Mathematicians From The United States

"THE great American who has departed from us was much more than an economist. But the vast realm over which he held sway and the intellectual climate of the epoch that nourished his thought have been admirably surveyed in Econometrica, and I shall confine myself to Fisher’s purely scientific work in our field. This will restrict our subject. But it will not lower it — at least, it could do so only through my own fault. For whatever else Fisher may have been—social philosopher, economic engineer, passionate crusader in many causes that he believed to be essential to the welfare of humanity, teacher, inventor, businessman — I venture to predict that his name will stand in history principally as the name of this country’s greatest scientific economist."

"For the third law.., Kepler had... 6 data points..: every planet.., the length of the orbit and the distance to the sun.., and he did... regression. ...He could fit a curve to these 6 data points and he got a square cube law. Amazing, but he was... lucky. That's not enough data to be... reliable."

"There was a later astronomer, Johann Bode who took the same data... and... had a prediction that the distances to the planets formed a shifted ... He also fit a curve, except there was one point missing, a... gap... His law predicted... a missing planet. ...[W]hen Uranus was discovered by Herschel, the distance fit... this pattern. Then Ceres was discovered... in the astroid belt, and it also fit the pattern. ...But then Neptune was discovered, and it was... way off. ...[I]t was a numerical fluke."

"Gauss... created one of the first mathematical data sets. He computed the first [~]100,000 prime numbers. ...He found a statistical pattern ...they get sparser and sparser, but the drop-off in the density was inversely proportional to the of the range of numbers. So he conjectured... the the number of primes up to x is... x divided by the natural log of x, and he had no way to prove this. It was a conjecture. It was revolutionary because it was maybe the first important conjecture of math that was statistical in nature. ...It just gave you an approximation that got better... as you went further... out. ...It started the field of... . ...[I]t was the first of many ...which ...started consolidating the idea that the prime numbers ...really didn't have a pattern, that they behaved like random sets of numbers with a certain density. They have some patterns. They're almost all odd. ...They're not actually random. They're... pseudorandom."

"[P]rofessional mathematics is not a sport (in sharp contrast to mathematics competitions). The objective in mathematics is not to obtain the highest ranking, the highest score, or the highest number of prizes and awards; instead, it is to increase understanding of mathematics (both for yourself, and for your colleagues and students), and to contribute to its development and applications."

"Understand the problem. What kind of problem is it? There are three main types of problems: ‘Show that ...’ or ‘Evaluate ...’ questions, in which a certain statement has to be proved true, or a certain expression has to be worked out; ‘Find a...’ or ‘Find all...’ questions, which requires one to find something (or everything) that satisfies certain requirements; ‘Is there a ...’ questions, which either require you to prove a statement or provide a counterexample (and thus is one of the previous two types of problem)."

"Over time it became more... productive to think of the primes as if they were generated by some... random set, and this allowed us to make... other predictions. ...[B]ecause of this statistical random model.., we are ...absolutely convinced it's true. ...[W]e have, over time, developed this very accurate conceptual model of what primes should behave like.., but it's mostly and non-rigorous, but extremely accurate. It's the same reason we believe the Riemann hypothesis is true, and why we believe that cryptography based on the primes is... mathematically secure."

"Relying on intelligence alone to pull things off at the last minute may work for a while, but generally speaking at the graduate level or higher it doesn't. One needs to do a serious amount of reading and writing, and not just thinking, in order to get anywhere serious in mathematics."

"[O]ne reason why we care about the Riemann hypothesis is that if... we knew it was false, ...it would mean there was a secret pattern to the primes that we were not aware of, and I think we would very rapidly abandon any cryptography based on the primes, because if there was one pattern... there are probably more, and these... can lead to exploits in crypto."

"what looks like a permanently high plateau"

"Molecular dynamics is not primarily about making movies of molecules. More often it is about developing quantitative predictions of molecular size and shape, flexibilities, interactions with other molecules, behavior under pressure, and the relative frequency of one state or conformation compared to another."

"A variety of studies in widely different fields show that supervisors who are getting the best production, the best motivation, and the highest levels of worker satisfaction are employee-centred appreciably more often than production-centred."

"Interaction and decision making rely heavily on group processes."

"To be effective in leading his own work group, a superior must be able to influence his own boss, that is he needs to be skilled both as a superior and as a subordinate."

"All component parts of any system of management must be consistent with each of the other parts and reflect the system's basic philosophy."

"The preceding analysis shows that a manager who has high performance goals and excellent job organization but who relies solely on economic needs and direct pressure to motivate his men is very likely to be to be disappointed by their achievements. The noneconomic motives must be used fully, along with the economic needs, to create high performance goals and establish the level of motivational forces which yield high productivity. Since the principle of supportive relationships and group methods of supervision enable a manager to make effective use of the noneconomic motives, some valuable insights can be obtained by examining how these managerial principles appear to affect the motivations, satisfactions, and behavior of the members of an enterprise. A substantial body of research findings demonstrates that the greater the loyalty of the members of a group toward the group, the greater is the motivation among the members to achieve the goals of the group, and the greater is the probability that the group will achieve its goals."

"The management system of an organization must have compatible component parts if it is to function effectively. This conclusion has a very important implication; experiments in organizations must involve internally consistent changes. The traditional atomistic research design is not appropriate for experiments involving organizational theory or management systems. Every aspect of a management system is related to every other part and interacts with it. The results obtained by altering a single variable or procedure while keeping all others the same usually will yield quite different results from those obtained when that variable is changed along with simultaneous and compatible changes in all other aspects of the management system. The true influence of altering one aspect of the system cannot be determined by varying it and it alone... In experiments involving organizational theory and management systems, therefore, a systems approach must be used. The organic integrity of each system must be maintained while experimental variations are being made."

"The superior in one group is a subordinate in the next group, and so on through the organization."

"[It remains to be seen whether the newer management theories, based as they are on research in industrial concerns, will be] equally applicable to other kinds of organized human activity, such as schools, voluntary organizations, unions, hospitals, governmental agencies, scientific and professional organizations, and the like."

"How best to organize the efforts of individuals to achieve desired objectives has long been one of the world's most important, difficult, and controversial problems."

"The leadership and other processes of the organization must be such as to ensure a maximum probability that in all interactions and all interactions and all relationships with the organization each member will, in the light of his background, values, and expectations, view the experience as supportive and one which builds and maintains his sense of personal worth and importance."

"As tasks become more varied and require greater training and skill, the relationship (between job attitudes and performance) appears to change progressively from the negative to positive."

"Research reveal that managers achieving better performance (i.e., greater productivity, higher earnings, lower costs, etc. ) differ in leadership principles and practices from those achieving poorer performance."

"A number of statistical assumptions are made in the application of his (Thurstone's) attitude scale e.g. that the scale values of the statements are independent of the attitude distribution of the readers who sort the statements assumptions which as Thurstone points out, cannot always be verified. The method is more over laborious. It seems legitimate to enquire whether it actually does its work better than the simple scales which may be employed and the same breath to ask also whether it is not possible to construct equally reliable scales without making unnecessary statistical assumptions."

"Conflict is viewed as the active striving for one's own preferred outcome which, if attained, precludes the attainment by others of their own preferred outcome, thereby producing hostility."

"As people acquire more education, their expectations rise as to the amount of responsibility, authority, and income they receive."

"[Each person] is a member of one or more functioning workgroups that have a high degree of group loyalty, effective skills of interaction and high performance goals."

"Hodge cohomology, algebraic de Rham cohomology, crystalline cohomology, the étale ℓ-adic cohomology theories for each prime number ℓ ... A strategy to encapsulate all the different cohomology theories in algebraic geometry was formulated initially by Alexandre Grothendieck, who is responsible for setting up much of this marvelous cohomological machinery in the first place. Grothendieck sought a single theory that is cohomological in nature that acts as a gateway between algebraic geometry and the assortment of special cohomological theories, such as the ones listed above—that acts as the motive behind all this cohomological apparatus. ..."

"Number theory swarms with bugs, waiting to bite the tempted flower-lovers who, once bitten, are inspired to excesses of effort!"

"Sometimes the mathematical anti-Platonist believes that headway is made by showing Platonism to be unsupportable by rational means, and that it is an incoherent position to take when formulated in a propositional vocabulary. It is easy enough to throw together propositional sentences. But it is a good deal more difficult to capture a Platonic disposition in a propositional formulation that is a full and honest expression of some flesh-and-blood mathematician’s view of things. There is, of course, no harm in trying—and maybe its a good exercise. But even if we cleverly came up with a proposition that is up to the task of expressing Platonism formally, the mere fact that the proposition cannot be demonstrated to be true won’t necessarily make it vanish. There are many things—some true, some false— unsupportable by rational means."

"Vladimir Voevodsky and his collaborators have provided us with a very interesting candidate-category of motives: a category (of sheaves relative to an extraordinarily fine Grothendieck-style topology on the category of schemes) which in some intuitive sense “softens algebraic geometry” so as to allow for a good notion of homotopy in an algebro-geometric setup and is sufficiently directly connected to concrete algebraic geometry to have already yielded some extraordinary applications. The quest for a full theory of motives is a potent driving force in complex analysis, algebraic geometry, automorphic representation theory, the study of L functions, and arithmetic. It will continue to be so throughout the current century."

"Mazur suggests that it’s possible to glimpse the essence of Grothendieck’s approach to mathematics by looking at two concepts—categories and functors. A category can be thought of almost as a grammar: take triangles, perhaps, and understand them in terms of their relationship to all other triangles. The category consists of objects, and relationships between objects. The objects are nouns and the relationships are verbs, and the category is all the ways in which they can interact. Grothendieck’s discoveries opened up mathematics in a way that was analogous to how Wittgenstein (and Saussure) changed our views of language."

"Catholics... believed that the grace of God was bestowed upon sinners only through the holy Church and its sacraments, enacted by an ordained priest. Protestants, conversely, believed in a "priesthood of all believers," meaning that God would bestow his grace directly upon them. Catholics believed that Christ was physically present in the bread and wine during the sacrament of the Mass. Protestants believed that Christ was either present everywhere (Luther) or that the Mass was a mere commemoration of his sufferings (Zwingli). Catholics believed that God would take into account a man's good works in this world in determining whether be saved or lost. Protestants believed that only faith and divine grace mattered. Catholics believed that the Bible required interpretation by the hierarchy and the traditions of the Church. Protestants believed that the Bible was a clear guide for righteous behavior, accessible to anyone. What these arguments had in common were that they were entirely inconclusive."

"Theological and philosophical disputes could rage forever, he Christopher Clavius] believed, because there was no universally accepted way to decide who was right and who was wrong. ...But mathematics was different: with mathematics, the truth forces itself upon its audience whether they like it or not. One could dispute the Catholic doctrine of the sacraments, but one could not deny the Pythagorean theorem; and no one could deny the correctness of the new calendar, based as it was on the detailed mathematical calculations."

"It was clear to Clavius that Euclid's method was successful in doing precisely what the Jesuits were struggling so hard to accomplish: imposing a true, eternal, and unchallengeable order upon a seemingly chaotic reality."

"The fact that the tragic story of Évariste Galois, the mathematical genius who burned brightly but all too briefly, is not as unusual as one might think among mathematicians of his and subsequent generations. It is, rather, the most famous and dramatic of an entire genre of mathematical stories that originated in the early decades of the nineteenth century but is still going strong today. Consider, for example... Niels Henrik Abel.. János Bolyai... Srinivasa Ramanujan... John Nash... Kurt Gödel... Alexander Grothendieck... Grigory Perelman... Among modern mathematicians, it seems, extreme eccentricity, mental illness, and even solitary death are not a matter of random misfortune. They are, rather... reserved only for the most outstanding members of the field."

"Where the Jesuits insisted on clear and simple postulates, the new mathematicians relied on a vague intuition of the inner structure of matter, whereas the Jesuits celebrated absolute certainty, the new mathematicians proposed a method rife with paradoxes, and seemed to revel in them; and whereas the Jesuits sought to avoid controversy at all cost, the new method was mired in intractable controversies... It was everything that the Jesuits thought mathematics must never be, and yet it flourished... It was known as the method of indivisibles."

"...the whole point of studying and teaching mathematics was that it demonstrated how universal truth imposed itself upon the world—rationally, hierarchically, and inescapably. Ideally, the Jesuits believed, the truths of religion would be imposed on the world just like geometrical theorems, leaving no room for avoidance or denial by Protestant or other heretics and leading to the inevitable triumph of the Church."

"We're rapidly creating an extraordinary silicon-based petri dish for evolution of intelligence. By the year 2025... we're likely to have computers whose raw processing power exceeds that of the human brain. Also, we're likely to have more computers than people..."

"Once we can manipulate our genome, Lamarckian fashion, the rate of change will be staggering."

"One of the factors that caused Spencer's ideas to lose popularity was social Darwinism... Social evolution is different from biological evolution: it's faster, it's Lamarckian, and it makes even heavier use of altruism and cooperation than biological evolution does. None of this was well understood at the time."

"If we choose, we can use genetic engineering to alter the character of our offspring. ...the motivation to do this will eventually become overwhelming."

"It seemed that fundamental physics was stuck. The particle physicists were smashing particles into each other with ever increasing force, trying to figure out how many quarks could dance on the head of a pin. The cosmologists were working with very few facts... on what seemed to me to be mainly religious grounds. And most of physics was still focused on pushing and pulling, on the material properties... rather than on its informational properties. ...those that relate to order and disorder."

"Speaking crudely, a living system—an organism—consists of a symbiotic relationship between a metabolism and a replicator. ...the replicator contains the blueprint of the organism, with the information needed to grow, make repairs, and reproduce. ...the metabolism provides the energy and raw materials needed to build and run the replicator."

"The basic principle of an autocatalytic network is that even though nothing can make itself, everything in the pot has at least one reaction that makes it, involving only other things in the pot. It's a symbiotic system in which everything cooperates to make the metabolism work—the whole is greater than the sum of the parts."

"On learning more astronomy, in the phenomenon of "averted vision," I found a justification for my rationale about the roundabout path to metaphysics via physics: to see a faint star, it's necessary to look away from it; as soon as one looks at it directly, it vanishes."

"Our goal is to build a broad-based model of key components of the economy: households, firms, banks and government... The failure to embrace things like simulation has inhibited progress in economics."

"Rational adventure in a physical setting is becoming increasingly rarer in the world. The evolution of technology and infrastructure has altered the kind of adventures that we can have, so that people with adventurous spirits either take on risk for its own sake, or they embark on rational adventures in the mental domain."

"The paradox that immediately bothers everyone who learns about the second law is this: If systems tend to become more disordered, why, then, do we see so much order around us? ...It seems to conflict with our "creation myth": In the beginning, there was a big bang. ...no one is saying that the second law of thermodynamics is wrong, just that there is a contrapuntal process organizing things at a higher level."

"In a world of superintelligent humans, the account of a quest to prove a theorem may take on a universality in its drama that seems inconceivable to us now. I believe that rational adventure is fundamental to the human spirit, and that it won't go away. But it will evolve of necessity to take place in the increasingly abstract domains that characterize the boundaries and frontiers of an evolving and ever more complex and abstract world. As a result, it will evolve into forms that are difficult for us to even think about at this point in time."