I’ve started reading the volume of Buffon’s Histoire Naturelle where the Buffon needle problem is discussed, partly out of curiosity (why is it called Essai d’arithmétique morale? why is it included in a book of natural history?) and partly, I must say, because the book’s format is remarkably convenient to put in a coat pocket and read while travelling in the Zürich tramways. (It’s certainly much easier to carry along than any modern scientific book I’ve ever seen).

A few pages suffice to understand the meaning of the title and, at least, why probability theory is related. Buffon is interested here in various types of “truths” or “certainties” (vérités et certitudes, in French). In fact, he distinguishes three kinds: (1) intellectual (geometric or mathematical) certainties; (2) physical certainties; and (3) moral certainties, and the discussion of the third kind is really the topic of his essay (and probably it is indeed the most original — of course I am not knowledgeable about his possible precursors here, but whereas I could recognize at least partly his ideas on the first two kinds of truths, I do not remember encountering something similar to the third one before).

Geometric certainties, for Buffon, are those of mathematics. He states that there is not much to be said about them, because they refer, or derive, ultimately, from accepted truths (axioms, though I didn’t see the word in the text) using obvious rules of inference (… (elles) se réduisent toutes à des vérités de définition). In effect, he takes the point of view (which I’ve seen in other places) that mathematics, properly considered, is tautological and really should be trivial, except if irrelevant or badly-posed questions are raised. (I’ll probably come back to this, and other mathematical topics in the essay, because there are amusing remarks there).

Physical and moral truths are different because they refer to the “real” world. The distinction between the two is that a typical physical certainty is that “the sun will rise tomorrow”, which is knowledge based on experience and considered to be “true” because it has been verified in a very large number of cases, going back to the beginning of the world. Moral certainties, on the other hand, are based on much less data, and are more a matter of personal conviction and analogies.

Where things really become interesting is when Buffon decides to quantify these two types of truths, which are not absolutely certain, by assigning a numerical probability to them — indeed, this is were probability theory comes in. Geometric certainties have absolute probability (l’évidence n’est pas susceptible de mesure), being immune to any kind of doubt. But Buffon says that physical certainties are not absolute, rather they are those which have a probability comparable to that of the sun rising tomorrow (puisque le Soleil s’est toujours levé, il est dès-lors physiquement certain qu’il se lèvera demain). The latter, he estimates rather curiously, by (implicitly) assigning the probability 1/2 to each event “the sun rises” on any given day, and assuming that those events are independent. Since, he says, the sun has risen every day for about 6000 years (!), the probability of the negation of a physically certain event is about

$2^{-6000\cdot 365+1}=2^{-2189999},$

and hence it is zero for any practical purpose. (It’s not clear what Buffon really thinks of this estimate of the age of the earth; he remarks that 2000 more years would not change anything to the qualitative feature of this bound, but doesn’t go further).

Now for moral certainties, things are much more interesting. Buffon states that he considers that a good estimate for the probability of the negation of a “morally certain” event is the probability p that a person, aged 56 years old and in good health, will die within the next twenty four hours. He argues that this is a correct value because people typically dismiss this probability by going about on their normal activities every day. So, he says, if some other event has probability less than p, it can be considered to be morally impossible, and therefore should be dismissed from consideration in any practical matter. (Or comme tout homme de cet âge, où la raison a acquis toute sa maturité, & l’expérience toute sa force, n’a néanmoins nulle crainte de la mort dans les vingt-quatre heures … j’en conclus que toute probabilité égale ou plus petite, doit être regardée comme nulle…)

Then Buffon goes about finding a value for p. For this, he has very detailed and scrupulous mortality tables and statistics (about 350 pages at the end of this volume, in fact), and from this he concludes that p is of size roughly 1/10189 (which he rounds up to 1/10000; note that “homme” means “person” in his discussion, since the mortality tables he uses do not distinguish between men and women). There is then an interesting footnote (click on the image below to see it on the right-hand page), where he informs the reader of the comments of Daniel Bernoulli on this matter: Bernoulli is said to have approved of the principle, but objected that the correct value should be 1/100000 instead, because one should take into account that sick people are included in the statistics, and are likely to be much more afraid of dying on a given day.

After establishing this value of p, more developments follow of how knowing this could (or should) affect people’s actions, and in particular a long discussion involving gambling, which is mathematically rather weird — so I’ll discuss it in the next post… (Readers may also, of course, still be wondering what the needle has to do with any of this…)

(Note: The tables at the end make for melancholy reading; Buffon observes that

One fourth of the human race perishes, so to say, before having seen the light, since about a quarter dies within the first eleven months of life… A third perishes, before the age of twenty three months, that is to say, before having had the use of their limbs, and of most of their other organs…

There is no indication that he thinks these numbers to be anything but “moral certainties” (he calls them vérités) — facts of life, that will not change during the course of humanity’s existence on Earth.)