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0 of 1 found the following review helpful:
 Good only for massive "skip" reading May 04, 2010
Most boring, confusing and appalling book with all those dramatic and romantic story sections. May be if it was an audio book, I would have given some half star extra. Please make your next book precise and crispier if you want to write about serious stuff. If not stick to romantic novels or scifi's.
1 of 1 found the following review helpful:
 they don't get him Apr 18, 2010
David Berlinski, like God, is under-appreciated. He is too subtle, too wonderful, and mostly beyond us.
1 of 1 found the following review helpful:
There is a time and a place for everything Feb 18, 2010
I think I have had a normal relationship with this book. The first time I had a copy in my hands, I quit reading it. That was around 2000.
Tonight, it is brillant. This is around 2010. What changed?
Well, I have. And my need for the information has. This is not a book about algorithms. It is a book about the idea of an algorithm and where that idea came from, historically and logically (from a system of mathematical logic) that well, "was invented".
The book is what it is. It is a beautiful "Tour de Force", but only if you want to take the tour.
5 of 6 found the following review helpful:
David Berlinski on Science and Things Beyond Science Nov 07, 2009
(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 issued the prediction 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, making an attempt 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, is 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 sentence 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. 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 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 1 found the following review helpful:
 Good Read Jul 21, 2009
Berlinski is trying to make us feel the conceptual twist that percipitated the algoritm out of the failed project to produce a secure foundation for mathematics. I'm not sure what the point of this is. The two great ideas (calculus and algorithm) don't entirely work for me, but still, it's a nice piece of history with a zany cast of logicians. As for the Berlinkski character, with his digressions and hyper-prose, these literary devices are not pointless, but a disussion of them would be a different review.
Also, I think that the section on Church and his lambda calculus was a distraction, a homage to his thesis asviser, even though Church was admittedly a great logician.
Finally, I would have liked to have seen a little more on the need for mathematical certainly and the larger "modernist" context.
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