Ever wonder whether some of the greatest stories in mathematics are true? As an instructor I often include stories like the drowning of Hippasus for revealing the irrational numbers to liven up my lecture and get students to question the “facts” that we learn in mathematics. In The Cult of Pythagoras Math and Myths, Alberto Martínez presents a historically accurate discussion of the many great stories in mathematics. As the title suggests Martínez discusses Pythagoras of Samos and the discovery or proof of the hypotenuse theorem. It is surprising that even though grade school students call the hypotenuse theorem the Pythagorean Theorem there is little historical evidence that Pythagoras has anything to do with its proof or discovery. Although the book contains an interesting discussion relating to this theorem it tends to get a bit redundant and is not my favorite part of the book.
Fortunately the author also discusses many other stories in the history of mathematics including Archimedes solution to the weight of a golden crown and Gauss’s childhood achievement of adding numbers one to one-hundred. Many of these stories have been told inaccurately by many mathematicians over the years, including myself. One reason some of the embellishments on these stories persist is that they are entertaining. My Calculus class always chuckles when I tell the story of Archimedes running naked though the streets yelling “Eureka, Eureka” when he determines how to solve the problem.
In addition to stories, this book contains an interesting discussion of some of the mysteries of mathematics. For example, what happens when you divide by 0? Does 0.9999… = 1? Why do we define algebraic operations with negative numbers the way we do? These topics would be an excellent supplement for a course on mathematics for teachers. Many of the questions discussed in the book are strikingly similar to discussions I have had with students when they were first introduced to these topics. For example, with the negative integers one can logically explain the necessity of these numbers to represent concepts such as debt. However, is there a physical meaning to the rule that when you multiply two negative numbers you get a positive number? In fact the author shows that much of algebra would remain the same if you changed this rule. The most substantial difference is the loss of distributivity. As an algebraist I found this discussion extremely interesting, but what I liked most was that it was completely accessible to anyone. No special math background was required; as long as you are familiar with the basic rules of algebra this discussion is clear.
I think this book would be great for any undergraduate student. Martínez comments that by telling young students through their math training that rules dictate algebra, and they should accept these rules as unchanging facts stifles some creativity. Perhaps mathematics students would struggle less with topics in courses such as Abstract Algebra if we encouraged creativity and exploration with these rules at an earlier age.
The book concludes with the question “What is math?” This question seems like it should have a straight forward answer but as Martínez suggest the answer might not be clear at all. We might describe math as a tool that helps us to describe physical phenomenon that we observe. However one question presented was whether the objects in math actually exist. Was mathematics invented or discovered? Can we find a perfect circle or π in nature? Does mathematics exist independent of our imagination, or is it just a set of symbols, formulas, axioms and definitions that once discovered we later found uses for?
I really enjoyed this book and think it would be a worthwhile read for anyone, but especially a student interested in the history behind the rules mathematicians now take for granted.
Ellen Ziliak is an Assistant Professor of mathematics at Benedictine University in Lisle IL. Her training is in computational group theory. More recently she has become interested in ways to introduce undergraduate students to research in abstract algebra through applications.