David Berlinski

An Ill-Conceived Practical Joke?

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 show more 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). show less

On the wonderful PBS Kids math show called "Odd Squad", there's a character named Obfusco, whose shtick is ostensibly that he speaks in "word problems" but whose main characteristic is really that he is always using similes and metaphors that are bizarre, opaque, or meaningless.

As I read Berlinski's first essay in this collection, I was reminded that he and Obfusco are blood brothers, and you have to be willing to overlook this quirk if you're going to get anything out of his writings.

I thought I was getting some in-depth insight into calculus, technical insight I missed in my college course. I did get that, but got a lot of back-story re: the history of the development of calculus, which was usually pretty interesting, and also lyrical, allegorical musings on the nature of functions, limits, and calculus itself, which I had to be in the mood for and frequently was not. This was my fault, the author writes very well, and has clearly thought very deeply about his subject, show more but I was looking for more math and less philosophy.

I thought some of the proofs could have been more clear (for example, on page 215, why is f(a) a constant?), but in general I could follow along well enough not to lose the thread. show less

Whose side are you on, anyway?

As someone with a degree in mathematics, I of course know something of Euclid, but mostly for his contributions to number theory (an area of mathematics that Euclid didn't even have a name for). As far as geometry was concerned, I knew of his work, but didn't know much about it. And, of course, it is for geometry that Euclid is primarily known.

So let it be said, for starters, that this is not an introduction to Euclid. You'll learn some things about Euclid's show more system of postulates, but very little about the rules of logic that Euclid used, and only a tiny handful of results are covered.

The first part of the book, in particular, seems to have a peculiar love-hate relationship with its material. Author Berlinski is correct in stating that Euclid made more assumptions than he states -- in effect, that his system of postulates and axioms is incomplete. So what is the solution? The solution is to bring in the additional postulates that are needed. This is never done; Berlinski eventually gets around to David Hilbert and his formalization of Euclid, but we don't learn much beyond that Hilbert did so. We neither learn what Euclid left out nor what we need instead.

I'm also a little bothered that Berlinski just lays into Euclid without considering the matter of language. Euclid wrote in classical Greek, not English, and words in Greek do not automatically mean the same in Greek as in English. Take, as a very elementary example, the verb "to be" (is, are, am, was, were, etc.). In English, the use of this verb it is mandatory, even in the present indicative. In Greek, it is not. So in English, you say "This is geometry." In Greek, you can say "This is geometry" or merely "This geometry" -- or, possibly, depending on context, "is geometry" without "this." And which form you use affects, if not the meaning, at least the emphasis. "Is," in Greek, is a much stronger verb than in English. And "is" can mean an equality, so the exact use of "is" can result in different meanings in Greek and English. There are many, many other places where Greek and English words do not precisely align. It is not fair to go after Euclid's failures based on English translations of Euclid without reference to the Greek. How much this matters I don't know (I assuredly don't have enough Greek to understand Euclid in that language!), but it irked me a lot.

Oh, and by the way, Euclid did not know that the interior angles of a triangle add up to π. Euclid didn't measure angles in radians -- and most people don't, either; we measure angles in degrees. Euclid didn't even know that π was an irrational number, or call it by that name. Measurement in radians, as I understand it, arises from the unit circle, not geometry. Please, Dr. Berlinsky, if you're going to talk about the interior angles of a triangle, either measure them in degrees (which everyone knows) or at least say you're doing measurements in radians.

The latter part of the book is devoted to non-Euclidean geometry -- that is, the geometries that emerge if one decides that Euclid's infamous fifth postulate (the "parallel postulate") is not correct, or at least is not axiomatic. This is a vital result -- the universe is non-Euclidean -- but I felt that this was a bit weak, too. There are at least four ways to address the parallel postulate, which basically says that if you have a line and a point not on the line, there is only one parallel line that passes through the point. One can assume that the parallel postulate is correct, and there is only one parallel line (Euclid's approach). One can assume that parallel lines are more than one (probably infinitely many). One can assume there are none. Or one can just say, "I won't assume any of these; I don't know what a parallel line is." These produce different non-Euclidean geometries. I really wish this had been clearer.

If Berlinski had just done a history of gripes about Euclid, it would have been a useful book. If he had done a real job of giving the information Euclid lacked or didn't understand the need for, it would have been a very useful book. As it is, I feel as if this falls between two stools. I spent far too much time saying, "Tell me more" (and, once in a while, "Tell me less"). Almost never did I feel like what I read was just the right amount. show less