Mathematicians from Canada

"Advanced technology is great at tasks such as keeping an airliner flying several thousand feet in the air, but is easily crippled by social complications."

"The invisible hand is an emergent property of this system, which never reaches an optimal equilibrium, but instead is fundamentally dynamic and unstable, with complex effects on society."

"Mathematics was not just about keeping track of were the moon was going, but also where the money was going."

"There is something intrinsically upside down and counter-intuitive in the relationship between money and happiness."

"A society in which each person is hell bent on maximizing his or her own utility, may therefore have declining overall utility."

"If the aim of economics is really to provide the greatest happiness to the most people, than it is rather shocking to discover that economic growth has no net effect on happiness. So what has gone wrong with the neoclassical prospect?"

"For even money itself has no value if there is no network of people to recognize it."

"Money, having freed itself from the physical universe, has become number itself, and finance a strange form of mathematical alchemy."

"The race to the moon was never really about the moon - its utility didn't rest in samples of moon rock. It was about capitalism versus communism, right versus left."

"Perfect order is boring, perfect randomness is boring, but complex systems are interesting."

"Adam Smith's invisible hand does exist, but as an emergent property of a complex system. It has a fuzzy tendency to reduce big price discrepancies, but it acts in a rather haphazard way."

"Not only is Homo economicus self-centered, he is a murdering psychopath. However, as with much of neoclassical economic thought, there is an element of circularity here."

"Economists often talk about the benefits of choice, but rational economic man doesn't actually have much freedom to chose, because he is a slave to his own preferences."

"The strength of capitalism does not lie in the neoclassical idea of stability, but in its ability to unleash this creative energy. like a chaotic mathematical system, it is capable of producing surprise."

"The market can quickly seize up. Prices don't adjust in a smooth continuous fashion, but instead abruptly reconfigure themselves, like the earth's crust during an earthquake."

"It has become increasingly evident, at least among those who aren't currency traders or true believers in the optimality of free markets, that it would be nice if this system could calm down a touch. One option is the Tobin tax, which would impose a tax of around 0.1 percent on currency trades, and act as a kind of damper on speculation."

"To build a genuinely sustainable economy, we need to recognize and embrace the dynamic nature of the world, and free ourselves from the dead holds of static dogma."

"Orthodox tools based on a normal distribution therefore fail exactly where they are most needed, at the extremes."

"The economy is crooked not straight; and mainstream economists are like flat-earthers who keep saying the world is flat despite all the evidence to the contrary."

"The economy is a nonlinear fractal system, where the smallest scales are linked to the largest, and the decisions of the central bank are affected by the gut instincts of the people on the street."

"It can be annoying to find out the name of a famous local landmark has no significance other than belonging to some distant relation or drinking buddy of the explorer."

"The neoclassical dogma of diminishing returns is completely wrong - success begets success and wealth begets wealth."

"The idea that money begets money, and that the rich and powerful enjoy unfair advantages, goes against what we are taught - or like to believe - about the capitalist system."

"The entire thrust of neoclassical thought is to argue that the free market economy is an efficient system that will optimize utility for all mankind, if only government will get out of the way. It therefore departs from classical economists such as Adam Smith, who recognized the importance of governments for regulating markets and preventing monopolies."

"The real reason for the longevity of the neoclassical model have less to do with science, and more to do with the social dynamics of the universities that propagate it, and its appeal to a particular mindset."

"To balance the economy, we need first to balance our priorities, and abandon rigid ideologies."

"Although there are at present many occupations that require a good deal of skill and training in advanced mathematics, mathematics itself is still often regarded as a curious profession demanding singular talents and a singular personality."

"Mathematical maturity is anyhow an uncertain concept, for the mind’s natural competence seems to change with age, its purview variable."

"What I have achieved has been largely a matter of chance. Many problems I thought about at length with no success. With other problems, there was the inspiration—indeed, some that astound me today. Certainly the best times were when I was alone with mathematics, free of ambition and of pretense, and indifferent to the world."

"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."

"He was a visionary. He pointed us into a direction where we can go and find the truth, find out what’s really going on. It’s about seeing the world in the right light."

"He’s like a modern-day Einstein. But everybody knows about Einstein and nobody knows about Langlands. Why is that?"

"He’s clearly one of the most important living mathematicians. His legend precedes him. But the question is, ‘Do mathematicians really know what he has done?’ It’s like having a famous writer but no one has read his books."

"He would become fluent in French, Russian, German and Turkish, and well-versed in their literature. Frenkel, who exchanges emails with Langlands in Russian, speculates that his versatility with languages may have had something to do with his ability to see connections in disparate fields of mathematics."

"Langlands spent every morning, seven days a week, for five years working on the paper he delivered in Oslo. It is written entirely in Russian and dedicated in large part to reformulating the geometric program championed by Frenkel. This new paper is an attempt to shift the field toward a more traditional approach: it proposes a new mathematical basis for the geometric theory that relates more closely to Langlands’s own conjectures by using similar tools to the ones he used in the ’60s—in the process, restoring his work back to its original arithmetic purity."

"Intriguingly, the mathematics of randomness, chaos, and order also furnishes what may be a vital escape from absolute certainty—an opportunity to exercise free will in a deterministic universe. Indeed, in the interplay of order and disorder that makes life interesting, we appear perpetually poised in a state of enticingly precarious perplexity. The universe is neither so crazy that we can’t understand it at all nor so predictable that there’s nothing left for us to discover."

"The theory of probability combines commonsense reasoning with calculation. It domesticates luck, making it subservient to reason."

"Ramsey theory implies that complete disorder is impossible. Somehow, no matter how complicated, chaotic, or random something appears, deep within that morass lurks a smaller entity that has a definite structure. Striking regularities are bound to arise even in a universe that has no rules."

"In mathematics, in science, and in life, we constantly face the delicate, tricky task of separating design from happenstance."

"Most coincidences are simply chance events that turn out to be far more probable than many people imagine."

"Tversky was fond of describing his work as “debugging human intuition.”...Tversky could establish again and again the existence of mismatches between intuition and probability—between cognitive illusion and reality."

"Indeed, mathematics is full of conjectures—questions waiting for answers—with no assurance that the answers even exist."

"The aim of science is to reduce the scope of chance."

"Randomness, chaos, uncertainty, and chance are all a part of our lives. They reside at the ill-defined boundaries between what we know, what we can know, and what is beyond our knowing. They make life interesting."

"More often than not, a piece of mathematics worked out years before—and believed to be totally without practical value—finds a role in the “real” world."

"To an increasing number of practitioners, computer simulations rooted in mathematics represent a third way of doing science, alongside theory and experiment."

"As the mathematician Clifford Taubes noted, “Physics is the study of the world, while mathematics is the study of all possible worlds.” Thus, mathematics unveils the infinite possibilities; physics pinpoints the few that structure our universe and our existence."

"The theory of harmonic forms in Riemannian manifolds may be regarded as a generalization of potential theory. It is therefore natural that the boundary value problems of this theory which generalize the classical Dirichlet and Neumann problems should play an important role in the theory."

"The propagation of elastic waves in a homogeneous solid is governed by a hyperbolic system of three linear second-order partial differential equations with constant coefficients. When the solid is also isotropic, the form of these equations is well known and provides the foundation of the conventional theory of elasticity (Love 1944). The explicit solution of the initial value, or Cauchy, problem for the isotropic case was found by Poisson, and in a different way by Stokes (1883). If the initial disturbance is sharp and concentrated, the resulting disturbance at a field point will consist of an initial sharp pressure wave, a continuous wave for a certain period, and a final sharp shear wave. The disturbance then ceases."

"The magnitude of Fundy tides may be seen as having been reached by a balance between a dissipative mechanism, with assumed quadratic frictional forces, and an energy imparting mechanism in the deep ocean where work done by the tide raising force is proportional to distance travelled and hence to the first power of amplitude. Further, it now appears that the second and third North Atlantic modes are those primarily stimulated by the Fundian resonance. To represent these processes within one model both the continental shelf shallows and oceanic areas must be included, as well as their zone of interaction across the continental shelf."

"The mathematical theory of the Navier-Stokes equations has centered upon basic questions of the existence, uniqueness, and regularity of solutions of the initial value problem for fluid motions in all of space or in a subdomain of finite or infinite extent. Such solutions, when they can be constructed or shown to exist, represent flows of a viscous incompressible fluid. In two space dimensions the theorem of existence, uniqueness and regularity was essentially completed thirty years ago by the work of Leray ..., Lions ... and Ladyzhenskaya ... who showed that a smooth solution of the initial value problem exists for arbitrary square-integrable initial data. For viscous, incompressible fluid motions in three space dimensions, ... the theorem of existence uniqueness and regularity has been proved only for sufficiently small initial data or in special cases such as cylindrical symmetry that essentially reduce the problem to two space dimensions in some sense."

"Our knowledge of Babylonian mathematics is derived mainly from tablets in the British Museum, the Prussian State Museum of Berlin, the Ottoman Museum of Constantinople, the University of Strasbourg, the University of Pennsylvania and the Palais du Cinquantenaire of Brussels."

"In yesteryears there were two gloriously inspiring centers of mathematical study in America. One of these was at the University of Chicago, when , and and were in their prime. The other center was at The Johns Hopkins University, 1876–83, where scholars were " Led by soaring-genius'd Sylvester," as has expressed it in his " Ode to The Johns Hopkins University.""

"... in America before the end of 1888 there had been appreciable amount of mathematical research, some of it of first importance, even according to recent standards. There had been centers of mathematical inspiration. Such universities as Yale, The Johns Hopkins, and Harvard, had been sending out doctors in mathematics for a number of years, and many Americans had been getting degrees in Europe. The time was ripe for an organization to draw together many people scattered throughout the country who were especially interested in mathematical pursuits."

"About half a century after Thales came Pythagoras. Under his inspiration geometry was first pursued as a study for its own sake. A man of great ability and a most interesting and magnetic mystic, he finally settled at Crotona on the southeastern coast of Italy."

"The idea of constructing a table in which the logarithm of unity was zero originated with Napier. Napier and Briggs never thought of logarithms as exponents of a base. ... It was not till considerably later that our modern definition of a logarithm as an exponent was put forward by such mathematicians as , 1684; , 1742; , 1748, 1770."

"I think that mathematics will have to become more and more algorithmic if it is going to be active and vital in the creative life. This means it is necessary to rethink what we teach, in school, in college, and in graduate school. In our emphasis on deductive reasoning and rigor we have been following the Greek tradition, but there are other traditions—Babylonian, Hindu, Chinese, Mayan—and these have all followed a more algorithmic, more numerical procedure. After all, the word algorithm, like the word algebra, comes from Arabic. And the numerals we use come from Hindu mathematics via the Arabs. We can’t regard Greek mathematics as the only source of great mathematics, and yet somehow in the last half century there has been such emphasis on the greatness of “pure” mathematics that the other possible forms of mathematics have been put down. I don’t mean that it is necessary to put down the rigorous Greek style mathematics, but it is necessary to raise up the status of the numerical, the algorithmic, the discrete mathematics."

"For my topic today I have chosen a subject connecting mathematics and aeronautical engineering. The histories of these two subjects are close. It might appear however to the layman that, back in the time of the first powered flight in 1903, aeronautical engineering had little to do with mathematics. The Wright brothers, despite the fact that they had no university education, were well read and learned their art using wind tunnels but it is unlikely that they knew that airfoil theory was connected to the Riemann conformal mapping theorem. But it was also the time of and later who developed and understood that connection and put mathematics solidly behind the new engineering. Since that time each new geernation has discovered new problems that are at the forefront of both fields. One such problem is flight near the speed of sound. This one in fact has puzzled more than one generation."

"The women who earned their Ph.D.'s in mathematics during the forties and fifties include some of the most distinguished mathematicians and mathematics educators of this century. Julia Robinson, Cathleen Morawetz, and Mary Ellen Rudin are probably the best-known women mathematicians of this generation."