62 of 68 people found the following review helpful:
2.0 out of 5 stars
An Ill-Conceived Practical Joke?, October 31, 2005
This review is from: Infinite Ascent: A Short History of Mathematics (Modern Library Chronicles) (Hardcover)
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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30 of 35 people found the following review helpful:
2.0 out of 5 stars
Interesting But Flawed, November 10, 2005
This review is from: Infinite Ascent: A Short History of Mathematics (Modern Library Chronicles) (Hardcover)
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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18 of 23 people found the following review helpful:
1.0 out of 5 stars
passive-aggressive math?, December 1, 2005
This review is from: Infinite Ascent: A Short History of Mathematics (Modern Library Chronicles) (Hardcover)
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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