Extract from God Created the Integers by Stephen Hawking
INTRODUCTION
WE ARE LUCKY TO LIVE IN AN AGE IN WHICH WE ARE STILL MAKING DISCOVERIES. IT IS LIKE THE DISCOVERY OF AMERICA - YOU ONLY DISCOVER IT ONCE. THE AGE IN
WHICH WE LIVE IS THE AGE IN WHICH WE ARE DISCOVERING THE FUNDAMENTAL LAWS OF NATURE...
- AMERICAN PHYSICIST RICHARD FEYNMAN, SPOKEN IN 1964
Showcasing excerpts from thirty-one of the most important works in the history of mathematics (four of which have been translated into English for the very first time), this book is a celebration of the mathematicians who helped move us forward in our understanding of the world and who paved the way for our current age of science and technology.
Over the centuries, the efforts of these mathematicians have helped the human race to achieve great insight into nature, such as the realization that the earth is round, that the same force that causes an apple to fall here on earth is also responsible for the motions of the heavenly bodies, that space is finite and not eternal, that time and space are intertwined and warped by matter and energy, and that the future can only be determined probabilistically. Such revolutions in the way we perceive the world have always gone hand in hand with revolutions in mathematical thought. Isaac Newton could never have formulated his laws without the analytical geometry of René Descartes and Newton's own invention of calculus. It is hard to imagine the development of either electrodynamics or quantum theory without the methods of Jean Baptiste Joseph Fourier or the work on calculus and the theory of complex functions pioneered by Carl Friedrich Gauss and Augustin Louis Cauchy - and it was Henri Lebesgue's work on the theory of measure that enabled John von Neumann to formulate the rigorous understanding of quantum theory that we have today. Albert Einstein could not have completed his general theory of relativity had it not been for the geometric ideas of Bernard Riemann. And practically all of modern science would be far less potent (if it existed at all) without the concepts of probability and statistics pioneered by Pierre-Simon Laplace.
All through the ages, no intellectual endeavor has been more important to those studying physical science than has the field of mathematics. But mathematics is more than a tool and language for science. It is also an end in itself, and as such, it has, over the centuries, affected our worldview in its own right. Karl Weierstrass provided a new idea of what it means for a function to be continuous, and George Cantor's work revolutionized people's idea of infinity. George Boole's Laws of Thought revealed logic as a system of processes subject to laws identical to the laws of algebra, thus illuminating the very nature of thought and eventually enabling to some degree its mechanization, that is, modern digital computing. Alan Turing illuminated the power and the limits of digital computing, long before sophisticated computations were even possible. Kurt Gödel proved a theorem troubling to many philosophers, as well as anyone else believing in absolute truth: that in any sufficiently complex logical system (such as arithmetic) there must exist statements that can neither be proven nor disproven. And if that weren't bad enough, he also showed that the question of whether the system itself is logically consistent cannot be proven within the system.
This fascinating volume presents all these and other groundbreaking developments, the central ideas in twenty-five centuries of mathematics, employing the original texts to trace the evolution, and sometimes revolution, in mathematical thinking from its beginnings to today. Though the first work presented here is that of Euclid, C.300 B.C., the Egyptians and Babylonians had developed an impressive ability to perform mathematical calculations as early as 3,500 B.C. The Egyptians employed this skill to build the great pyramids and to accomplish other impressive ends, but their computations lacked one quality considered essential to mathematics ever since: rigor. For example, the ancient Egyptians equated the area of a circle to the area of a square whose sides were 8/9 the diameter of the circle. This method amounts to employing a value of the mathematical constant pi that is equal to 256/81. In one sense this is impressive - it is only about one half of one percent off of the exact answer. But in another sense it is completely wrong. Why worry about an error of one half of one percent? Because the Egyptian approximation overlooks one of the deep and fundamental mathematical properties of the true number 'pi': that it cannot be written as any fraction. That is a matter of principle, unrelated to any issue of mere quantitative accuracy. Though the irrationality of 'pi' wasn't proved until the late eighteenth century, the early Greeks did discover that numbers existed which could not be written as fractions, and this was both puzzling and shocking to them. This was the brilliance of the Greeks: to recognize the importance of principle plura in mathematics, and that in its essence mathematics is a subject in which one begins with a set of concepts and rules and then rigorously works out their precise consequences.
Euclid detailed the Greek understanding of geometry in his Elements, in Alexandria, around 300 B.C. In the ensuing centuries the Greeks made great strides in both algebra and geometry. Archimedes, the greatest mathematician of antiquity, studied the properties of geometric shapes and created ingenious methods of finding areas and volumes and new approximations for 'pi'. Another Alexandrian, Diophantus, looking over the clutter of words and numbers in algebraic problems, saw that an abstraction could be a great simplification. And so, Diophantus took the first step towards introducing symbolism into algebra. Over a millennium later, Frenchman Rene Descartes united the two fields: geometry and algebra, with his creation of analytic geometry. His work paved the way for Isaac Newton to invent calculus, and with it, a new way of doing science. Since Newton's day, the pace of mathematical innovation has been almost frenetic, as the fundamental mathematical fields of algebra, geometry, and calculus (or function theory) have fed on and in turn nourished one another, yielding insights into applications as diverse as probability, numbers, and the theory of heat. And as mathematics matured, so did the range of questions it addresses: Kurt Godel and Alan Turing, the last two thinkers represnted in this volume, address perhaps the deepest issue - the question of what is knowable. Like those of the past, future developments in mathematics are sure to affect, directly or indirectly, our ways of living and thinking. The wonders of the ancient world were physical, like the pyramids in Egypt. As this volume illustrates, the greatest wonder of the modern world is our own understanding.