- The Washington Times - Sunday, September 28, 2008

Talk of meltdowns and bailouts and bouncing stock market numbers are making mathematicians of us all. Not necessarily good mathematicians, but numbers in the news have a way of resonating with even the most word-bound.

Under the circumstances, Keith Devlin’s “The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern” could not be more timely. In its pages, largely centered on the pivotal correspondence between Blaise Pascal and Pierre de Fermat in the mid-1600s, one can gather insights on how the world of numbers works and how it allows us to plan with precision the most important events of our lives.

On the book’s cover, though not part of title or subtitle, just left of a portrait of a heavy-lidded Pascal, is the tout: “A Tale of How Mathematics is Really Done.” More to the point, the book is a tale of how mathematics is applied, that is, how the story of math over time has changed our lives. Mr. Devlin, who is the “Math Guy” on National Public Radio, takes us back in history to explore the early days of probability theory. For this math-challenged reader, the book was a bumpy but exhilarating ride. I predict, but cannot know for sure, that math junkies will love this book.

And prediction is, of course, the name of the game here. It all traces back to what the author describes as “a single mathematics document that changed the course of history.” Mr. Devlin identifies this as “the letter that Pascal wrote to Fermat in the seventeenth century that established modern probability theory,” but he was equally as interested in gaining “insight into how two of the world’s greatest mathematical minds grappled with a problem, stumbling, making ‘elementary errors,’ eventually being rewarded with flashes of brilliance.”

In the early pages of the book, Mr. Devlin writes, “On Monday, August 24, 1654, the famous French mathematician Blaise Pascal sat at his desk and composed a letter to his equally famous countryman Pierre de Fermat. When completed, the letter would come to less than three thousand words, but it would change human life forever. It set out for the first time, a method whereby humans can predict the future.

“Not in the sense of saying what will happen; no one can do that, not even if you are Blaise Pascal, a child prodigy who at sixteen wrote his first original paper and while still a teenager had invented, manufactured, and sold a mechanical desk calculator (the Pascaline) that largely anticipated the commercial desk calculators that became commonplace some three hundred years later. Rather, Pascal’s letter showed how to predict the future by calculating, often with extraordinary precision, the numerical likelihood of a particular event’s occurring.”

At the center of the Pascal / Fermat correspondence was the search for a solution to the “unfinished game” problem: How do you divide the pot when players are forced to end a game of dice before someone has won? The idea turned out to be far more groundbreaking than Pascal realized. From it came probability theory and a world of thought forever changed.

Before the correspondence between Pascal and Fermat, scholars generally agreed that it was impossible to predict something by calculating mathematical outcomes. As Mr. Devlin writes, “Events that were not in some way predetermined were thought to be beyond rational analysis. The future is a matter determined by God, people felt; what will happen will happen, and there is nothing anyone can do about it other than pray and show allegiance to whatever deity you believe controls your destiny.”

What Pascal and Fermat worked out was something we now know as risk management, a process that figures into the way we think about buying insurance or buying a house. As Mr. Devlin writes, “Managing risk is now fundamental to almost every aspect of our lives. Yet when you read the letter from Pascal to Fermat and see the enormous difficulty these two great mathematicians had in grasping the very idea of predicting the likelihood of future events, let alone how you do so, you realize that what we nowadays take for granted was a huge advance in human thinking that came only through significant intellectual effort.”

Mr. Devlin shares the great mathematicians’ correspondence, walks readers through critical mathematical problems and contextualizes it all in a lively narrative. The book is a refreshing testimony to the rewards of thinking rationally about how future events might unfold. It is what we wish for in lawmakers and economists at work today, and it is what we wish for in ourselves.

“The Unfinished Game” is a rewarding read, focusing not only on the struggles Pascal and Fermat overcame, but taking time to follow up their work with insights into the lives of those who came after them. There is the brilliant but bedeviled Girolamo Cardano, who helped pave the way to the eventual solution with the insight that the apportionment of the pot depended not on how many rounds each player had already won, but on how many each player must still win in order to win the contest. There is John Graunt, who developed techniques that moved Pascal and Fermat’s probability theory out of the gambling world and into the everyday world. Also, from the Bernouli family of mathematicians, there is Jakob Bernouli whose introduction of the concept of a “numerical degree of certainty” marked a significant step toward risk management.

And, finally, there is Thomas Bayes, who developed Bayes’ formula, a method for revising a probability in light of new information that is used today in everything from interpreting a seemingly dire medical diagnosis to predicting the likelihood of a terrorist attack.

Mr. Devlin does a remarkable job of showing just how much derived from the history-changing Pascal-Fermat correspondence. He writes, “We may not know what tomorrow will bring, but we can quantify what it might bring and act accordingly.”

Wall Street take heed.


By Keith Devlin

Basic Books, $24.95

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