November 20, 2006
Astonished
An obscure passage from Plutarch’s De Stoicorum repugnantiis:
But now he [Chrysippus] says himself that the number of conjunctions produced by means of ten assertibles exceeds a million, though he had neither investigated the matter carefully by himself nor sought out the truth with the help of experts. [...] Chrysippus is refuted by all the arithmeticians, among them Hipparchus himself who proves that his error in calculation is enormous if in fact affirmation gives 103049 conjoined assertibles and negation 310952.
Hipparchus flourished in the second century BC. Today we would probably call him a mathematical physicist. Plutarch wrote some two hundred fifty years later.
In 1870, the mathematician Friedrich Wilhelm Karl Ernst Schröder showed that the number of possible ways to bracket a row of ten symbols, e.g.
(AA)(AAA(AA)A)(AA)and not including singletons or the whole row, was equal to 103049. This is sometimes written as s(10) = 103049.
In 1994, David Hough at George Washington University noticed that these two numbers were the same.
In 1998, Habseiger, Kazarian and Lando noticed that Plutarch's second number, 310952, was very close to half the sum of the tenth and eleventh of Schröder's numbers: (s(10) + s(11))/2 = 310954.
The modern branch of mathematics which deals with enumerating combinations is called combinatorics. It was once thought that the ancient Greeks had no interest in the subject.
(Hat-tip to Alexandre Borovik, who comes from the opposite end of the Tea Road as some of my forebears. The passage from Plutarch comes from F. Acerbi, "On the Shoulders of Hipparchus: A Reappraisal of Ancient Greek Combinatorics", Arch. Hist. Exact Sci. 57 (2003) 465-502.)
Posted by coyu at November 20, 2006 05:10 AMThat really is astonishing. Sadly, I don't really know enough to say any more.
Posted by: King-Walters at November 23, 2006 01:35 PMI think today the problem would be a mid-level Olympiad question, but that's only because partitions and recursion are now everyday concepts in math.
In ancient Greek mathematics, there's very little evidence of either the idea of a partition -- the number of ways a whole number can be put together additively, like 3 = 2 + 1 = 1 + 1 + 1 -- or the idea of recursion, the way a Fibonacci number is the sum of the previous two Fibonacci numbers.
What would be truly freaky is if there were an ancient manuscript of Hipparchus which posited something like, "the number of partitions of four objects, nine objects, and every fifth collection of objects further, is divisible by five; and the number of partitions of five objects, twelve objevts, and every seventh collections of objects further, is divisible by seven." (Ramanujan's congruences, restated in verbal form.) This is probably not going to happen.
Incidentally, one can guess what method Chrysippus used to calculate the number of ten element statements in Stoic logic: he probably estimated it as 2^10 * 2^10 = 1024 * 1024 = 1048576, which is slightly over a million. But as Plutarch remarks, it's not a careful investigation of the problem.
(Also, look at those names: Chrysippus, "Golden Horse"; Hipparchus, "Horse Master". Sometimes, while parsing Greek, you want to break out the war bonnets.)
Posted by: Carlos at November 28, 2006 03:02 AM