Many topics

A mathematical essay turns out to have much more than mathematics in it.

From somewhere, probably a University Book Store, our astronomer years ago picked up a paperback edition of A Philosophical Essay on Probabilities by the Marquis de Laplace.  That name is known to anyone familiar with advanced mathematics, meaning science and engineering majors at least.  Anyway, our astronomer finally got around to reading it recently, and found it far more wide-ranging than he expected.

A book by that title published today, or even in the last century, would dig deeply into what probability actually is and what it means, maybe with some examples of how to apply it.  It might treat in depth of the controversy between the Bayesians and the Frequentists that our astronomer remembers from his graduate school days.  If written by a mathematician it would cover something called measure theory and might mention Borel sets.  But “philosophy” in Laplace’s time (1749-1827) had a somewhat different meaning from what we understand today.  What we call “science” was then known as “Natural Philosophy,” and “philosophy” itself might be taken to mean any organized inquiry into a subject.  So Laplace does not feel bound to confine himself to what we recognize as mathematical probability.

He does of course start there, with card games and dice, and what is a fair stake in a game of chance.  He includes some more advanced problems; here is one, which we have dressed up a bit to make it more interesting to students:

“To proceed on your Quest, you are given a leather pouch with two amulets.  An amulet may be either silver or gold.  You may not look into the pouch, but to pass a Gate you must draw one at random, replacing it afterward.  The amulet you draw for the first Gate is silver.  The one for the second Gate is also silver.  What is the probability that the one you draw for the third Gate is again silver?”

Laplace goes on to apply his mathematics of probability to scientific measurements, something that one would also expect nowadays, showing that careful analysis can tease out features that are not obvious.  He looks at the orbit of the Moon and makes a start on the tides (a difficult subject to do quantitatively). He discusses methods of showing whether vaccination against smallpox is worthwhile, a topic with a modern ring.

He continues beyond that to discuss (what we would call) the scientific method, noting that coming up with hypotheses is an essentially creative act.  He even proceeds to methods of selecting among candidates for public office.  And he is acutely aware of the fallibility of human beings.  They are not good at estimating probabilities; they accept implausible things merely on authority; they are more sensitive to possible losses than equal possible gains (the latter is a conundrum of Economics nowadays).  There is a plea for tolerance, and an admonition for nations to act in good faith.

So our astronomer found a good deal more than he expected.  That is one of the pleasures of digging into old books.