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on October 31, 2005
At the time that I ordered this book, I had a natural inclination to be sympathetic with its author, since his reputation indicated that he and I had similar views about politics and the philosophy of science. That only increased my disappointment when this ended up being one of the least enlightening and most annoying books I've ever encountered. If Berlinski is as talented as I'd been led to believe, it's hard not to interpret _Infinite Ascent_ as either some sort of practical joke or a rush job to fulfill a contract.

In _Infinite Ascent_, Berlinski has a tendency to wax grandiloquent, using metaphors and similes that serve no evident purpose and are sometimes downright bizarre, as when, for example, he likens sets and their elements to the male anatomy (p. 129). Following this up one page later with Berlinski's fantasy about schoolgirls with "their starched shirt fronts covering their gently heaving bosoms" (p. 130) does nothing to ameliorate concern about the author's tendency to get distracted.

One of Berlinski's running themes is the use of "..." in mathematics to represent the continuation of a pattern. He likes to joke about this so much that he starts inserting these dots in his formulas needlessly, just to get to comment on them. For example, instead of just writing down the (extremely short) formula for subtracting complex numbers (p. 69), he leaves an ellipsis and then states that "the crutch of three dots [covers] the transmogrification of a plus to a minus sign and nothing more."

Some of Berlinski's comments are real head-stratchers: "[The Elements] is very clear, succint as a knife blade. And like every good textbook, it is incomprehensible." (p. 14); "[Exponential functions] mount up inexorably, one reason that they are often used to represent doubling processes in biology, as when undergraduates divide uncontrollably within a Petri dish." (p. 71). Huh?

_Infinite Ascent_ has few formulas or other concrete mathematical details, and what there is is often wrong. The formulas for the solutions to quartic equations of quadratic type are botched (p. 93), roots of equations are confused with zeros of functions (p. 80), inscribed rectangles are described while circumscribed rectangles are drawn (p. 56), and g12*du1*du2 is misidentified as a formula for the infinitesimal distance between the points u1 and u2 (p. 120). The sections on logic are the ones Berlinski handles most competently, but even that has been covered better by many others.

Berlinski thinks that Weierstrass's definition of limit is "infinitely wearisome" (p. 145) and is "promptly forgotten" by mathematicians after they have learned it. I think most analysts would disagree strongly with his opinion, and would classify the definition of limit among those things they couldn't forget if they wanted to. (That Berlinski himself very well might have forgotten it is suggested by his unconventional decision to use the letter delta to represent a *large* index (p. 61) in his definition of the limit of a sequence.)

Berlinski opines that the Fundamental Theorem of Calculus (connecting differentiation to definite integration) is something that "no one at all would expect". On the contrary, I consider it to be eminently plausible. Berlinski also describes the classic math book _Counterexamples in Analysis_ as consisting of "a series of misleading proofs supporting theorems that are not theorems." _Counterexamples in Analysis_ actually contains nothing of the sort. Rather than containing fallacious "proofs" of non-theorems, it contains exactly what its title says it does: Counterexamples (i.e., examples that show why the hypotheses of (true) theorems are necessary and why stronger conclusions are unwarranted).
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on November 10, 2005
It's difficult to determine whom this book is addressed to. A lay reader will come out none the wiser after reading the chapters on complex numbers and groups. Just dressing up powerful general ideas in vague, mystifying, and allusive prose serves no purpose. For instance (p.81) he refers to the heart-breaking charm of complex analysis. Yeah, so? These statements don't edify a lay reader. The same can be said for the discussion of Lie groups (pp. 100-101).

A mathematician on the other hand, will find the book redundant, and annoying -- both for its inaccuracies and general, loose vagueness.
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on May 27, 2007
Just for the record, Paul Dirac was British, not French as asserted by Berlinski on page 8. Dirac was born in Bristol and held the Lucasian Chair of Mathematics at Cambridge and built the mathematical foundation for quantum electrodynamics. Dirac has been written about extensively. It is amazing that a book that purports to be "a short history of mathematics" doesn't have anyone checking facts, proof reading, or editing. One loses interest after encountering a major flub so early in the book.
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on December 1, 2005
I've never come across such a work or author before. This author is plainly venting his passive-aggressive tendencies in this less than enlightening work. It took me until page 108 to finally figure what he's up to.

There he puts Euclid's axioms in such a format as to be quite deliberately obscure. Then two pages later he suddenly jumps to measuring angles in radians, though he's never done it before and makes no statement that he's doing so. If you are not already ahead of him, you are lost. So it goes with the rest of the book. Meanwhile Berlinski stands to the side saying, "What did I do? What did I do? Oh, well, perhaps you should read something simpler if you cannot follow me."

Berlinski is plainly a person of wit and intelligence. Alas, he's allowed another side of his persona to pop up here.
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on December 28, 2009
"The sum total of what a man knows is vanishingly small. What seems in the end more important is that one should pursue knowledge." Bertrand Russell

Discussed mostly are mathematical things which will always be beyond me. But, to my credit, I keep trying to understand them. To David Berlinski's credit he keeps trying to explain them. (Yes, he can be obtuse--but can anyone make some of this stuff clear? . .Yes, he can be flippant. . .Yes, he can be arrogant. . .) (He can be refreshing and funny, too!) But instead of finding fault with any of the presentation, I would rather praise him for bringing me some insight and for bringing me somewhat closer to understanding. He ends this book with a reference to the tingle that mathematicians sometimes sense. That, perhaps, is how Berlinski should be judged; and there are times this book tingles.
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on October 17, 2005
A dazzling series has taken a bit of a misstep in Infinite Ascent, A Short History of Mathematics by David Berlinski. Like others in the series it is short but, unlike the rest, it does not make its subject very readable for anyone not already well-versed in the topic, which is a primary joy of this series of books on such varied topics. Granted, this is one of the more difficult areas to cover in such a short span of pages and, at times, the author does dramatically bring out the passion of the subject and the various historical figures throughout the book in an entertaining fashion, he just as often overwhelms the general reader with difficult information lacking an understandable context and, what is far worse, makes the situation wholly untenable by interjecting rather lame humour reflecting his own social and political issues and not being in any way helpful in illuminating an already complex subject. A rather tough and joyless read with occasional flashes of heart when a character from the past such as Galois or Gobel enters the story. Definately not a book for the casual reader.
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on February 26, 2014
Other reviewers have pointed out the excesses in the use of metaphors and other uses of language in this book. In addition, there are a fair number of errors, some of them egregious, as also noted in reviews. The book is written, as well, at a fairly superficial level. Unlike many reviewers, I regard these deficiencies as more than balanced by the presentation of history in the book. In my view, the noted negative points are sufficient to give the book a somewhat lower rating than five stars. On the other hand, I like the way the book was structured, and I felt that, for such a vast field as mathematics has become, Dr. Berlinski has encompassed some important and characteristic aspects of mathematics. He transmits this information in a short book that is easy to read and understand. I thought he gave a pretty good picture of various topics, and I liked his discussion of modern mathematics at the end of the book. Some of what he writes borders on the brilliant. Such a short history obviously needs to be supplemented with other readings and studies, but I enjoyed reading the book, and can recommend his treatment of history as insightful.
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on August 7, 2009
This book irritated me so much that I'm going to just rant and abandon any hope of writing a 'helpful' review. I picked the book up on a remainder table for $1, and I clearly paid too much. Berlinski doesn't appear to know any more than a smattering of mathematical history (or math for that matter), but he dearly loves to hear himself talk as if he does. His flowery, grandiloquent delivery style accomplishes nothing remotely useful. An example from Chapter two: "Not until the twentieth century would mathematics and logic, having for so long exchanged their moist breath (sic), fuse ecstatically into the single subject of mathematical logic." Come on - "moist breath"?, "fuse ecstatically"? Nothing is being accomplished here except for the author going on a linguistic ego trip. Other reviewers have commented on the sloppy fact-checking and needlessly obscure presentation of the mathematics, and I agree with every one of those comments. Don't waste your time or money on this clunker.
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on February 16, 2006
I like poetry, but I don't think it works well when writing a book on mathematical history. The point of this book seems to be the showcasing of the author's formidable command of language. OK - now that we know how great the author is in his use of language - what's left? There is some good information, but nothing so extraordinary that it merits the work you have to go through to glean it. SUMMARY: Long on beautiful language, short on information.
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on March 21, 2008
In this book David Berlinski takes you on a trip through the ages into the hidden mystical history of mathematics. The book is relatively short so Berlinski gets right to the point using Biblical like precision and compactness in the array of language he chooses to employ. This book is for anyone who wants to know and understand the history of mathematics from a deep intuitive perspective but doesn't want or need to learn mathematics itself. Belinski does the finest job here of simplifying the concepts of advanced mathematics to such a degree that any high school graduate (with a dictionary handy) can ascertain their meaning and importance. A concise history of mathematics is this book.

For the more "self declared" intelligent and learned out there, this book does an excellent job describing in poetic detail the significance of the story imbedded in the enigmatic history of mathematics, as well as explaining the significance of math itself. Berlinski uses a wonderful bouquet of words that will help to titillate even the brainiest and experienced of readers. If you are looking for a little mathematical enlightenment and wouldn't mind a transcending journey, be prepared to expierence Infinite Ascent!
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