The Advent of the Algorithm: The Idea that Rules the World

Overview

The algorithm is just code, but it makes things happen. It's the set of abstract, detailed instructions that makes computers run. Any programmer can invent a new algorithm-and many have become millionaires doing just that. Computers, the Internet, virtual reality-our world is being transformed before our eyes, all because some quirky logicians and mathematicians followed the dream of ultimate abstraction and invented the algorithm. Beginning with Leibniz and culminating in the middle of this century with the work...

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Overview

The algorithm is just code, but it makes things happen. It's the set of abstract, detailed instructions that makes computers run. Any programmer can invent a new algorithm-and many have become millionaires doing just that. Computers, the Internet, virtual reality-our world is being transformed before our eyes, all because some quirky logicians and mathematicians followed the dream of ultimate abstraction and invented the algorithm. Beginning with Leibniz and culminating in the middle of this century with the work of little-known geniuses and eccentrics like Gödel and Turing, David Berlinski tells this epic tale with clarity and imaginative brilliance. You don't have to be a programmer or a math buff to enjoy his book. All you have to do is be fascinated by the greatest innovation of the twentieth century.

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Editorial Reviews

Publishers Weekly - Publisher's Weekly
Berlinski's successful A Tour of the Calculus displayed his spectacular talent for explaining math and its various real-world consequences. This hefty follow-up explores what Berlinski considers "the second great scientific idea of the West. There is no third." Calculus gave us modern physics, but the algorithm gave us--is still giving us--the computer (or, more precisely, the computer program). In short, densely intertwined, lyrically constructed chapters, Berlinski describes the discoveries of major algorithmic thinkers. We hear of Gottfried von Leibniz, one of the founders of formal logic; of Gottlob Frege, David Hilbert and Bertrand Russell, who set out to draw up formal, mathematical criteria for truth; of Kurt G del, who proved that it couldn't be done; of computer pioneer, code breaker and gay martyr Alan Turing; of programs, undecidability, DNA and entropy. We see equations and graphs, but we also hear tales from Isaac Bashevis Singer and bizarre anecdotes of Berlinski's own travels. A novelist (The Body Shop) as well as a mathematician, Berlinski has composed energetic, intertwined tales that make it nearly impossible for readers, once drawn in, to lose interest or to get lost among flying abstractions. (He may well attract the same readers who gravitated, 20 years ago, to Douglas Hofstadter's G del, Escher, Bach, though the books' personalities and prose styles have little in common.) Although not perfect--the book can be hyperbolic or too aphoristic and digressive for those who just want to learn about math (or the philosophy of computing)--this captivating volume is nevertheless an uncommon achievement of both style and substance. Agent, Susan Ginsburg; author tour. (Mar.) Copyright 2000 Cahners Business Information.|
Library Journal
In his newest work, which complements his A Tour of the Calculus (Pantheon, 1996), professional writer and sometime mathematician Berlinski traces some of the highlights in the development of modern mathematical logic and shows how they have converged on the algorithm--which may be defined as a prescription for carrying through a computation in a finite series of steps. Berlinski compares and contrasts the triumph of the algorithm with the earlier successful career of calculus. His writing style is vivid and dynamic--almost too much so. However, he succeeds in carrying his readers through the basic notation of mathematical logic in a fashion that should work well even for lay readers. Thumbnail biographical sketches of several major logicians and several fragments of fiction further enliven this zesty and unusual book. Recommended for public and academic libraries.--Jack W. Weigel, formerly with Univ. of Michigan Lib., Ann Arbor Copyright 2000 Cahners Business Information.\
From The Critics
David Berlinski is a brave man. His last book, A Tour of the Calculus, explored in laymen's terms (as much as that's possible), the subject that caused most laymen great pain during their student years. Now he has returned to explain "the second great scientific idea of the West. There is no third."

An algorithm is a procedure, written in a symbolic vocabulary, that gets something done step-by-step without the need for any intelligent assistance. Even if the word rings some dusty bells, it comes as a surprise to most of us that it is a revolutionary concept. But without it, the amazingly powerful, and equally dumb, machine that sits on our desktops - the digital computer - would be completely useless.

Rest assured, you no more need to master the technical details of this book to run your computer than you need to know the ignition temperature of unleaded gasoline to run your car. But if you master the details, there are satisfactions aplenty.

The algorithm is one of the astounding intellectual achievements of modern times. Until the late 19th century, the rules of logic were much the same as they had been in ancient Greece. Aristotle invented the syllogism, which comprises a major premise (for example, "All Microsoft products are overpriced"), a minor premise ("Word is a Microsoft product") and a conclusion ("therefore, Word is overpriced").

The scope of logic was pushed several thousand years ahead by one remarkable man, Gottlob Frege, whose aim, along with the handful who comprehended his work, was to find a logic within mathematics. Mathematics (and eventually the world, which runs according to the mathematical laws discovered by Isaac Newton and others) would then be subject to proofs, which determine truth or falsehood beyond the possibility of doubt.

Several generations later, in one of the mighty intellectual thunderbolts of all time, Kurt Godel proved that the procedure Frege had sought would not be found - because it could not exist. No matter how the truth-seeking algorithm is devised, there will always be truths that elude it.

This was a devastating conclusion: From then on something less than certainty would be the best that logicians, mathematicians and scientists could ever hope to find. Berlinski walks us through this remarkable "incompleteness" theorem, and this is where he is at his best, illuminating complex concepts with clarity.

In the course of his proof, Godel demonstrated how complex reasoning can be reduced to a series of mechanical steps. Alan Turing, another tortured genius (as most of these logicians were), invented an imaginary "machine" that could use such mechanical steps to solve virtually any intellectual problem. This became the basis of the digital computer, long before any such device existed. A few years later Turing would help crack the Enigma machine, the Nazis' cryptographic code during World War II. Fifty years later, it was largely his insight that yielded the information revolution.

The algorithm extends beyond even the furthest reaches of computing. Berlinski sees DNA as an algorithm: The molecule of life replicates exactly by automatic biochemical command, sending information not only throughout an organism, but also forward from generation to generation. Will human thought itself be reduced to an algorithm?

Now the bad news. Berlinski intersperses informative sections of his book with short stories of elusive significance, imaginary conversations with historical personalities, and even accounts of cute meetings among himself, his agent and his editor. Berlinski's writing often gets so mannered that you can imagine him typing with one hand and patting himself on the back with the other.

The workings of Turing's machine had only two commands: what to write and what to erase. This fascinating book would have been far better if Berlinski had devised such a machine for it.


Daniel Evan Weiss is the author of Honk If You Love Aphrodite, and other novels.
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Product Details

  • ISBN-13: 9780151003389
  • Publisher: Houghton Mifflin Harcourt
  • Publication date: 3/1/2000
  • Edition number: 1
  • Pages: 416
  • Product dimensions: 6.00 (w) x 9.00 (h) x 1.20 (d)

Meet the Author

David Berlinski is the author of three novels and four works of nonfiction, including the bestselling A Tour of the Calculus . Berlinski received his Ph.D. from Princeton University and is a regular contributor to Commentary and Forbes ASAP . He lives in Paris.

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Read an Excerpt

The Jeweler's Velvet

Two ideas lie gleaming on the jeweler's velvet. The first is the calculus, the second, the algorithm. The calculus and the rich body of mathematical analysis to which it gave rise made modern science possible; but it has been the algorithm that has made possible the modern world.

They are utterly different, these ideas. The calculus serves the imperial vision of mathematical physics. It is a vision in which the real elements of the world are revealed to be its elementary constituents: particles, forces, fields, or even a strange fused combination of space and time. Written in the language of mathematics, a single set of fearfully compressed laws describes their secret nature. The universe that emerges from this description is alien, indifferent to human desires.

The great era of mathematical physics is now over. The three-hundred-year effort to represent the material world in mathematical terms has exhausted itself. The understanding that it was to provide is infinitely closer than it was when Isaac Newton wrote in the late seventeenth century, but it is still infinitely far away.

One man ages as another is born, and if time drives one idea from the field, it does so by welcoming another. The algorithm has come to occupy a central place in our imagination. It is the second great scientific idea of the West. There is no third.

An algorithm is an effective procedure, a way of getting something done in a finite number of discrete steps. Classical mathematics is, in part, the study of certain algorithms. In elementary algebra, for example, numbers are replaced by letters to achieve a certain degree of generality. The symbols are manipulated by means of firm, no-nonsense rules. The product of (a + b) and (a + b) is derived first by multiplying a by itself; second, by multiplying a by b twice; and third, by multiplying b by itself. The results are then added. The product is a2 + 2ab + b2 and that is the end of it. A machine could execute the appropriate steps. A machine can execute the appropriate steps. No art is involved. And none is needed.

In the wider world from which mathematics arises and to which the mathematician must like the rest of us return, an algorithm, speaking loosely, is a set of rules, a recipe, a prescription for action, a guide, a linked and controlled injunction, an adjuration, a code, an effort made to throw a complex verbal shawl over life's chattering chaos.

My dear boy, Lord Chesterfield begins, addressing his morganatic son, and there follows an extraordinary series of remarkably detailed letters, wise, witty, and occasionally tender, the homilies and exhortations given in English, French, Latin, and Greek. Dear boy is reminded to wash properly his teeth, to clean his linen, to manage his finances, and to discipline his temper; he needs to cultivate the social arts and to acquire the art of conversation and the elements of dance; he must, above all, learn to please. The graceful letters go on and on, the tone regretful if only because Lord Chesterfield must have known that he was volleying advice into an empty chamber, his son a dull, pimpled, rather loutish young man whose wish that his elegant father would for the love of God just stop talking throbs with dull persistence throughout his own obdurate silence.

The world the algorithm makes possible is retrograde in its nature to the world of mathematical physics. Its fundamental theoretical objects are symbols, and not muons, gluons, quarks, or space and time fused into a pliant knot. Algorithms are human artifacts. They belong to the world of memory and meaning, desire and design. The idea of an algorithm is as old as the dry humped hills, but it is also cunning, disguising itself in a thousand protean forms. With his commanding intelligence, the seventeenth-century philosopher and mathematician Gottfried Leibniz penetrated far into the future, seeing universal calculating machines and strange symbolic languages written in a universal script; but Leibniz was time's slave as well as her servant, unable to sharpen his most profound views, which like cities seen in dreams, rise up, hold their shape for a moment, and then vanish irretrievably.

Only in this century has the concept of an algorithm been coaxed completely into consciousness. The work was undertaken more than sixty years ago by a quartet of brilliant mathematical logicians: the subtle and enigmatic Kurt Godel; Alonzo Church, stout as a cathedral and as imposing; Emil Post, entombed, like Morris Raphael Cohen, in New York's City College; and, of course, the odd and utterly original A. M. Turing, whose lost eyes seem to roam anxiously over the second half of the twentieth century.

Mathematicians have loved mathematics because, like the graces of which Sappho wrote, the subject has wrists like wild roses. If it is beauty that governs the mathematicians' souls, it is truth and certainty that remind them of their duty. At the end of the nineteenth century, mathematicians anxious about the foundations of their subject asked themselves why mathematics was true and whether it was certain and to their alarm discovered that they could not say and did not know. Working mathematicians continued to work at mathematics, of course, but they worked at what they did with the sense that some sinister figure was creeping up the staircase of events. A number of redemptive schemes were introduced. Some mathematicians such as Gottlob Frege and Bertrand Russell argued that mathematics was a form of logic and heir thus to its presumptive certainty; following David Hilbert, others argued that mathematics was a formal game played with formal symbols. Every scheme seemed to embody some portion of the truth, but no scheme embodied it all. Caught between the crisis and its various correctives, logicians were forced to organize a new world to rival the abstract, cunning, and continuous world of the physical sciences, their work transforming the familiar and intuitive but hopelessly unclear concept of an algorithm into one both formal and precise.

Their story is rich in the unexpected. Unlike Andrew Wiles, who spent years searching for a proof of Fermat's last theorem, the logicians did not set out to find the concept that they found. They were simply sensitive enough to see what they spotted. But what they spotted was not entirely what they sought. In the end, the agenda to which they committed themselves was not met. At the beginning of the new millennium, we still do not know why mathematics is true and whether it is certain. But we know what we do not know in an immeasurably richer way than we did. And learning this has been a remarkable achievement--among the greatest and least-known of the modern era.

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Table of Contents

Preface The Digital Bureaucrat    xi
Introduction The Jeweler's Velvet   xv
The Marketplace of Schemes     1
Under the Eye of Doubt   21
Bruno the Fastidious   46
Cargoload and Crack-Up   64
Hilbert Takes Command   96
Godel in Vienna 116
The Dangerous Discipline 146
Flight into Abstraction 155
The Imaginary Machine 181
Postscript 198
The Peacock of Reason 205
Time Against Time 215
An Artifact of Mind 249
A World of Many Gods 275
The Cross of Words 306
Epilogue The Idea of Order at Key West                   332
Acknowledgments 335
Index 337
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  • Anonymous

    Posted March 24, 2003

    This book is brilliant

    David Berlinski¿s, The Advent of the Algorithm, is one of the most undervalued pieces of writing I have ever seen. Filled with understanding and insight, it is sweet relief for the mathematical layman. <BR> I was given a copy of it as a Christmas present and it has helped me to understand things which I thought I would go to my grave with never so much as a glimpse. Unlike the great majority of `understand-it-yourself¿ math books¿which deluge the reader with an unrelieved monsoon of arcane concepts, the Advent of the Algorithm informs the reader. With patience and more than a little kindness, Berlinski¿s insightful approach gives the reader a good foothold into the nature and importance of one of the pillars of western reasoning, and it is a perfect book for those of us whose chests tightened on the last-mile walk to an algebra class. Berlinski teaches us to breathe. By interspersing clearly elucidated ideas with history, biography, humor and good storytelling, Berlinski soothes and cools the mathematical experience. He is an attentive lover of literary writers, like Vladimir Nabokov, Isaac Bashevis Singer and Jorge Borges¿men whose works and styles he often approximates quite decently¿using work styled after theirs to offer insight into the structures he presents to the reader. The Advent of the Algorithm is not a book for PhD¿s in Quantum Physics, or graduate students of Ring Theory. If you are burning with mathematical talent and are perfectly comfortable with the Calculus, the book is not for you and it will bring nothing to you. If, however, you are just someone, an intelligent man or woman who has always hated the impassable wall between yourself and mathematical understanding, The Advent of the Algorithm is a real treasure. A good, and wonderfully decent introduction to concepts in mathematical logic and their implications for philosophy, The Advent of the Algorithm is a book for the rest of us; one for those of us who sat through math classes and felt everything but hope.

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  • Anonymous

    Posted March 24, 2000

    A fascinating story from an utterly original author

    David Berlinski has delivered another fascinating tale of a subject understood by relatively few. This book will allow the algorithm to become a topic of educated conversation outside the circles of mathematicians and computer scientists. The text preserves all of Berlinski's florid, idiosyncratic and sometimes difficult style, shifting between careful analysis, historical drama, insightful explanation, and obscure fictional aside. Readers will either love it or hate it. (I love it.) Unfortunately, some readers will misunderstand Berlinski's subtlety and insight. For instance, the official trade review of the book complains that Berlinski never really defines 'algorithm.' This is incorrect. The introduction concludes with an offset definition: 'In the logician's voice: an algorithm is a finite procedure, written in a fixed symbolic vocabulary, governed by precise instructions, moving in discrete steps, 1, 2, 3,..., whose execution requires no insight, cleverness, intuition, intelligence, or perspicuity, and that sooner or later comes to an end.' It doesn't get much clearer than that. But Berlinski doesn't ponder long over what he takes to be obvious, and he doesn't always speak in the logician's voice. The Advent of the Algorithm demonstrates that a seemingly dull concept can have unimaginably profound implications. Those implications illuminate everything from computing and information technology to the nature of life and the universe. And ultimately (not to spoil the ending) Berlinski argues that the advent of the algorithm foretells the end of scientific materialism, suggesting nothing so much as a world permeated by the marks of intelligence and design. To paraphrase, we are shocked to discover information--something we had assumed was found exclusively in the domain of human activity--flourishing on the alien shores of biology.

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