John Derbyshire set for himself a daunting task in writting Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics: to bring the complex world of math down from its ivory tower and present a glimpse of its magic to the laymen reader. For his challenge Derbyshire picked a riddle, the Riemann Hypothesis (RH), that has been tantalizing mathematicians for over a century; a conundrum that Derbyshire labels as the â€œgreatest unsolved problem in mathematics.â€ In Prime Obsession, he not only attempts to tell the story of Bernhard Riemann and his famous hypothesis but to communicate the complex and high level math involved down in such a way that a laymen reader might glimpse its meaning. It is to his immense credit that Derbyshire makes this interwoven tale of math and history both interesting and illuminating.
A description of the RH should reveal the challenge Derbyshire faced:
All non-trivial zeroes of the zeta function have real part one-half.
Huh? As I said, it is a daunting task to attempt to unpack this dense subject so that someone outside a graduate program in math can understand it. Obviously, the author was well aware of this challenge when he set out:
To an ordinary reader, even a well-educated one, who has had no advanced mathematical training, this is probably quite incomprehensible. It might as well be written in Old Church Slavonic. In this book, as well as describing the history of the Hypothesis, and some of the personalities who have been involved with it, I have attempted to bring this deep and mysterious result within the understanding of a general readership, giving just as much mathematics as is needed to understand it . . . I think I have pitched my book to the level of a person who finished high school math satisfactorily and perhaps went on to a couple of college courses.
I can speak to this issue with some personal experience. After all, the last math class I had was as a sophomore in high school and, as I recall, I didnâ€™t do that well. Despite a medium to high level of math phobia, I was able to work my way through (i.e. read some sections whole, skim others) the math and follow the arc of the story and the math. I wonâ€™t pretend I â€œunderstandâ€ the RH but I have a greater appreciation for the concepts and, in fact, know a lot more about math in general as a result. All of this highlights the fact that the author is a patient, witty, and gentle teacher. If he hadnâ€™t been, I wouldâ€™ve given up on the book very early.
To spread the pain, Derbyshire alternates between math heavy and history focused chapters throughout the book. The result is a unique blend of careful calculation and logical building blocks with the history of mathematics during the last 150 years. In its basic form the book tries to build up the readerâ€™s foundational knowledge to the point where he can understand the â€œwho, what, and whereâ€ of the RH. The â€œwhoâ€ is Bernhard Riemann and a number of other key mathematicians of the last century and a half. Derbyshire tries to give the reader a glimpse into the life and mind of this quite, shy, and pious visionary not just to provide the back drop to his famous hypothesis but also, I think, to extol his virtues as a man and a scholar. Part of this backdrop includes Riemannâ€™s mentors, contemporaries, and the men who took up the challenge he discarded almost casually in his 1859 paper. Derbyshire touches on a host of interesting men and relationships: Carl Friedrich Gauss and his relationship with the Duke of Brunswick; Leonhard Euler – author of what Derbyshire calls the â€œGolden Keyâ€ â€“ and of Russia under Peter the Great; the interestingly named Pafnuty Lvovich Chebyshev and â€œChebyshevâ€™s first Resultâ€; Jacques Hadamard and Charles de la Vallee Poussin and their simultaneous but separate work proving the Prime Number Theory; and the inenigmatic David Hilbert who spoke the inspirational words: â€œWe must know, we shall knowâ€ (in German). In other words, after having read this book you will be familiar with most of the important mathematicians of the last 150 years and their contributions to the field.
The â€œwhatâ€ is of course the RH and the concepts that are needed to understand it. As noted above, Derbyshire outlines these concepts with wit and pace despite the heavy subject. The chapter titles give a hint of his tone: Nine Zulu Queens Ruled China, The Argument Ant and the Value Ant, Big Oh and Mobius Mu, and the aptly titled A Little Algebra. Those whose better math days are behind them â€“ or who never had better math days â€“ will enjoy Derbyshireâ€™s brevity and straightforwardness. Those with a deeper knowledge and love of math will surely enjoy his guided tour through the increasingly complex topics as he leads you toward the payoff.
The â€œwhereâ€ is the history in which all this takes place. What makes Prime Obsession so interesting is the way it blends the math with the history so that one complements the other. Luckily for the reader, Derbyshire is just as skillful at historical sketches as he is at unpacking advanced math. It takes a skillful hand to provide historical background and important details without becoming tedious or distracting the reader from the main story. Having a background in history myself, I was impressed with his ability to not only set the stage but also bring the period to life.
As I hope I have made clear above, Prime Obsession is a unique work. The math involved is serious and difficult but presented as cleanly and as simply as is possible. The history is fascinating and neatly tied into the ever-progressing math. I would recommend this work to anyone with a love of math. The larger your knowledge base in the subject the greater ease with which you can work your way through the math; and likely the greater enjoyment you will receive. For those with less knowledge, the history and personalities remain fascinating but the math will be a challenge. Like most things in life, however, you get what you put into it.
Be sure to check out my interview with John here.