mathematicians-from-russia

"Revolutions in mathematics are quiet affairs. No clashing armies and no guns. Brief news stories far from the front page. Unprepossessing. Just like the raw damp Monday afternoon of April 7, 2003, in Cambridge, Massachusetts. Young and old crowded the lecture theater at the Massachusetts Institute of Technology (MIT). They sat on the floor and in the aisles, and stood at the back. The speaker, Russian mathematician Grigory Perelman, wore a rumpled dark suit and sneakers, and paced while he was introduced."

"By the end of 2006 it was generally believed that Perelman’s proof was correct. That year, the journal Science named Perelman’s proof the “Breakthrough of the Year.” Like Smale and Freedman before him, the forty-year old Perelman was tapped to be a Fields Medals recipient for his contributions to the Poincaré conjecture (in fact, Thurston also received a Fields Medal for his work that indirectly led to the final proof). The countdown for the $1 million prize had begun (some wonder if Perelman and Hamilton will be offered the prize jointly)."

"If the proof is correct then no other recognition is needed."

"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."

"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."

"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."

"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."

"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."

"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."

"Mathematics allows you to see the invisible."

"... physics, one could say, is in sort of a crisis, in some sense, because of a current gap between the sophisticated theories, which come from applying sophisticated mathematics, and the actual universe. ... I think perhaps new ideas are needed, and I wouldn't be surprised if Witten is one those people who come up with those ideas."

"... looking back, ... I have not hurt people personally, but ... I could be mean, for instance, I could be harsh. And now I see it as sign of weakness, as a sign of insecurity."

"People tend to think that mathematicians always work in sterile conditions, sitting around and staring at the screen of a computer, or at a ceiling, in a pristine office. But in fact, some of the best ideas come when you least expect them, possibly through annoying industrial noise."

"I was proud of my work with Yakov Isaevich, and he of me. Despite our good relationship, however, I kept my “other” mathematical life – my work with Fuchs and Feigin and all of that – secret from him as I did from most other people. It was as though applied mathematics was my spouse, and pure mathematics was my secret lover."

"We should all have access to the mathematical knowledge and tools needed to protect us from arbitrary decisions made by the powerful few in an increasingly math-driven world. Where there is no mathematics, there is no freedom."

"In mathematics, it's not a game where the fastest wins. But rather, it's more like who can see farther, who can see deeper. That's the one who achieves more."

"The stakeholder concept was originally defined as "those groups without whose support the organization would cease to exist." The list of stakeholders originally included shareowners, employees, customers, suppliers, lenders and society. Stemming from the work of Igor Ansoff and Robert Stewart (in the planning department at Lockheed) and, later Marion Doscher and Stewart at SRI, the original approach served an important information function in the SRI corporate planning."

"The publication of the book, Corporate Strategy, by H. Igor Ansoff was a major event in the 1965 world of management. As early as it came in this literature, the book represented a kind of crescendo in the development of strategic planning theory, offering a degree of elaboration seldom attempted since."

"Ansoff (1965) gives only some indication of what he means by environment. This can be explained by Ansoff's classification of decisions into strategic. administrative and operating decisions. He (1965, p. 8) defines these types of decision as follows:"

"A natural companion to the competitive advantage is the synergy component of strategy. This requires that opportunities within the scope possess characteristics which will enhance synergy."

"The triplet of specifications - the product-market scope, the growth vector and the competitive advantage - describes the firm's product-market path in the external environment."

"By searching out opportunities which match its strengths the firm can optimize the synergistic effects."

"We shall approach practical objectives through a series of approximations. Keeping the maximization of the rate of return as the central theoretical objective, we shall develop a number of subsidiary objectives (which the economists call proxy variables) which contribute in different ways to improvement in the return and which are also measurable in business practice. A firm which meets high performance in most of its subsidiary objectives will substantially enhance its long-term rate of return. (The defect in our approach is that we cannot prove that the result will be a ‘‘maximum’’ possible overall return.) As will be seen, this road has its own obstacles: the difficulties of long term maximization are replaced by the problem of reconciling claims of conflicting objectives."

"This theory maintains that the objectives of the firm should be derived by balancing the conflicting claims of the various 'stakeholders' in the firm: managers, workers, stockholders, suppliers, vendors. The firm has a responsibility to all of these and must configure its objectives so as to give each a measure of satisfaction. Profit which is a return on investment to the stockholder is one of such satisfactions, but does not receive special predominance in the objective structure,"

"At the beginning of this century a self-destructive democratic principle was advanced in mathematics (especially by Hilbert), according to which all axiom systems have equal right to be analyzed, and the value of a mathematical achievement is determined, not by its significance and usefulness as in other sciences, but by its difficulty alone, as in mountaineering. This principle quickly led mathematicians to break from physics and to separate from all other sciences. In the eyes of all normal people, they were transformed into a sinister priestly caste . . . Bizarre questions like Fermat's problem or problems on sums of prime numbers were elevated to supposedly central problems of mathematics."

"Let me just say that Arnold was a geometer in the widest possible sense of the word, and that he was very fast to make connections between different fields."

"The axiomization and algebraization of mathematics, after more than 50 years, has led to the illegibility so such a large number of mathematical texts that the threat of complete loss of contact with physics and the natural sciences has been realized."

"Mathematics is the part of physics where experiments are cheap."

"Such axioms, together with other unmotivated definitions, serve mathematicians mainly by making it difficult for the uninitiated to master their subject, thereby elevating its authority."

"When you are collecting mushrooms, you only see the mushroom itself. But if you are a mycologist, you know that the real mushroom is in the earth. There’s an enormous thing down there, and you just see the fruit, the body that you eat. In mathematics, the upper part of the mushroom corresponds to theorems that you see. But you don’t see the things which are below, namely problems, conjectures, mistakes, ideas, and so on. You might have several apparently unrelated mushrooms and are unable to see what their connection is unless you know what is behind."

""In almost all textbooks, even the best, this principle is presented so that it is impossible to understand." (K. Jacobi, Lectures on Dynamics, 1842-1843). I have not chosen to break with tradition."

"All mathematics is divided into three parts: cryptography (paid for by CIA, KGB and the like), hydrodynamics (supported by manufacturers of atomic submarines) and celestial mechanics (financed by military and by other institutions dealing with missiles, such as NASA.)."

"In the last 30 years, the prestige of mathematics has declined in all countries. I think that mathematicians are partially to be blamed as well—foremost, Hilbert and Bourbaki—the ones who proclaimed that the goal of their science was investigation of all corollaries of arbitrary systems of axioms."

"A person, who had not mastered the art of the proofs in high school, is as a rule unable to distinguish correct reasoning from that which is misleading. Such people can be easily manipulated by the irresponsible politicians."

"It is almost impossible for me to read contemporary mathematicians who, instead of saying “Petya washed his hands,” write simply: “There is a t_1<0 such that the image of t_1 under the natural mapping t_1 \mapsto {\rm Petya}(t_1) belongs to the set of dirty hands, and a t_2, t_1, such that the image of t_2 under the above-mentioned mapping belongs to the complement of the set defined in the preceding sentence.”"

"In the middle of the twentieth century it was attempted to divide physics and mathematics. The consequences turned out to be catastrophic. Whole generations of mathematicians grew up without knowing half of their science and, of course, in total ignorance of any other sciences. They first began teaching their ugly scholastic pseudo-mathematics to their students, then to schoolchildren (forgetting Hardy's warning that ugly mathematics has no permanent place under the Sun)."

"I discovered that a whole range of problems of the most diverse character relating to the scientific organization of production (questions of the optimum distribution of the work of machines and mechanisms, the minimization of scrap, the best utilization of raw materials and local materials, fuel, transportation, and so on) lead to the formulation of a single group of mathematical problems (extremal problems). These problems are not directly comparable to problems considered in mathematical analysis. It is more correct to say that they are formally similar, and even turn out to be formally very simple, but the process of solving them with which one is faced [i.e., by mathematical analysis] is practically completely unusable, since it requires the solution of tens of thousands or even millions of systems of equations for completion. I have succeeded in finding a comparatively simple general method of solving this group of problems which is applicable to all the problems I have mentioned, and is sufficiently simple and effective for their solution to be made completely achievable under practical conditions."

"It is difficult to distinguish another scholar in the history of the twentieth century who contributed as much as him to the fusion of mathematics and economics, the sciences with the antipodal standards of scientific thought. pointed out that he can list only John von Neumann and alongside Leonid Kantorovich among those few of his contemporaries who synthesized the mathematical and humanitarian cultures."

"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 economics 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."

"The accounting methods based on mathematical models, the use of computers for computations and information data processing make up only one part of the control mechanism, another part is the control structure."

"The method of successive approximations is often applied to proving existence of solutions to various classes of functional equations; moreover, the proof of convergence of these approximations leans on the fact that the equation under study may be majorised by another equation of a simple kind. Similar proofs may be encountered in the theory of infinitely many simultaneous linear equations and in the theory of integral and differential equations. Consideration ofjkbni semiordered spaces and operations between them enables us to easily develop a complete theory of such functional equations in abstract form."

"A solution of newly appearing economic problems, and in particular those connected with the scientific-technical revolution often cannot be based on existing methods but needs new ideas and approaches. Such one is the problem of the protection of nature. The problem of economic valuation of technical innovations efficiency and rates of their spreading cannot be solved only by the long-term estimation of direct outcomes and results without accounting peculiarities of new industrial technology, its total contribution to technical progress."

"In planning the idea of decentralization must be connected with routines of linking plans of rather autonomous parts of the whole system. Here one can use a conditional separation of the system by means of fixing values of flows and parameters transmitted from one part to another. One can use an idea of sequential recomputation of the parameters, which was successfully developed by many authors for the scheme of Dantzig-Wolfe and for aggregative linear models."

"In our time mathematics has penetrated into economics so solidly, widely and variously, and the chosen theme is connected with such a variety of facts and problems that it brings us to cite the words of which are very popular in our country: "One can not embrace the unembraceable". The appropriateness of this wise sentence is not diminished by the fact that the great thinker is only a pen-name."

"The university immediately published my pamphlet, and it was sent to fifty People’s Commissariats. It was distributed only in the Soviet Union, since in the days just before the start of the World War it came out in an edition of one thousand copies in all."

"Once some engineers from the veneer trust laboratory came to me for consultation with a quite skilful presentation of their problems. Different productivity is obtained for veneer-cutting machines for different types of materials; linked to this the output of production of this group of machines depended, it would seem, on the chance factor of which group of raw materials to which machine was assigned. How could this fact be used rationally?"

"The twentieth century return to Middle Age scholastics taught us a lot about formalisms. Probably it is time to look outside again. Meaning is what really matters."

"Lobachevski had to teach Geometry; and began by an intensive critical study of Euclid, acting as devil's advocate. This approach to the anomalous parallel axiom resulted in his non-Euclidean Geometry."

"There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world."