Magnificent Mistakes in Mathematics

We all learn from our mistakes (and not only the mathematical ones). This is one of the reasons these authors wrote the book. By showing what kind of mistakes can be made in mathematics, and to what absurd conclusions that may lead, it is hoped that the reader understands better the rules of the game and be more careful in jumping to conclusions.

However, we may learn not only from our own mistakes. There have been many historical mistakes, made by leading mathematicians. The first chapter is a collection of such examples. Pythagoras was mistaken when he thought that nature could be completely explained with natural numbers and their ratios. There have been historical mistakes in the calculation of $\pi$, and many wrong attempts have been made to prove Fermat's last theorem, Goldbach's conjecture, or solve the 4 colour problem, and many other such famous problems. Galileo, Euler, Fermat, Legendre, Poincaré, Einstein, they all made mistakes and often in published papers. Gauss seems to be a glorious exception to this rule. No errors are known in his published papers. This chapter is an enumeration of summaries of these historical errors, although a complete book could be devoted to each of them, why and how the wrong conclusion was made and what kind of research this has started. For example, a wrong calculation of a notorious gambler Chevalier de Mérimé caused him to loose repeatedly. He asked Pascal to explain what seemed to him a paradox, and the correspondence between Pascal and Fermat on this problem can be considered to be the start of probability theory. And we all know that the attempts to prove Fermat's last theorem has given rise to a many new mathematical results.

The subsequent chapters discuss arithmetical, algebraic, geometrical and statistical mistakes. Here we find many obvious errors that are commonly committed by students like division by zero, or violating the rules of distributivity ($\sqrt{a + b} = \sqrt{a} + \sqrt{b}$ or $\frac{a}{b}+\frac{c}{d}=\frac{a+c}{b+d}$ and the likes). Also jumping too soon to a general conclusion is common. Infinite sums, and working with $\infty$ often leads to false results. Of a somewhat different nature are the rounding errors of digital calculators or computers that may play dirty tricks on us. Sometimes wrong logic may lead to correct answers. Arguments are then needed to convince the student of the low score notwithstanding the fact that the final result was correct. All these are familiar and teachers are desperate when students keep persistently sinning against them. Many examples of mistakes can however be reduced to the same error made under a slightly disguised form. So there is basically a lot of repetition which makes these chapters a bit dull from time to time for readers that are well beyond these rookies mistakes. There are however also errors that are counter intuitive or that have some pitfalls and that are often used in quizzes or to astonish the innocent reader with an apparent paradox. For example suppose the earth is a perfect sphere. Put a rope around the equator and enlarge it by 1 meter. Then keep this longer rope at an equal distance above the surface. Can a mouse pass under the rope? Our intuition says no. However, computation reveals a quite different result. The cat can pursue the mouse at the other hemisphere passing below the rope quite easily. Amusing and puzzling examples are the optical illusions and impossible figures in a geometrical context. A famous geometrical missing square puzzle showing that 65 = 64 is attributed by the authors to Lewis Carroll, although the principle is much older. It has happened that mistakes were deliberately introduced as a prank. Martin Gardner presented in his April 1975 column of Scientific American a map that would require 5 colours, which turned out to be an April fools joke on his readers. Stories like this and the more recreational pitfalls will keep you reading to the end.

All the mistakes are discussed, but sometimes this is not really deep, and sometimes it feels like there is still an untold story behind. The broadness of the examples that are covered (the examples mentioned above are just a tiny sample from a vast set) prevents the inclusion of further details, but an author like Martin Gardner for example could have written a full column including the history, background, generalizations and variations on some of the issues, while here it's more like a sequential enumeration, i.e., a catalog, of the same. All historical facts e.g., are essentially restricted to the first chapeter. In brief: the authors present a not always very deep, but a broad and diverse collection of examples of what people can, and unfortunately often will, do wrong when playing with mathematics. May the reader be wiser after finishing this book.

A. Bultheel
KU Leuven
Book details

It's mistakes galore. Mistakes of all ages and of all sorts and in all areas of mathematics, logic, and statistics. Historical mistakes made by great mathematicians (yes, they are not immune). Mistakes regularly made by students (the typical beginners mistakes much to the exasperation of their hopeless teachers). Mistakes made on purpose for the fun of it (mistakes with a pun). Easily mistaken results because they are counter intuitive (the puzzling ones). You will certainly meet some that you made yourself, but many others that you did not even think of.



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