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Simply put, an algorithm is a set of instructions-it's the code that makes computers run. A basic idea that proved elusive for hundreds of years and bent the minds of the greatest thinkers in the world, the algorithm is what made the modern world possible. Without the algorithm, there would have been no computer, no Internet, no virtual reality, no e-mail, or any other technological advance that we rely on every day.
In The Advent of the Algorithm, David Berlinski combines science, history, and math to explain and explore the intriguing story of how the algorithm was finally discovered by a succession of mathematicians and logicians, and how this paved the way for the digital age. Beginning with Leibniz and culminating in the middle of the twentieth century with the groundbreaking work of Gödel and Turing, The Advent of the Algorithm is an epic tale told with clarity and imaginative brilliance.
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.
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.
|Preface: The Digital Bureaucrat||xi|
|Introduction: The Jeweler's Velvet||xv|
|Chapter 1||The Marketplace of Schemes||1|
|Chapter 2||Under the Eye of Doubt||21|
|Chapter 3||Bruno the Fastidious||46|
|Chapter 4||Cargoload and Crack-Up||64|
|Chapter 5||Hilbert Takes Command||96|
|Chapter 6||Godel in Vienna||116|
|Chapter 7||The Dangerous Discipline||146|
|Chapter 8||Flight into Abstraction||155|
|Chapter 9||The Imaginary Machine||181|
|Chapter 11||The Peacock of Reason||205|
|Chapter 12||Time Against Time||215|
|Chapter 13||An Artifact of Mind||249|
|Chapter 14||A World of Many Gods||275|
|Chapter 15||The Cross of Words||306|
|Epilogue: The Idea of Order at Key West||332|
Posted May 8, 2007
Following up his Tour of the Calculus tour-de-force, David Berlinski scores again with his history of algorithms. More so than the calculus book, though, this one bogs down in spots and gets positively murky at the end when he takes on DNA. Nonetheless,a great intro for the mathematically challenged and literately-oriented.Was this review helpful? Yes NoThank you for your feedback. Report this reviewThank you, this review has been flagged.
Posted June 18, 2001
David Berilnski is one of the most gentile of men. I have experienced him working, and he is amazing. A genius, and he helped me discover the genius in myself!Was this review helpful? Yes NoThank you for your feedback. Report this reviewThank you, this review has been flagged.
Posted July 16, 2001
Mr. Berlinski has written an excellent work detailing the numerous developments in mathematical logic which eventually led to the birth of the modern computer. We meet all the characters in this incredible journey; Leibniz, Peano, Frege, Godel, Church, Kleene, Turing, Von Neumann, and Shannon. Not only does Mr. Berlinski bring all these individuals to life, he also does a wonderful job of introducing the reader to the intricacies of mathematical logic. Moreover, he uses a number of intriguing fictional episodes,which are interspersed throughout the book, to illustrate many of the more technical points in logic. In conclusion, The Advent of the Algorith allows the reader to grasp how modern mathematical logic led to the birth of the computer which has forged the modern world. A first rate introduction to mathematical logic.Was this review helpful? Yes NoThank you for your feedback. Report this reviewThank you, this review has been flagged.