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The Advent of the Algorithm: The 300-Year Journey from an Idea to the Computer, by David Berlinski
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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.
- Sales Rank: #1176757 in Books
- Color: Other
- Published on: 2001-05-03
- Original language: English
- Number of items: 1
- Dimensions: 8.00" h x .93" w x 5.25" l, .87 pounds
- Binding: Paperback
- 368 pages
- ISBN13: 9780156013918
- Condition: New
- Notes: BRAND NEW FROM PUBLISHER! 100% Satisfaction Guarantee. Tracking provided on most orders. Buy with Confidence! Millions of books sold!
Amazon.com Review
Francis Sullivan of the Institute for Defense Analysis said "Great algorithms are the poetry of computation"; David Berlinski calls the algorithm "the idea that rules the world." The Advent of the Algorithm is not so much a history of algorithms as a historical fantasia. Berlinski spins freely between semifictional accounts of historical figures, personal reminiscence, and mathematical proofs--without ever really defining an algorithm in so many words.
This is not the book for those who were maddened by Berlinski's A Tour of the Calculus; his style remains quirky, digressive, self-referential, and dense:
And then, by some inscrutable incandescent insight, Leibniz came to see that what is crucial in what he had written is the alternation between God and Nothingness. And for this, the numbers 0 and 1 suffice.
Twinkies and Diet Coke in hand, computer programmers can now be observed pausing thoughtfully at their consoles.
Berlinski's argument seems to be that algorithms--step-by-step procedures for getting answers--superceded logic, and will be superceded in turn by more biological, empirical, fuzzy methods. The structure of the book reflects this argument--sketches of people like Leibniz, Hilbert, Gödel, and Turing are interwoven with proofs and with characters of Berlinski's own invention. Berlinski's voice, closer to Hofstadter than to Knuth, remains unique. --Mary Ellen Curtin
From Publishers 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 Reed Business Information, Inc.
From 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 Reed Business Information, Inc.
Most helpful customer reviews
5 of 15 people found the following review helpful.
OK for those interested in 'nontechnical history of ideas'.
By ralph kelsey
UPGRADE to 2 1/2 stars. Wordy discussion of historical figures with some relation to algorithms. Little information about any actual algorithms. I retract my previous review (1 star) because that regarded this book as an algorithms book, but it is more of a general interest book that I probably would have liked when I was in highschool.
15 of 18 people found the following review helpful.
David Berlinski on Science and Things Beyond Science
By Raymond C. Togtman
(I wrote this review of Berlinski's The Advent of the Algorithm, A Tour of the Calculus, and Newton's Gift in 2001 but could find no one to publish it; so I am posting it here. Someone who is interested in his thought might learn from it.)
In 1979, in the lecture that he delivered upon assuming the Lucasian chair of mathematics at Cambridge University, Stephen Hawking predicted that the end of theoretical physics was in sight and that there was a fifty per cent chance that a theory of everything would be produced within twenty years. The theory that he had in mind involved supergravity in a universe of eleven dimensions, ten of space and one of time. Recently he has repeated his prediction--another twenty years, another fifty per cent chance. This time the potential theory is a string theory in eleven-dimensional space-time. By conceiving of particles as infinitesimal points, earlier quantum physics had been plagued with infinities in such quantities as mass and charge, since any finite quantity, when compressed to a point, becomes infinitely dense. String theory eliminates these infinities by imagining particles and forces as lines or loops. Although they may look and act like infinitesimal points, they really are not, because their extension is hidden in tiny, curled-up, extra dimensions that are present at every point of our familiar three dimensional space.
David Berlinski's response to all this, in his book The Advent of the Algorithm, is to declare "The great era of mathematical physics is now over . . . The understanding that it was to provide is infinitely closer than it was when Isaac Newton wrote . . . but it is still infinitely far away." In his A Tour of the Calculus, he adds that "the era in thought that the calculus made possible is coming to an end. Everyone feels that this is so, and everyone is right."1
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1. These are by no means extravagant statements. In his The End of Physics, David Lindley states that even some prominent particle physicists, such as Richard Feynman and Sheldon Glashow, think that their discipline is moving into a world of "unverifiable mathematical invention" and turning into "recreational mathematics."
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In the last several years Berlinski has written a trio of books--A Tour of the Calculus, The Advent of the Algorithm, and Newton's Gift--that are about much more than their titles suggest. A Tour of the Calculus is, in the main, an introduction, both historical and systematic, to the mathematical discipline of the calculus, and a very creative, if somewhat odd, introduction at that. Although it contains no exercises and reads more like a novel than a book on mathematics, it is also uncommonly clear and thorough. What concerns me in this review, however, is the view of science that it begins to suggest and the religious murmurings of that view.
Berlinski tells us that modern mathematical science began with the revolutionary thesis that the real world can be represented by a system composed of the real numbers. This was an idea unknown to the ancient Greeks, an idea that medieval monks would have "regarded as superstitious mummery (as perhaps it is)." We are told also that mathematical theories apply only to mathematical facts (i.e., numbers), that mathematics can no more be applied to non-numerical facts than "shapes can be applied to liquids." The curtain is thus lifted onto a view of mathematical physics that regards it as a very limited endeavor which, however powerful within its own terrain, certainly cannot provide us with a theory of everything by conceiving of the fundaments of reality as mathematical entities known as points or lines or loops.
After his tour through the calculus Berlinski ends the book by dwelling briefly on the differences between mathematical physics and the worlds of biological science and human affairs. Whereas physics, based as it is on the calculus and its conception of the continuity of the real world in terms of the continuity of the infinitesimally progressing real numbers, attempts to penetrate downward to the bedrock of reality found, it is thought, in infinitesimal particles, biology trades depth of insight for adequacy of description and remains on the surface of experience. It desires not theories but facts. And so, its attitude of thought corresponds to that of everyday life, where notions of purpose, design, and ends are found everywhere. While Steven Weinberg, peering downward into the subatomic realm, remarks sourly in The First Three Minutes that "the more the universe seems comprehensible, the more it seems pointless," biology and human culture are suspended above that realm in a circle of meaning.
These ideas are taken up at much greater length in The Advent of the Algorithm. As its title suggests, this book is first of all about the algorithm, the intellectual artifact that made the computer possible. Algorithms are formal, mechanical procedures written in a fixed symbolic vocabulary (which can be embodied in material objects) that produce a definite result through a finite series of steps that can be followed without insight or intelligence. Within mathematics they are mechanical computation procedures. Berlinski's book is about much more than this, however. Together with its predecessor it is a philosophical reflection on important aspects of the development of scientific thought since the seventeenth century. It contains dashes of biography and history surrounding its characters, usually to remind us of the dark twentieth century backdrop to the developments discussed. The technical discussion is interwoven with stories, sometimes giving an imaginative presentation of ideas dealt with, other times providing an entertaining diversion. It contains veiled ruminations on the inscrutability of life, mind, and divinity. And, for a book on scientific matters, it is a literary gem.
According to Berlinski, the first intimation of the idea of the algorithm was Gottfried Leibniz's notion in the seventeenth century of a universal symbolic formal system, independent of language, that would embody the arts of discovery and judgment and would allow its users to reason together on all matters from mathematics to metaphysics to morals. Needless to say, this notion got no further than the notion itself. In the nineteenth century Guiseppe Peano drew the idea closer with his arithmetical axioms, by means of which he attempted to subordinate arithmetic to the logic of axiom and theorem used in Euclidean geometry.
Developments gathered steam in the late nineteenth and early twentieth centuries, when mathematicians anxiously realized that they did not know why mathematics is true and whether it is certain. The attempt to banish doubt from mathematics took two different tacks, both of which, in essence, attempted to reformulate mathematical thought in terms of formal, mechanical procedures that are completely explicit and transparent. First, Gottlob Frege developed his predicate calculus, which is basically the symbolic logic studied by philosophy undergraduates, and married this to set theory, which attempts to ground arithmetic in the utterly primitive notion of a set (i.e., any arbitrary collection). This direction foundered in 1903 on Bertrand Russell's paradox. The next redemptive scheme was developed by David Hilbert with his notion of formal arithmetic and metamathematics. Arithmetic would be mechanized, that is, reduced to a formal game played with formal symbols according to formal rules, all of which ostensibly have no mathematical meaning. The meaning of the game, the validity of its procedures and proofs, would be seen by the mathematician after the fact when he regards it from the metamathematical level. But of course that meaning was present beforehand as well, because the mathematician devised the symbols and the rules of the game with mathematical meaning in mind. Such a rigorously formalized arithmetic, in which all thinking is reduced to a finite number of explicit mechanical steps, would, it was hoped, allow all mathematical conclusions to be established with unassailable certainty.
Hilbert's program was defeated in 1931 by the incompleteness theorems of Kurt Godel. In Godel's first incompleteness theorem (from which the second, pertaining to consistency, is derived) he writes in symbolic logic a self-referential statement that says "I cannot be demonstrated in the axiom system used here" and then transforms this into a statement of arithmetic by means of his coding scheme. The result is that we have at least one true statement of arithmetic that transcends formal proof; and that one true statement is enough to demontrate that mathematical reasoning is not, at bottom, axiomatic. Many thinkers regard Godel's theorems as equal in importance to Einstein's relativity theories, since they demonstrate conclusively that a net of axiomatic certainty cannot be thrown around arithmetic (and if not around arithmetic then not around any science that incorporates arithmetic, e.g., mathematical physics).
As part of his incompleteness theorems Godel defined a class of mathematical functions that he called primitive recursive. The primitive recursive functions are those functions that can be derived by a finite number of specified mechanical operations that proceed upward from the presumed bedrock of arithmetic--the number zero, the notion of succession (one is the successor of zero), and the notion of identity. As an example Berlinski gives the Fibonacci sequence--0, 1, 1, 2, 3, 5, 8, 13 . . . --where each number is the sum of its two predecessors. He calls this the "inferential staircase," an expression that will return often in the book. The primitive recursive functions were the first fully clear, fully precise expression of the idea of the algorithm. Other expressions soon followed, from Alonzo Church, Emil Post, and Alan Turing. These I pass over, except to note that Turing's algorithms were imaginary machines that could compute any computable function, machines that are now embodied in all digital computers.
The rest of the book is a meditation on the algorithm and its implications for our understanding of biological life, of the attempt to conceive the mind as a computer, and of the grand project of mathematical physics to create a theory of everything. Whereas mathematical physics, by virtue of the infinitesimal calculus, comprehends a continuous world and moves in an infinity of space and time, the algorithmic computations performed by computers are discrete and apply only over finite intervals; they move crab-wise and can only approximate a continuous reality to which they are ultimately alien. The partial differential equations used, for example, in quantum physics to describe the behavior of the atom, are analytically intractable when applied to any atom more complex than helium, and their solutions therefore can only be approximated by algorithmic means. Such solutions do not return us to the continuous world in which we live but remain within the makeshift reality represented within the computer. Similarly, the second law of thermodynamics, that supposedly most general and well-established of physical theories, is based upon algorithmic thinking applied to an imagined "toy" universe. Berlinski thus asks how we can know that there is behind all this "one waiting world, one system of description, one resolution of experience." Again, "if the entropy of the physical world is running down, it must at some time have been running up. There are world and there are worlds, and physics describes one of them, but only one of them."
Algorithms are not only discrete and finite. They also involve symbols and information, and therefore meaning and intelligence, things that are immaterial and cannot be reduced to physical reality. For meaning is always relative to meaning, intelligence to intelligence. There is no way to break through this circle to a bedrock of material or physical fact. Indeed, there are no physical facts to reach. But, Berlinski asks, if the construction of the "inferential staircase" that scientists are trying to erect in order to rise from the primitive simplicity of Democritus's "atoms and the void" to meaning and the mind presupposes the algorithm, and therefore meaning and the mind, what good is that staircase?
Finally, there are the implications of the algorithm for our understanding of biological life and the computational theory of the mind. The DNA code and the processes of transcription, translation, and replication that are involved in everything from cellular replacement to the growth of an embryo into an infant are, for good reason, understood by molecular biologists after the model of the algorithm. Daniel Dennett and other Darwinian evolutionists seize upon this analogy between molecular biology and the algorithm, with its capacity of be followed mindlessly, and tell us that that evolution is an algorithmic process that produces life, mind, meaning, and purpose through the mindlessness of random mutation and natural selection. The mathematical logicians who formulated the concept of the algorithm, however, designed it as an intelligent procedure. Although the steps that constitute it are mechanical and can be followed without intelligence, it produces an intelligent result because it embodies an aspect of the intelligence of its maker. The algorithm thus is an artifact of mind.2 Moreover, the DNA code that is analogous to it has no clear text behind it, and this code and the processes it controls operate in a world of meaning where "something is given and something read, something ordered and
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2. It is amusing to note that by describing natural selection as an algorithmic process in Darwin's Dangerous Idea, Dennett is unwittingly asserting that evolution has intelligence.
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something done. But just who is doing the reading and who is executing the orders, this remains unclear." Here is a very great mystery indeed, and Berlinski thus asks "How does intelligence gain ascendancy over matter? How does it?"
Not by means of the formal systems, modeled by computers in such things as parallel processing neural networks, that are almost omnipresent in contemporary cognitive theory. Such projects are nothing more than descendants of the Hilbert program of the game of formal arithmetic described above, where mathematical meaning can only be made to supervene on the game from the metamathematical level because the rules of the game were devised from the start with mathematical meaning in mind. Similarly, all formal, material, algorithmic models of mind do nothing to explain mind because they are devised with mind in mind. Moreover, it cannot be understood how the unity of consciousness, which embraces both the I and the world perceived by the I, could arise from the sequential, algorithmic operations of a formal system. As many people have come to realize, to rebuke contemporary cognitive theory it is enough simply to open one's eyes. Berlinski ends this discussion with a quotation from the Greek philosopher Heraclitus: "You could not discover the limits of the soul, not even if you traveled down every road. Such is the depth of its form [logos]." In other words, the mind is inscrutable, just as divinity is inscrutable.
Whereas A Tour of the Calculus and The Advent of the Algorithm complement each other, Newton's Gift, a biography of Sir Isaac, stands on its own. It is once again a remarkably literate book, and although it does not try to be exhaustive, it provides a good and compelling account both of Newton's life and of his intellectual odyssey into rational mechanics (with a fair number of mathematical examples and derivations). Here I will only note that, like The Advent of the Algorithm, it confronts common scientific dogma with question upon question, doubt upon doubt.
For a long time philosophers believed that the answers to the book of nature lay just behind the familiar, that "things occur in nature as the result of the exercise of some will" directed by some emotion, that "if as in life nothing comes about without some form of intelligent agency, so perhaps in nature as well." Berlinski then adds, "This way of thinking is very plausible; it may at the end of time turn out to be correct." He also maintains that Newton's laws of motion and law of gravity do not really grasp forces, the "secret springs" of reality, as Newton called them. They only express the mathematical form of those secret springs. Berlinski thus comments, "This may well seem to represent a disturbing retreat [from Cartesian mechanics], one moving backward to the dark night of medieval powers and potentialities. And so it is." Finally, and most important, Berlinski observes in this book, as he did in The Advent of the Algorithm, that there is one aspect of Newton's world that is not explained by Newton's theory, one aspect of the world of contemporary physics that is not explained by contemporary physical theory, and that is the theories themselves. They are transcendent, "tantalizing traces in matter of an intelligence that has so far hidden itself in symbols," and they cannot be explained in material terms. Newton therefore said that they "could only proceed from the counsel and domination of an intelligent and powerful Being."
It seems to me that behind the veil of David Berlinski's doubts and questions, his subtlety, elusiveness, and playfulness, there lies the outline of something that resembles a broadly premodern--or at least, pre-Darwinian--view of reality. We have an intelligence that is ascendant over matter. We have a hierarchy going upward from physical reality to biological reality to mind. And we have a world in which "things are as they are for no better reason than that they are what they are."
Since, for Berlinski, divinity is inscrutable, even though all deities reveal themselves by means of trivialities, he ends The Advent of the Algorithm with another of his stories, "The Cardinal at Dinner." When the professore dottore says to the cardinal that "the more the universe seems comprehensible, the more it seems to have no meaning," the cardinal responds "if the universe is comprehensible, surely that is evidence that it is not entirely lacking in meaning, no?" When he says that "everything in the world can be explained by the behavior of matter," the cardinal asks, "even the laws of physics themselves? Even the reason for the world's existence?" The professor simply shrugs and says, "Eminence, every chain of explanation must come to an end." The cardinal points slyly toward the ornate ceiling of the dining room. Then he abandons the German in which he and the professor had been conversing and says something in his slangy Italian dialect to the men seated around him, who throw up their hands and laugh.
0 of 0 people found the following review helpful.
A Frequently Unhelpful Attempt To Be Helpful
By Donald Mitchell
From the book's introduction: "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." Everyday examples include computer programs and the functioning of our genetic systems to produce aspects of life.
Dr. Berlinski works from the conceptual questions and quests in philosophy (especially logic) and mathematics that led to the development of the mental concept of the algorithm that we know and think about consciously today. The path is an interesting one, and full of randomness and false leads. Like most progress, it is two steps forward and one step back.
Today, use of algorithms is at the base of our most powerful forms of progress. For example, some classes of problems are all but impossible to solve and we use algorithms to create simulations that allow us to approximate the answers. Mathematicians can now solve vastly more classes of problems using computers than they ever could before, rapidly expanding our knowledge. The speed up in progress in many scientific areas is also related to the use of algorithms to ask and answer questions. For example, the human genome decoding was greatly accelerated by using computer-based decoding algorithms to locate the likely sections of meaningful DNA information.
With so much intellectual richness and potential to work with, how did Dr. Berlinski end up with a 3 star instead of a 5 star rating from me?
First of all, he has one of the most irritating writing styles I have ever had the displeasure to persist through. Most annoying were the long fictional diversions he uses to faintly illuminate minor aspects of the point that he is making. These are denoted by four shadowed boxes before and after them. My advice is, SKIP THESE SECTIONS! As your faithful reviewer, I read almost all of them. I found little to reward me in these sections, and much to annoy. I also resented the space that could have gone to better use. The second annoying characteristic was the overuse of symbolic logic and mathematical expressions. Long after the point was made, he was still wandering around filling in little nuances that added almost nothing to one's understanding of the development of algorithms and their development.
The reason for deducting the second star was that he spent far too little time speculating on the future implications of algorithms. Dr. Berlinski is actually quite good in this area, and his writing on this aspect of the book is spare and effective. He also seems to have good ideas. Clearly, as new users of algorithms in the last few decades, we are only beginning to understand their implications. What a wonderful opportunity to inform our next quest! But alas, it was a too short trip in this book.
I was tempted to grade the book down one more star for unnecessarily complicated explanations of symbolic logic conventions and of mathematical functions. I studied these areas in college, and the explanations I received then were much simpler, shorter, and clearer. Many people who are unfamiliar with these subjects will find the material here unnecessarily confusing. But then I realized that writing a book about algorithms is a brilliant concept, deserving of a star for itself. This negative and this positive canceled one another out, leaving me at 3 stars. Now your understand the algorithm I used to rate the book.
The stories of Liebniz, Godel, and Turing were especially interesting for me. These were quite well told. If you cannot bear the whole book, be sure to look at these. You can use the index to find them.
I hope someone with a better sense of what the reader would like to examine and how to communicate will follow this book with a better one on this subject.
If you decide to read this book, despite its drawbacks, please use the experience to think about how you can simplify important communications in your life so they will be better understand and acted upon.
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