How Math Explains the World (03/2011)

[...]... heavily on mathematics However, when we think of the social sciences, we do not usually think of mathematics Nonetheless, Arrow’s theorem is completely mathematical, in a sense even more so than Heisenberg’s uncertainty principle, which is a mathematical result derived from hypotheses about the physical world Arrow’s theorem is as “pure” as the “purest” of mathematics—it deals with functions, one of the most... degree four or less, and in the sixteenth century the general solution of the quintic (the polynomial of degree five) was the goal of the best algebraists in the 8 How Math Explains the World world The physics community was likewise poised at the turn of the twentieth century, seeking a way out of the ultraviolet catastrophe the prediction that a perfectly black object in thermal equilibrium would emit... deterministic vision of the universe At the same mathematics conference that David Hilbert, the leading mathematician of the day, was describing to a rapt audience his vision of how mathematical truth might some day be automatically ascertained, in a back room far from the limelight Gödel was showing that there were some truths whose validity could never be proven Social scientists had searched for the ideal method... idea The same thing can happen to mathematicians and scientists, but there is another type of block that exists for the mathematician or scientist for which there is no analogy from the arts A mathematician or scientist may work on a problem that has no answer A composer might be able to come to grips with the idea that, at the moment, he is incapable of composing music, but he would never accept the. .. million for the solution of each Some of the problems, such as the Birch and Swinnerton-Dyer conjecture, are highly technical and even the statement of the problem is comprehensible only to specialists in the field Two of these problems, the Navier-Stokes equation and the Yang-Mills theory, are in the realm of mathematical physics Solutions to these problems will enable a better understanding of the physical... translate the preferences of the individual voters into the preferences of the society to which those voters belong The second half of the twentieth century witnessed a profusion of results in a number of areas, demonstrating how our ability to know and to do is limited, but these are unquestionably the Big Three There are a number of common elements to these three results The first is that they are all mathematical... in their careers, and classes in which relatively low-level material is taught to students who, given the choice of taking the class or a root canal without anesthesia, might well opt for the latter The second type of class includes the math courses required by the business school—most of the students in these classes believe they will someday be CEOs, and in the unlikely event they ever need a math. .. distinguished Swedish mathematician Yes, mathematics has its Fields Medal, awarded every four years, but it is awarded only to mathematicians under forty If you win it you are set for life, prestige-wise, but you’re not going to be able to put your kids through college on the proceeds At the turn of the millennium, the Clay Mathematics Institute posted seven critical problems in mathematics—and offered... surveying The properties of the functions discussed in Arrow’s theorem are clearly motivated by the problem Arrow initially started to investigate how to translate the preferences of individuals (as expressed by voting) into the results of an election The utility of mathematics is due in large measure to the wide variety of situations that are amenable to mathematical analysis The following tale has been... precisely scheduled so as to make optimal use of the avail- 2 How Math Explains the World able time while simultaneously making sure that outside constraints were also satisfied—such as making sure the space capsule was rotated so it did not overheat Thus was born the branch of mathematics known as scheduling, and with it the discovery of how improving the individual components that go into an ensemble . a mathematical result derived from hypotheses about the physical world. Arrow’s theorem is as “pure” as the “purest” of mathematics—it deals wi th functions, one of the most important mathematical. deterministic vision of the universe. At the same mathematics conference that David Hilbert, the leading mathematician of the day, was describing to a rapt audience his vision of how mathematical truth. mathematician who lived during the first half of the twentieth century. Hardy wrote a fascinating book (A Mathematician’s Apology), in which he described his passion for the aesthetics of mathe- matics.