Journey through Genius

PREFACE

In his autobiography, Bertrand Russell recalled the crisis of his youth:

There was a footpath leading across fields to New Southgate, and I used to go there alone to watch the sunset and contemplate suicide. I did not, however, commit suicide, because I wished to know more of mathematics.

Admittedly, few people find such absolute salvation in mathematics, but many appreciate its power and, more critically, its beauty. This book is designed for those who would like to probe a bit more deeply into the long and glorious history of mathematics.

For disciplines as diverse as literature, music, and art, there is a tradition of examining masterpieces-the "great novels," the "great symphonies," the "great paintings"-as the fittest and most illuminating objects of study. Books are written and courses are taught on precisely these topics in order to acquaint us with some of the creative milestones of the discipline and with the men and women who produced them.

The present book offers an analogous approach to mathematics, where the creative unit is not the novel or symphony, but the theorem. Consequently, this is not a typical math book in that it does not provide a step-by-step development of some branch of the subject. Nor does it stress the applicability of mathematics in determining planetary orbits, in understanding the world of computers, or, for that matter, in balancing your checkbook. Mathematics, of course, has been spectacularly successful in such applied undertakings. But it was not its worldly utility that led Euclid or Archimedes or Georg Cantor to devote so much of their energy and genius to mathematics. These individuals did not feel compelled to justify their work with utilitarian applications any more than Shakespeare had to apologize for writing love sonnets instead of cookbooks or Van Gogh had to apologize for painting canvases instead of billboards.

In this book I shall explore a handful of the most important proofs and the most ingenious logical arguments-from the history of mathematics, with emphasis on why the theorems were significant and how the mathematician resolved, once and for all, the pressing logical issue. Each chapter of Journey Through Genius has three primary components:

The first is its historical emphasis' The "great theorems" on the pages ahead span more than 2300 years of human history. Before discussing a particular result, I shall set the scene by describing the state of mathematics, and perhaps the state of the world generally, prior to the theorem. Like everything else, mathematics is created within the context of history, and it is of interest to place Cardano's solution of the cubic two years after the publication of Copernicus's heliocentric theory and two years before the death of England's Henry VIII, or to emphasize the impact of the Restoration upon Cambridge University when a young scholar named Isaac Newton entered it in 1661.

The second component is the biographical Mathematics is the product of real, flesh-and-blood human beings whose lives may reflect the inspirational, the tragic, or the bizarre. The theorems contained here represent the work of a number of individuals, ranging from the gregarious Leonhard Euler to the pugnacious Johann Bernoulli to that most worldly of Renaissance characters, Gerolamo Cardano. Understanding something of the lives of these diverse individuals can only enhance an appreciation of their work.

The final component, and the primary focus of the book, is the creativity evident in these "mathematical masterpieces." just as one could not hope to understand a great novel without reading it, or to appreciate a great painting without seeing it, so one cannot really come to grips with a great mathematical theorem without a careful, step-by-step look at the proof. To acquire such an understanding requires a good bit of concentration and effort, and the chapters to follow are meant to serve as a guide in that undertaking.

There is a remarkable permanence about these mathematical landmarks. In other disciplines, the fads of today become the forgotten discards of tomorrow. A little over a century ago, Sir Walter Scott was among the most esteemed writers in English literature; today, he is regarded considerably less enthusiastically. In the twentieth century, superstars come and go with breathtaking speed, and ideas that seem destined to change the world often end up on the intellectual scrap heap.

Mathematics, to be sure, is also subject to changes of taste. But a theorem, correctly proved within the severe constraints of logic, is a theorem forever. Euclid's proof of the Pythagorean theorem from 300 B.C. has lost none of its beauty or validity with the passage of time. By contrast, the astronomical theories or medical practices of Alexandrian Greece have long since become archaic, slightly amusing examples of primitive science. The nineteenth-century mathematician Hermann Hankel said it best:

In most sciences one generation tears down what another has built, and what one has established another undoes. In mathematics alone each generation adds a new story to the old structure.

In this sense, as we examine the timeless mathematics of great mathematicians, we come to understand Oliver Heaviside's wonderfully apt observation: "Logic can be patient, for it is eternal."

A number of factors have gone into the selection of these few theorems to represent the best of mathematics. As noted, my chief consideration was to find arguments that were particularly insightful or ingenious. This, of course, introduces an element of personal taste, and I recognize that a different author would certainly generate a different list of great theorems. That aside, it is an extraordinary experience to behold, first-hand, the mathematician gliding through clever deductions and making the seemingly incomprehensible become clear. It has been said that talent is doing easily what others find difficult, but that genius is doing easily what others find impossible. As will be evident, there is much genius displayed on the pages ahead. Here are genuine classics the Mona Lisas or Hamlets of mathematics.

But other considerations influenced the choice of theorems. For one, I wanted to include samples from history's leading mathematicians. It was a must, for instance, to have selections from Euclid, Archimedes, Newton, and Euler. To overlook such figures would be like studying art history without mentioning the work of Rembrandt or Cezanne.

Further, for the sake of variety, I have sampled different branches of mathematics. The propositions in the book come from the realms of plane geometry, algebra, number theory, analysis, and the theory of sets, The variety of these topics, and the occasional links and interplays among them, may add a note of freshness to this work.

I also wanted to present important mathematical theorems, rather than merely clever little tricks or puzzles. Indeed, most of the results in the book either resolved long-standing problems in mathematics, or generated even more profound questions for the future, or both. At the end of each chapter is an Epilogue, usually addressing an issue raised by the great theorem and following it as it echoes down through the history of mathematics.

Then there is the question of level of difficulty. Obviously, mathematics has many great landmarks whose depth and complexity render them incomprehensible to all but experts. it would be foolish to include such results in a book aimed at the general, scientifically literate reader. The theorems that follow require only the tools of algebra and geometry, of the sort one acquires in a few high school courses. The two exceptions are a brief use of the sine curve from trigonometry in discussing the work of Euler in Chapter 9 and an application of elementary integral calculus in the work of Newton in Chapter 7; many readers may already be acquainted with these topics, and for those who are not, there is a bit of explanation to smooth over the difficulty.

I should stress that this is not a scholarly tome. There are certainly questions of great mathematical or historical subtlety that cannot be addressed in a work of this kind. While I have tried to avoid including false or historically inaccurate material, this was simply not the time nor place to investigate all facets of all issues. This book, after all, is meant for the popular, not the scientific, press.

Along these lines, I must add a word about the authenticity of the proofs. In preparing the book, I have found it impossible to avoid the need for some compromise between the authors' original notation, terminology, and logical strategy and the requirement that the mathematical material be understandable to the modern reader. A complete adherence to the originals would make some of these results very difficult to comprehend; yet a significant deviation from the originals would conflict with my historical objectives. In general, I have tried to retain virtually all of the spirit, and a good bit of the detail, of the original theorems. The modifications I have introduced seem to me to be no more serious, than, say, performing Mozart on modern instruments.

And so, we are about to begin our journey through two millennia of mathematical landmarks. These results, old as they are, retain a freshness and display a sparkling virtuosity even after so many centuries. I hope that the reader will be able to understand these proofs and to recognize what made them great. For those who succeed in this venture, I expect there will be not only a sense of awe that comes from appreciating the greatness of others, but also a sense of personal satisfaction that one can, indeed, comprehend the works of a master.

W. Dunham
Columbus, Ohio

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