly successful at predicting the behavior of atoms and subatomic particles, the quantum laws looked askance at Einstein's formulation of gravity. This set the stage for more than a half-century of despair as physicists valiantly struggled, but repeatedly failed, to meld general relativity and quantum mechanics, the laws of the large and small, into a single all-encompassing description. Such was the case until December 1984, when John Schwarz, of the California Institute of Technology, and Michael Green, then at Queen Mary College, published a once-in-a-generation paper showing that string theory could overcome the mathematical antagonism between general relativity and quantum mechanics, clearing a path that seemed destined to reach the unified theory."

"As I once told a newspaper reporter, in order to be sure to be quoted: discovery of supersymmetry would be more profound than life on Mars."

"String theory is an ambitious approach to the construction of a mathematical description of the physics that governs the properties of elementary particls and their interactions as well as the structure of space and time. It incorporates (and maybe even explains) well-established principles such as quantum mechanics and relativity. In fact, many string theorists (myself included) believe that string theory constitutes the third big physics revolution of the century, following relativity and quantum mechanics. It certainly requires conceptual advances every bit as bizarre and unexpected as was the case in the prior two revolutions."

"In the early 1960s there existed a successful quantum theory of the electromagnetic force (QED), which was completed in the late 1940s, but the theories of the weak and strong nuclear forces were not yet known. In UC Berkeley, where I was a graduate student during the period 1962 – 66, the emphasis was on developing a theory of the strong nuclear force. I felt that UC Berkeley was the center of the Universe for high energy theory at the time. Geoffrey Chew (my thesis advisor) and Stanley Mandelstam were highly influential leaders. Also, Steve Weinberg and Shelly Glashow were impressive younger faculty members. David Gross was a contemporaneous Chew student with whom I shared an office."

"The second superstring revolution (1994-??) has brought non-perturbative string physics within reach. The key discoveries were the recognition of amazing and surprising "dualities." They have taught us that what we viewed previously as five theories is in fact five different perturbative expansions of a single underlying theory about five different points! It is now clear that there is a unique theory, though it may allow many different vacua. ... Three different kinds of dualities, called S, T, and U have been identified.""

"Among the problems of the known string theories, as a theory of hadrons, was the fact that the spectrum of open strings contains massless spin 1 particles, and the spectrum of closed strings contains a massless spin 2 particle (as well as other massless particles), but there are no massless hadrons. In 1974, Joël Scherk and I decided to take string theory seriously as it stood, rather than forcing it to conform to our preconceptions. ... Specifically, Scherk and Schwarz (1974) proposed trying to interpret string theory as a unified quantum theory of all forces including gravity. Neveu and Scherk (1972) had shown that string theory incorporates the correct gauge invariances to ensure agreement at low energies (compared to the scale given by the string tension) with Yang-Mills theory. Yoneya (1973,1974) and Scherk and Schwarz (1974) showed that it also contains gauge invariances that ensure agreement at low energies with general relativity."

"Supergravity theories generically contain non-compact global symmetry groups. The general rule is that the scalar fields of the theory in question parametrize a symmetric space. Thus, if the non-compact symmetry group is G, and its maximal compact subgroup is H, the scalar fields map the space-time into the symmetric space G/H, and the number of scalar fields is dim G – dim H. The first supergravity example of this type to be found, N = 4 supergravity is one of the most interesting. In this case there are two scalar fields and the symmetric space is SL(2,R)/SO(2)."

"The generally accepted computation is that 2½ acres of land are required to provide a minimum adequate diet for each person, by Western standards anyhow. On a vegetarian diet it has been estimated that 1½ acres per head may provide enough. The reason for this difference is that animals grazed for meat-eating purposes require about 15 times more land than is necessary to raise an equivalent amount of nutrition in the form of grains, vegetables and fruit for human consumption. This means that India, on a vegetarian diet, is living far more wisely within its own land resources than are the meat-eating peoples."

"The predecessors of Newton and Leibnitz knew perfectly well how to determine tangents and areas, but they had to approach each problem from first principles. The great contribution of Newton and Leibnitz was precisely to make the procedures for finding tangents, areas, etc. into a calculus, that is, a systematic way of calculating—a collection of algorithms, to use the currently fashionable word."

"Suppose you want to teach the "cat" concept to a very young child. Do you explain that a cat is a relatively small, primarily carnivorous mammal with retractible claws, a distinctive sonic output, etc.? I'll bet not. You probably show the kid a lot of different cats, saying "kitty" each time, until it gets the idea. To put it more generally, generalizations are best made by abstraction from experience. They should come one at a time; too many at once overload the circuits."

"The fact that N Bourbaki was not a real person and represented a group of mathematicians was known to very few, for example, Ralph Boas, but the world of mathematics had come to believe in his existence. In an article for the Encyclopaedia Britannica, Boas revealed the truth; he was severely reprimanded in a letter to him "From my ashram in the Himalayas", beginning with "You miserable worm, how dare you say that I do not exist?" and signed 'Nicolas Bourbaki'!"

"Why do I paint? In my work as a mathematician, form, structure, and economy of expression are important. In music, add tone quality, balance and harmony. And now in painting, these aesthetic values are enhanced by color, contrast, and composition."

"Algebraic geometry has developed in waves, each with its own language and point of view. The late nineteenth century saw the function-theoretic approach of Riemann, the more geometric approach of Brill and Noether, and the purely algebraic approach of Kronecker, Dedekind, and Weber. The Italian school followed with Castelnuovo, Enriques, and Severi, culminating in the classification of algebraic surfaces. Then came the twentieth-century "American" school of Chow, Weil, and Zariski, which gave firm algebraic foundations to the Italian intuition. Most recently, Serre and Grothendieck initiated the French school, which has rewritten the foundations of algebraic geometry in terms of schemes and cohomology, and which has an impressive record of solving old problems with new techniques. Each of these schools has introduced new concepts and methods."

"You can ask the question about these ancient topics, such as s and ... and ask, are these good problems... I'd like to give a small amount of evidence... that they are... [S]tudying them helped us develop all of elementary number theory and from elementary number theory we developed the rest of number theory, and also you can argue that from elementary number theory came algebra.."

"Maybe so, but something is going on with the primes."

"Moreover, according to Alexander, the “troubled mathematical martyr,” exemplified not only by Galois but also by Abel, János Bolyai, Riemann, Cantor, Gödel, Turing, John Nash, Grothendieck, Perelman, and even, in a certain sense, Cauchy, remains to this day the dominant image of the “ideal mathematician,” long after the romantic paradigm was exhausted in the arts."

"Within mathematics itself, Voevodsky's proposal, if adopted, will create a new paradigm. In his “fairy tale” and some of his other papers, Langlands made deft use of categories and even 2-categories, but number theory is only superficially categorical, and so is the Langlands program. In the event that Univalent Foundations could shed light on a guiding problem in number theory — the Riemann hypothesis or the Birch Swinnerton-Dyer conjecture, which is not so far removed from Voevodsky's motives — then we could easily see Grothendieck's program absorbing the Langlands program within Voevodsky's new paradigm."

"A “good theorem,” as Tate puts it, lasts forever. Once proved, it will always stay proved, and other mathematicians are free to use it and build on it as they please, sometimes to great effect."

"Langlands and Grothendieck are both (at least) Giants by any measure, and both were consciously successors of Galois."

"Langlands' life has been by no means as extravagant as Grothendieck's, but his romanticism is evident to anyone who reads his prose; the audacity of his program, one of the most elaborate syntheses of conjectures and theorems ever undertaken, has few equivalents in any field of scholarship."

"It doesn't matter how long it takes, if the end result is a good theorem."

"Just as Weil's conjectures were about counting solutions to equations in a situation where the number of solutions is known to be finite, the BSD conjecture concerns the simplest class of polynomial equations—elliptic curves—for which there is no simple way to decide whether the number of solutions is finite or infinite."

"Though he died in 1946, Keynes still dominated economics at Cambridge in the mid-1960s. It wasn't just that all the leading figures among the faculty, Richard Kahn, Joan Robinson, Brian Reddaway, David Champernowne, Nicholas Kaldor, and James Meade, had been his pupils and/or collaborators. It was the tone of the place. The study of economics, theoretical or empirical, was driven by the desire to improve the conduct of economic policy. This did not mean that pure theory was neglected. Indeed, in those years Robinson was fighting a stirring battle in the realms of high theory with Paul Samuelson and Robert Solow from the Massachusetts Institute of Technology. Nor was Cambridge innocent of the cutting edge of econometric technique. Richard Stone, who, following Keynes's suggestions, had created the first modern national income accounts, was still director of the Department of Applied Economics where much of modern econometrics was pioneered. Nonetheless, we were taught that theory and technique should serve the higher cause of rational economic policies, and to use our newly learned econometric expertise to write essays on unemployment, or inflation, or the balance of trade, or some other topic at the top of the current political agenda."

"Skidelsky tells us Keynes was also practical, absorbed in questions of economic policy, argumentative, benevolent and intolerant, often rude, and had an intellectual arrogance that would allow positions previously held with great passion to be calmly abandoned. Well, that is exactly what the Cambridge faculty was like in the 1960s. Not only did the ghost of Keynes dominate the content of economics education at Cambridge, it also dominated the style. That style could be sustained with substance only by the extraordinarily gifted. So it is not surprising that the Cambridge faculty, although still very "Keynesian," looks much more conventional these days."

"Since I was a teenager I had an interest in number theory. Fortunately, I came across a good number theory book by L. E. Dickson, so I knew a little number theory. Also I had been reading Bell’s histories of people like Gauss. I liked number theory. It’s natural, in a way, because many wonderful problems and theorems in number theory can be explained to any interested high-school student. Number theory is easier to get into in that sense. But of course it depends on one’s intuition and taste also."

"Today we face a problem that involves two difficult to satisfy conditions. On the one hand we have to find a way for computer assisted verification of mathematical proofs. This is necessary, first of all, because we have to stop the dissolution of the concept of proof in mathematics. On the other hand, we have to preserve the intimate connection between mathematics and the world of human intuition. This connection is what moves mathematics forward and what we often experience as the beauty of mathematics."

"Voevodsky's construction makes it possible to obtain an “incarnation” of the motivic cohomology but it does not, however, find a solution to the standard conjectures, which are still today – along with the Hodge conjecture – the fundamental open question in modern algebraic geometry."

"Within mathematics itself, Voevodsky's proposal, if adopted, will create a new paradigm. In his “fairy tale” and some of his other papers, Langlands made deft use of categories and even 2-categories, but number theory is only superficially categorical, and so is the Langlands program. In the event that Univalent Foundations could shed light on a guiding problem in number theory — the Riemann hypothesis or the Birch Swinnerton-Dyer conjecture, which is not so far removed from Voevodsky's motives — then we could easily see Grothendieck's program absorbing the Langlands program within Voevodsky's new paradigm."

"It soon became clear that the only real long-term solution to the problems that I encountered is to start using computers in the verification of mathematical reasoning."

"Infinity-groupoids encode all the paths in a space, including paths of paths, and paths of paths of paths. They crop up in other frontiers of mathematical research as ways of encoding similar higher-order relationships, but they are unwieldy objects from the point of view of set theory. Because of this, they were thought to be useless for Voevodsky’s goal of formalizing mathematics. Yet Voevodsky was able to create an interpretation of type theory in the language of infinity-groupoids, an advance that allows mathematicians to reason efficiently about infinity-groupoids without ever having to think of them in terms of sets. This advance ultimately led to the development of univalent foundations."

"A technical argument by a trusted author, which is hard to check and looks similar to arguments known to be correct, is hardly ever checked in detail"

"Mathematical research currently relies on a complex system of mutual trust based on reputations. By the time Simpson's paper appeared, both Kapranov and I had strong reputations. Simpson's paper created doubts in our result, which led to it being unused by other researchers, but no one came forward and challenged us on it."

"This newest, largest and best collection of illustrative documents and programs, tables, and charts, extracts from federal and state commission and departmental reports, selections from trade and commercial newspaper and journals, and experts from the masters, old and new, is clearly the most important book of the year for students and teachers of economics."

"The materials collected in this volume are intended to acquaint the student with economic principles as they are manifested in the tangible facts of economic life. A few extracts of primarily theoretical character have been included to represent important aspects of contemporary or historic thought; but for the most part the selections are not so much authoritative formulations of economic laws as concrete case-material embodying such laws, or affording a background of information which the systematic treatises on economics can hardly give and which the teacher certainly cannot often assume that his students will possess."

"Briefly stated, the Follett philosophy is that any enduring society, any continuously productive industrial organization, must be grounded upon a recognition of the motivating desires of the individual and of the group. Consistently, Miss Follett sought to force home a realization of the fact that the democratic way of life, implemented by intelligent organization and administration of government and of industry, is to work toward an honest integration of all points of view, to the end that every individuality may be mobilized and made to count both as a person and as an effective part of his group and of society as a whole."

"Follett was always preoccupied with the dynamic view of organization, with the thing in process, so to speak. Authority, Power, Leadership, the Giving of Orders, Conflict, Conciliation — all her keywords are active words. There is a static or structural approach to the problem of organization which has its value; but those who are most convinced of the importance of such structural analysis would be the first to admit that it is only a step on the journey, an instrument of thought; it is not and cannot be complete in itself; it is only the anatomy of the subject. As in medicine, the study of anatomy may be an essential discipline, but it is in the physiology and psychology of the individual patient that that discipline finds its working justification."

"Fifteen years ago Follett expounded a philosophy of management that even to-day is a generation ahead of practice, and one can find therein a significant parallel with the pioneering work of Frederick W.Taylor. In his teaching, the adoption of a new philosophy of management—“the mental revolution” as he put it—was a fundamental part of his new technique; but, as knowledge of his methods spread, the practice of scientific management was allowed to develop and expand shorn from its underlying philosophy. Mary Follett was not concerned merely with the technique, the methods of management; her contacts with business and its leaders gave her ample chance of seeing the new technique emerge and grow. But with deeper insight she could see what too many business leaders missed, that the philosophy of management and its psychological foundations were still unheeded, perhaps even unknown. In this lies the greatness of her contribution. It is as modern and applicable to-day as it was when first she spoke; it will be as modern and applicable tomorrow."

"Miss Follett was not a “business woman,” in the sense of having actually managed any sort of business. But her lucid and illuminating ideas about organization were of priceless interest and value to the many industrial leaders and students of human relations problems, organization and politics with whom she came in contact. Her conceptions were in advance of her time. They are still in advance of current thinking. But they are a gold-mine of suggestion for anyone who is interested in the problems of establishing and maintaining human co-operation in the conduct of an enterprise. They have the added advantage of being presented with remarkable simplicity and clarity."

"Mary Parker Follett was a prominent business philosopher of the period, who agreed with Sheldon about the need to emphasize human factors in management, but placing greater stress on the need to develop a science of cooperation. According to Follett, what she called her 'Law of the Situation' could be a means for bridging the gap between an ideal of scientific management and the unilateral position that it seemed to involve in practice. In effect she was proposing the same collaboration between leaders and subordinates that was usually to be found between leaders of the same rank."

"This theme of mismeasurement interacts with the designation of the one hundred years between 1870 and 1970 as the “special century.” Measurement errors are greatest in the early years, both in the scope of the standard of living and in the extent of price index bias. Clearly the welfare benefits to consumers in the categories of life entirely omitted from GDP were greatest long ago: the transition from the scrub board to the automatic washing machine was a more important contributor to consumer welfare than the shift from manual to electronic washing machine controls or from a twelve-pound tub to an eighteen pound tub. The most important unmeasured benefit of all, the extension of life expectancy, occurred much more rapidly from 1890 to 1950 than afterward."

"During the relatively brief period in the late 1960s when economists were pondering the possible obsolescence of business cycles, the scholarly discipline of macroeconomics showed signs of becoming fragmented into speciality areas devoted to components of the then popular large-scale econometric models-for example, consumption, investment, money demand, and the Phillips curve. But more recently the revival of severe real world business cycles, together with the revolutions associated with Milton Friedman's monetarism and Lucas's classical equilibrium models, has brought about a revival of interest in economic analysis that focuses on a few broad aggregates summarizing activity in the economy as a whole-nominal and real income, the inflation rate, and the unemployment rate."

"The original Lucas version of the new-classical macroeconomics combined the undeniable appeal of rational expectations with two more dubious assumptions inherited from Friedman (1968), that is, continuous market clearing and imperfect information, to form the foundation of the famous “Lucas supply function” (more justly, the Friedman-Lucas supply function). Soon Sargent and Wallace (1975) extracted from Lucas’s model its implication for monetary policy, the famous “policy-ineffectiveness proposition.” The demonstration by Barro (1977) that one could interpret historical U.S. data to be consistent with the proposition and the theory brought new-classical economics to its shortlived period of peak influence."

"Part of the downfall came early and on theoretical grounds, with the realization that real-world information lags for aggregate variables like the price level and money supply were much too short to rationalize the persistent multiyear deviations from equilibrium that seemed to characterize business cycles in most industrialized countries. The second dubious assumption, continuous market clearing, was viewed more critically once it was recognized that it was not an inextricable concomitant of rational expectations, especially when Stanley Fischer (1977) and Edmund Phelps and John Taylor (1977) showed that rational expectations could be embedded in a model containing real-world institutional features like multiperiod wage and price contracts to generate nonmarket-clearing behavior. Once Fischer and Phelps-Taylor had shown that rational expectations by itself was a necessary but not a sufficient condition to validate new-classical policy conclusions, the race was on to develop the new-Keynesian theory based on rational expectations and one or another institutional impediment to continuous market clearing."

"But business cycles as a subject for study have enjoyed a revival for at least a decade now, stimulated in part by the severity of the 1974-75 and 1981-82 recessions and in part by the intellectual ferment surrounding the development of the "equilibrium business cycle model" and the attention paid to the seminal work of Robert E. Lucas, Jr., contained in his book Studies in Business Cycle Theory (1981). Indeed, there is no longer any need to lament the passing of economics courses explicitly carrying the title "Business Cycles," since the topic of business cycle behavior and analysis has so infiltrated courses carrying the title "Macroeconomics" that the two subjects have become almost interchangeable. 2 In this light it is fitting that the major research program of the NBER in this area is called "Economic Fluctuations" rather than "Macroeconomics.""

"Since the late 1960s macroeconomic debates in the United States have centered on the competing interpretations of the new classical and new Keynesian macroeconomics. The initial new classical model developed in the early 1970s by Robert E. Lucas, Jr., combined market-clearing, imperfect information, and rational expectations. After much testing, it was eventually rejected in the late 1970s for failing to explain why business cycles lasted on average four years while information delays lasted only a few weeks. It was soon replaced by a second new classical approach, the Real Business Cycle (RBC) model, which was also based on continuous market clearing and competitive equilibrium, but now generated the business cycle through serially correlated procyclical technology shocks."

"There are four headwinds that are just hitting the American economy in the face. They're demographics, education, debt and inequality. They're powerful enough to cut growth in half. So we need a lot of innovation to offset this decline. And here's my theme: Because of the headwinds, if innovation continues to be as powerful as it has been in the last 150 years, growth is cut in half. If innovation is less powerful, invents less great, wonderful things, then growth is going to be even lower than half of history."

"New-classical economics has been undeniably influential, but not in the way that its three prominent creators originally imagined. Its most important contribution to macroeconomics, the assumption of rational expectations, was stolen almost immediately, and applied more fruitfully, by the new Keynesians."

"The postwar era has not surprised Arthur Burns, for business cycles have continued their "unceasing round." although the United States recession of 1981-82 was the eighth since World War II and the deepest postwar slump by almost any measure, the 1983-84 recovery displayed an upward momentum sufficient to befuddle forecasters and delight incumbent politicians. Nor would a reincarnated Joseph Schumpeter be disappointed in the current status of business cycle research in the economics profession. To be sure, interest in business cycles decayed during the prosperity of the 1960s, as symbolized in the 1969 conference volume, Is the Business Cycle Obsolete? and in Paul Samuelson's remark the same year that the National Bureau of Economic Research "has worked itself out of one of its first jobs, namely, the business cycle.""

"The century of revolution in the United States after the Civil War was economic, not political, freeing households from an unremitting daily grind of painful manual labor, household drudgery, darkness, isolation, and early death. Only one hundred years later, daily life had changed beyond recognition. Manual outdoor jobs were replaced by work in air-conditioned environments, housework was increasingly performed by electric appliances, darkness was replaced by light, and isolation was replaced not just by travel, but also by color television images bringing the world into the living room. Most important, a newborn infant could expect to live not to age forty-five, but to age seventy-two. The economic revolution of 1870 to 1970 was unique in human history, unrepeatable because so many of its achievements could happen only once."

"Much time was wasted and ink spilled in the late 1960s and early 1970s trying to interpret the lagged effect of prices on wages as reflecting adaptive lags in the formation of expectations. But if we have learned anything from the new Keynesian economics of Fischer, Taylor, Blanchard, and their younger followers, it is that price and wage inertia is compatible with rational expectations."