Plutarch Numbers

In Moralia, the Greek biographer and philosopher Plutarch states ``Chrysippus says that the number of compound propositions that can be made from only ten simple propositions exceeds a million. (Hipparchus, to be sure, refuted this by showing that on the affirmative side there are 103,049 compound statements, and on the negative side 310,952.)'' These numbers are known as the Plutarch numbers. 103,049 can be interpreted as the number $s_{10}$ of Bracketings on ten letters (Stanley 1997), Habsieger et al. 1998). Similarly, Plutarch's second number is given by $(s_{10}+s_{11})/2=310,954$ (Habsieger et al. 1998).

References

Biermann, K.-R. and Mau, J. ``Überprüfung einer frühen Anwendung der Kombinatorik in der Logik.'' J. Symbolic Logic 23, 129-132, 1958.

Biggs, N. L. ``The Roots of Combinatorics.'' Historia Mathematica 6, 109-136, 1979.

Habsieger, L.; Kazarian, M.; and Lando, S. ``On the Second Number of Plutarch.'' Amer. Math. Monthly 105, 446, 1998.

Heath, T. L. A History of Greek Mathematics, Vol. 2: From Aristarchus to Diophantus. New York: Dover, p. 256, 1981.

Kneale, W. and Kneale, M. The Development of Logic. Oxford, England: Oxford University Press, p. 162, 1971.

Neugebauer, O. A History of Ancient Mathematical Astronomy, Vol. 1. New York: Springer-Verlag, p. 338, 1975.

Plutarch. §VIII.9 in Moralia, Vol. 9. Cambridge, MA: Harvard University Press, p. 732, 1961.

Stanley, R. P. Enumerative Combinatorics, Vol. 1. Cambridge, England: Cambridge University Press, p. 63, 1996.

Stanley, R. P. ``Hipparchus, Plutarch, Schröder, and Hough.'' Amer. Math. Monthly 104, 344-350, 1997.