The Broken Dice and Other Mathematical Tales of Chance [Paperback]

If you had to go through it in graduate school you'd probably agree with me that behaviour under uncertainty is usually handled rather mechanically by professors. You may even have considered that, anyway, not much intuition could be behind those theories. Ivar Ekeland shows in this book that dealing with the fundamentals of expectations, probabilities, games, and risk can be fun, and you get the intuition to start thinking by yourself as well. Early morning reading, though.

Ivar Ekeland is one of the better writers of popular mathematics. In "The Broken Dice," he continues with the themes explored in "Mathematics and the Unexpected." Divided into six chapters (Chance, Fate, Anticipation, Chaos, Risk and Statistics) the book is an elegant examination of the human struggle to find order in the seeming contingency that is the natural world. Mixed with the mathematical discussions are excerpts from Icelandic sagas, the Bible, and Shakespeare that reinforce the message that our analytical search for meaning is still fundamentally a humanistic endeavor.

The Broken Dice, and Other Mathematical Tales of Chance

This is a well-written book, and, unlike some translations of mathematical writing, the translation of technical terms into English is good. There are occasional references in footnotes, but no bibliography or suggestions for further reading, at least in the English translation; perhaps French references were not judged to be very useful for English speakers. I also found the work vaguely unsatisfying, for similar reasons to those given by David Aldous -your experience may vary:) Some specific comments and reasons for giving only three stars follow:

1) Mathematicians sometimes have an eccentric sense of humor. I must warn other readers that there are a couple of passages where the author identifies "notional inventions" by attributing them to Jorge Luis Borges, who, besides being a famous author of magical realism, is (according to Wikipedia) an admitted hoaxer and forger. (Perhaps magical realism escaped the confines of literature and entered his day-to-day life:) Do not waste time looking for the story Ekeland heard where "in a famous lecture on the tales in A Thousand and One Nights Borges tells ... that I was unable to find in my edition." in a more complete edition; do not believe in the otherwise unknown "Brother Edvin" whose very modern-sounding discussion of chance is "debated in a remarkable manner in a manuscript that unfortunately has disappeared."; do not look for the (infinitely long) book "that Borges misplaced on the shelves of the Argentine National Library" -Oh those disappearing manuscripts and misplaced books, oh those unfindable stories and infinitely long books, oh those unknown monks:) Fortunately Ekeland is decent enough to tip us off with the references to Borges -I wonder whether Jorge Luis Borges is a French cultural analog of Baron Münchhausen.

If you really want to look for the purported tale from A Thousand and One Nights I would suggest Borges' collection Historia Universal de la Infamia, translated as A Universal History of Iniquity (or Infamy), under Tale of the Two Dreamers.

2) On P-43 Ekeland claims that "The notation log N denotes the logarithm N in binary form, that is, a number that appears on all hand-held calculators..." -at least all hand-held calculators sold by the Jorge Luis Borges calculator company:) Mine (from Hewlett-Packard) do not produce binary logarithms.

I suspect this is a translation infelicity, and a better reading would have been: "The notation log N denotes, in binary form, the logarithm [of] N, a number that is [available] on all hand-held calculators,..." -although even today some cheaper ones may not have logarithmic functions and the binary form is uncommon.

3) On P-78 Ekeland muses: "...who can know how alone the Athenians felt before the battle of Salamis? The other Greek cities surrendered to the Medes with hardly a fight. The Athenians alone abandoned their city, setting off to sea despite their inexperience, they took their wives and children to the island of Salamis, and now they are in the strait about to confront an army and navy ten times larger than their own..."

I hope Ekeland's other information is better than his Greek history -or perhaps the translator overlooked another reference to Borges:) Wikipedia confirms my recollections from those long-ago required freshman courses:

Twenty Greek allies remained with Athens at Salamis, including Sparta, Athens provided slightly fewer than half of the allied triremes involved in the battle of Salamis. The Athenian women and children were evacuated en masse to the Peloponnesian city of Troezen, not to Salamis. And I don't know where Ekeland gets the idea that Greek/ Athenian sailors were inexperienced. The Greeks were very experienced traders on the sea, which, in that era, required an ability to defend oneself (and frequently to prey on others who were not as able), and the less affluent citizens, who were unable to purchase the arms and armor of a hoplite or even a skirmisher, would join the navy for their required military service. And, the lowest modern estimate for the Greek forces is 366 triremes, the highest for the Persians is 1207; even with a Jorge Luis Borges calculator that does not come out anywhere near a "navy ten times larger". Finally, this is one of many passages where Ekeland provides no references -not even to Borges:)

Other comments:

4) Chapter 7 "Statistics" is more about the application of probabilities, or using our knowledge of chance events to manage risks than about statistics-but Ekeland already has chapters on "Chance" and "Risk". In very simple terms, probability is about what we can say about the results (data) we expect to get, when we know the underlying processes, while statistics is about what we can say about the underlying processes when we have some results (data).

5) Finally, I am happy to note that Ekeland has flouted the "old publisher's adage" that each equation halves the sales of the book. By my count there are over 20 algebraic equations or expressions in the book. If in doubt buy the book just to combat this pernicious belief!