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Magnificent Mistakes in Mathematics

Prometheus Books
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This is a “mathematics appreciation” book, consisting of numerous mathematical vignettes, all of them accessible to people with little (say, high-school level) mathematical training, and all of them loosely — sometimes very loosely — organized under the theme of mathematical mistakes. Accuracy of the title requires, however, that all three terms “magnificent,” “mistakes,” and “mathematics” be interpreted fairly broadly.

When I first heard that I was going to be reviewing this book, I idly amused myself by trying to guess what mistakes would be covered. I had assumed from the title that the book would concentrate on famous mistakes in mathematics that, although incorrect, nevertheless led to enhanced insight into, or development of, the subject. In other words, I thought that this book would do for mathematics what Livio’s book Brilliant Blunders: From Darwin to Einstein — Colossal Mistakes by Great Scientists That Changed our Understanding of Life and the Universe apparently (based on the description; I haven’t read it) does for science.

The one example that jumped immediately to mind- perhaps because I had just finished reviewing a book that approached abstract algebra from the perspective of Fermat’s Last Theorem — was Lamé’s famous incorrect proof, using factorization in the ring of cyclotomic integers, of that result; the error was the assumption of unique factorization in a ring that did not (necessarily) have it, and that error is directly related to the development of algebraic number theory.

In fact, however, my assumption was incorrect; the book’s focus is on much more elementary mistakes in mathematics. Lamé’s error is mentioned in this book, but only in passing: while the authors discuss Fermat’s Last Theorem — the primary “mistake” here being Fermat’s likely error in assuming that he had a proof — it is stated that Lamé thought he had a correct proof, but didn’t; the details of his mistake are not provided, presumably because they would be tough going for the book’s intended audience. Most of the mistakes referred to in this book, in fact, are not the kind of errors that significantly advanced the development of our subject, but are instead of the easy “brain teaser” variety, the analysis of which could easily show up as an exercise in a college mathematics text.

The first chapter of the book is entitled “Noteworthy Mistakes by Famous Mathematicians”, and recounts about 20 errors. The discussions range from a few paragraphs to a few pages in length, and vary also in quality and interest. In some cases, the mistake in question is discussed substantively, as in, for example, Chevalier de Méré’s mistaken idea that the probability of tossing a six on four tosses of a single die is the same as the probability of tossing a double-six on 24 tosses of a pair of dice. This mistake actually did prompt serious mathematical discussion and helped advance the modern theory of probability.

Unfortunately, however, in a number of the vignettes of this chapter, the notion of “mistake” seems almost an afterthought, an excuse for shoe-horning a discussion of some interesting mathematics into a book entitled Magnificent Mistakes. For example, there is a brief discussion of Catalan’s Conjecture — which turned out not to be a mistake; the conjecture was eventually proved — in which the authors state that there “were many mistaken attempts at proving this conjecture”, and leave it at that. Likewise, the authors mention several currently open problems in number theory (including Goldbach’s conjecture, the twin primes conjecture and the Collatz conjecture) and simply say that mistakes have been made in attempting to prove these. There may undoubtedly have been a number of mistaken attempts at proving these results, but absent some indication of what the mistakes were, it remains unclear what value, if any, these mistakes had.

Some other mistakes mentioned here are rather dry and technical. We learn, for example, that Legendre mistakenly asserted that there do not exist positive rational numbers r and s satisfying r3 + s3 = 6, but in fact the equation does have the solution r= 17/21, s= 37/21. That’s nice, but other than the interest of seeing that even somebody of Legendre’s skills could be wrong about something, why would anybody care about this? The authors don’t say.

Finally, at least one of the mistakes mentioned in this first chapter is nothing more than a typo. Enrico Fermi once wrote down an equation on a blackboard in which he interchanged two symbols; unfortunately, this was photographed in 1948 and that photograph then wound up on a postage stamp (actually two postage stamps — the authors reproduce a picture of a 2001 American stamp commemorating the one hundredth anniversary of Fermi’s birth, but there is also an Italian postage stamp with the same error appearing on it. You can see them both at This was an amusing bit of trivia that I hadn’t known about previously, but I would hardly call it a “magnificent mistake”.

The rest of the book is divided into four chapters, discussing mistakes in each of the areas of arithmetic, algebra, geometry and probability, respectively. The mistakes mentioned here are, as previously noted, quite elementary (lots of illustrations, for example, of why it’s not good to divide by 0, or to forget the possibility of negative square roots), and are comparable to the errors discussed in the book Paradoxes and Sophisms in Calculus by Klymchuk and Staples, reviewed some months back in this column. A number of the mistakes discussed here consist of obviously false results (1=0, all triangles are isosceles, etc.) with so-called “proofs” containing hidden flaws in them; the geometry chapter, in particular, has quite a few of these. Many of them could probably be used as fodder for homework assignments or brief comments in lectures. However, one of the really historic mistakes in geometry (the mistaken idea that the Euclidean parallel postulate can be proved as a theorem) is unfortunately not mentioned.

In these four chapters, as in the first one, I sometimes had the impression that the authors were stretching things to bring something within the ambit of the book’s title. For example, about half a dozen mistakes recounted in the chapter on geometry were nothing more than standard optical illusions, including, for example, the old chestnut of determining which of two lines (both the same length, but one with arrows at the end pointing in and the other with the arrows pointing out) is longer. I’ll grant that this arguably is an example of a mistake, but I’m not even sure how it amounts to “mathematics”, let alone “magnificent”. And, in the chapter on probability, there were a few examples that were completely divorced from mistakes: at one point, for example, the authors devote a short section to questions like “how many ways can the sum of two prime numbers add up to 999?” They correctly point out that for the sum of two integers to be odd, one must be even and the other odd; since the only even prime is 2, the only possible sum is 2 + 997. This is a cute, albeit pretty obvious, example and I can see using it as a homework problem in an introductory course in abstract algebra or number theory, but where is the “mistake” here? If it is “not thinking before you attempt the problem”, then just about any example of mathematical reasoning could be used as an illustration.

I thought the last chapter, on probability and statistics, was probably the most successful and in keeping with the promise of the book’s title. Probability is notorious for being counter-intuitive, of course, and several mistakes in this area actually have had the effect of advancing substantive knowledge. There were nice discussions in this chapter on the birthday and Monty Hall problems, as well as other mistakes, the analysis of which enhance the reader’s understanding of combinatorics, probability, or statistics. For example, the authors point out how to count the number of ways of filling in a mini-Sudoku board (with four rows and columns rather than the traditional nine). Similar, but much more complicated, reasoning can be used to determine the total number of possible “full” Sudoku puzzles.

I do think the authors missed an opportunity in this chapter, though, by not mentioning some of the well-known cases in which probability has been misused in court. See, for example, the first two chapters of Math on Trial by Schneps and Colmez, for discussions of how the probabilities of non-independent events were multiplied in court in two criminal trials, quite possibly to the detriment of the defendants (although one never knows just how much the jury was influenced by the mathematical errors, a point that was made by Paul Edelman in his review of that book in the August 2013 issue of the Notices of the AMS.)

By and large, I thought the level of exposition in the book was quite high. The material was, generally speaking, presented at a level that could be understood by the intended audience of mathematically-unsophisticated people, even high school students. On one or two occasions, I had some concern with the way things were done, because I thought the discussion actually had the potential to misinform. For example, the authors pose the following question, to which they offer both an incorrect solution and a correct one: consider a machine “that contains countless non-transparent balls”, every fifth one of which contains a $5 bill; the others have no money in them. If three balls are chosen simultaneously, what is the probability that none of them contain any money? The “correct” answer provided by the authors is (4/5)3, which is correct only when the number of balls is literally infinite, which of course is not possible in practice. When the number of balls is finite the answer is not (4/5)3, but the discussion may lead some people to believe otherwise. At the end of the discussion the authors do give the correct answer when the total number of balls is five but they fail, I think, to adequately explain just where the “countless” assumption is used and why it is necessary.

These kinds of issues were the exception rather than the rule, however. Because of the generally good writing style, and also because of the short length of most of the vignettes presented here (few exceed three pages), this book might prove useful as leisure reading by a bright high school student or by a teacher looking for an interesting thing to talk about in class or assign as a homework problem.

The book could, however, benefit from a better table of contents. The current one lists only the five chapters and their titles and makes no effort to list the individual errors. That may be understandable, because there are lots of these, but some effort to provide a listing would, say, facilitate the efforts of a teacher looking for a particular example to use in class, especially since the various vignettes seem to be randomly strew around. (There is, admittedly, a decent index, but that doesn’t seem to be as useful as a table of contents would be.)

Summary: although I think the title of the book should be taken with a huge grain of salt, there are some interesting things to be found here. I doubt very many people will read this book cover-to-cover, but it is certainly worth flipping through.

Mark Hunacek ( teaches mathematics at Iowa State University, and is grateful that nobody has ever photographed some of the things that he has occasionally written on a blackboard.

Date Received: 
Friday, August 9, 2013
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Alfred S. Posamentier and Ingmar Lehmann
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Mark Hunacek
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Friday, August 9, 2013