The Poincarè Conjecture
by Donal O’Shea
Writing popular books on mathematics is tough. There’s not much sex or violence, and the academic battles range from ethereal to esoteric. Many a reader will stop after a few chapters and decide that not only don’t they understand the math, but they care even less about the mathematicians.
Still, author O’Shea takes a brave stab at explaining the Poincarè Conjecture, its background, and its proof. The highlight of the book is the lucid and enjoyable account of the history and personalities of the field, particularity the subtle battle between Felix Klein and Poincarè. The fight unfolds slowly with an exchange of seemingly polite letters with small, sharp knives buried between the lines. Klein was a genius, but Poincarè a greater one, and both shaped 19th century mathematics. Between the two, topology became an important field, and Poincarè left us with one maddeningly simple looking problem that took over a century to solve – what are the possible shapes space can take on?
Questions like this don’t resonate well with the general public, mostly because you need a few advanced degrees to even complexly understand the problem. While O’Shea insists anyone with high school geometry can follow the arguments, I found his exposition occasionally opaque. That’s no fault of his; the problem is subtle, and he scrupulously avoids equations lest they scare the faint of heart. For those interested, he leaves 70 pages of footnotes, references and glosses, but I doubt you will explain this problem to your golf buddies after reading this book. What you might garner is a better understanding of how messy this rarefied field can be, and what exactly is meant by the word “proof”. Despite what you might think, mathematics is a field filled with politics and ambiguity.