How the Chevalier de Méré met Blaise PascalPosted: February 21, 2021
First, there’s mathematics. Obviously, you’ve got to be able to handle numbers and quantities—basic arithmetic. And the great useful model, after compound interest, is the elementary math of permutations and combinations. And that was taught in my day in the sophomore year in high school. I suppose by now in great private schools, it’s probably down to the eighth grade or so.
It’s very simple algebra. It was all worked out in the course of about one year between Pascal and Fermat. They worked it out casually in a series of letters.
so says Charlie Munger in his 1994 speech, “A Lesson on Elementary Worldly Wisdom as it Relates To Investment Management & Business.”
These letters between Pascal and Fermat sounded worth a read, so I went to check them out. The year in question was 1654. Up until that time, no one* had really worked out and set down the math of probability. You can’t blame them, if you think about it. Even in 1654 it was probably pretty hard to even get your hands on enough paper for working out math problems.
Struggling to really wrap my head around the contents of the letters (on top of everything, the first letter is now lost), I picked up The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern: A Tale of How Mathematics is Really Done by Keith Devlin. An interesting book and a great introduction to the mental blocks that had kept people from working out probability before these two weirdos started corresponding.
An even clearer articulation of the problem of points that set Pascal and Fermat to work can be found in Peter Bernstein’s Against The Gods: The Remarkable Story of Risk:
In 1654, a time when the Renaissance was in full flower, the Chevalier de Méré, a French nobleman with a taste for both gambling and mathematics, challenged the famed French mathematician Blaise Pascal to solve a puzzle. The question was how to divide the stakes of an unfinished game of chance between two players when one of them is ahead. The puzzle had confounded mathematicians since it was posed some two hundred years earlier by the monk Luca Paccioli. This was the man who brought double-entry bookkeeping to the attention of the business managers of the day, and tutored Leonardo da Vinci in the multiplication tables. Pascal turned for help to Pierre de Fermat, a lawyer who was also a brilliant mathematician. The outcome of their collaboration was intellectual dynamite. What might appear to have been a seventeenth century game of Trivial Pursuit led to the discovery of the theory of probability, the mathematical heart of the concept of risk.
Their solution to Paccioli’s puzzle meant that people could for the first time make decisions and forecast the future with the help of numbers.
Bernstein helpfully restates the problem of points in the form of a World Series situation. What is the probability your team will win the best of seven series after it has lost the first game? (assume the teams are, as in a game of chance, evenly matched)
Well, Pascal pointed out that we just need to list all the possible outcomes of the remaining six games, and calculate from there. There are 22 combinations in which your team would come out on top after losing the first game, and 42 combinations in which the opposing team would win. As the result, the probability is 22/64 = .34375
As Bernstein points out, there’s something here that trips a lot of people up, even Fermat. There aren’t really 64 possible outcomes, because why would we include possibilities like your team goes win-win-win-win-win-win for the remaining six games? The World Series would’ve been over after that fourth win. W-W-W-W-W-W is not a possible outcome of the remaining six games.
As Pascal remarked in the correspondence with Fermat, the mathematical laws must dominate the wishes of the players themselves, who are only abstractions of a general principle. He declares that “it is absolutely equal and immaterial to them both whether they let the [game] take its natural course.
So there you go. Win-win-win-win-win-win-win is one of the forked paths off win-win-win-win. It must be accounted for, or we won’t count the potential possibilities correctly.
Naturally enough I got bored with the math part and wanted to know more about the Chevalier de Méré. Who was this fun loving gambling nobleman who put two all-time math geniuses to work helping him win at dice?
Turns out he was a guy, named Antoine Gombaud, who dubbed himself Chevalier de Méré in his writing. Much of his writing was obsessed with the idea of honnête, and how to be l’homme honnête, which included honesty but also modesty, elegance, appropriateness, excellence, sociability. You can read all about it here in what appears to be an excerpt of Manning The Margins: Masculinity and Writing in Seventeenth-Century France by Lewis Seifert, a professor at Brown.
But still, how did this cool guy hook up with Pascal? Devlin says that the Chevalier and Pascal met at a gambling table. Pascal would go back and forth between somewhat extreme religious periods. During an early one of these, when he was getting pretty hard core, a doctor warned him off:
His doctor advised him that for the sake of his health, he should abandon the Jansenist ways and lead a life more normal for a young man. Although he would remain strongly religious for the remainder of his all-too-short life, Pascal resumed normal activities. Indeed, he did so with vigor, adding regular visits to the gaming rooms to his earlier academic pursuits. It was at the gambling table that Pascal met the Chevalier de Méré
Looking into this question of how, exactly, the Chevalier and Pascal met, I found a different, more detailed, and funnier, version. Here is the Chevalier de Méré himself describing how he met Pascal:
“I once made a trip with the Duke of Roannez, who used to express himself with good and just sense and whom I found good company. Monsieur Mitton, whom you know and who is liked by all at court was also with us, and because that trip was supposed to be a promenade rather than a voyage, we only thought of entertaining ourselves and we discussed everything. The duke was interested in mathematics, and in order to relieve tedium on the way he had provided a middle-aged man, who was then very little known, but who later certainly has made people talk about him. He was a great mathematician who knew little but that. These sciences gave little sociable pleasure, and this man, who had neither taste nor sentiment, could not refrain from mingling into all we said, but he almost always surprised us and made us laugh.” De Méré goes on to tell that Pascal carried strips of paper which he brought forth from time to time to write down some observations. After a few days Pascal came to enjoy the company and talked no more of mathematics.
so reports Oystein Ore, writing in the May 1960 issue of The American Mathematical Monthly (vol 67, No. 5, “Pascal and the Invention of Probability Theory”). Oystein Ore says:
Pascal and Fermat never met in person, which is kind of sad. In 1660 Fermat proposed that they meet, but at the time they were both too sick and miserable to travel very far. Within a few years they were both dead.
Pascal invented a kind of calculator, the Pascaline, but it was too expensive to produce them:
Late in life, in another religious phase, Pascal reflected on gamblers:
And that’s the story of Pascal and Fermat!
* it wouldn’t blow my mind if one of the great mathematicians of the Arab world had worked some of this out, written it down, and put a copy in the House of Wisdom in Baghdad, but most of those books were destroyed when Hulagu, Genghis Khan’s grandson, destroyed that city in 1258. Bummer!