Hipparchus on Compound Statements Michael Lahanas Ιππαρχος ο Ρόδιος

Griechische Mathematik: Kombinatorik von Hipparchos

Xenocrates of Chalcedon, (Ξενοκράτης ο Χαλκηδόνιος) who succeeded Speusippus (Σπεύσιππος ο Αθηναίος) as head of Plato's Academy (339 BC- 314 BC) solved a combinatorial analysis problem finding that 1002000000000 syllables can be written with the letters of the Greek alphabet. (Technology Museum of Thessaloniki ) Archimedes considered a combinatorial problem of the stomachion.

Richard Stanley discussed in [ST97] of how the classical Schröder numbers s(n) [Sch1870] are even more classical than has been believed before.

All lattice paths for N=1, N=2 and only 3 examples for N=3 from the 22 possible. The number of combinations, 2, 6, ... define the Schröder number S(N)

The Schröder number S(n) is the number of lattice paths in the Cartesian plane that start at (0, 0), end at (n, n), contain no points above the line y = x, and are composed only of steps (0, 1), (1, 0), and (1, 1). The corresponding small Schröder numbers s(n) are obtained using:

s(n) = S(n)/2 for n > 2, s(1) = 1

(s(1),s(2),s(3), s(4),...,s(10),...) = (1,1,3,11,...,103049,...)

The numbers can be obtained by the following relation:

(n+2)s(n+2)-3(2n+1)s(n+1)+(n-1)s(n) = 0, n greater or equal 1., s(1)=s(2)=1.

"In pure mathematics [Hipparchus] is said to have considered a problem in permutations and combinations, the problem of finding the number of different possible combinations of 10 axioms or assumptions, which he made to be 103049 (v.l. 101049) or 310952 according as the axioms were affirmed or denied. It seems impossible to make anything of these figures."

Heath, History of Greek Mathematics

According to [ST97] Plutarch (50-120 AD) said: "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 103049 compound statements, and on the negative side 310952.)"

Stanley's article points out that 103049 is the number of "bracketings" of a string of 10 letters, as shown by Schröder. This certainly suggests that Hipparchus was calculating something that was somehow related to the number of bracketing of n letters, although as Stanley says:

"...it remains to determine exactly what Hipparchus and Plutarch meant by a 'compound proposition'. In Stoic logic, compound propositions are built up from simple ones using such connectives as 'and', 'or', and 'if...then'. This does not seem like enough information to pinpoint exactly what Hipparchus had in mind."

According to ideas presented in Mathpages Hipparchus could consider the combinatorics of certain compound statements in Boolean logic involving n ordered elements with the connectives 'and', 'or', and parentheses.

For example, with the usual Boolean notation conventions on n simple statements A,B,..., we have

n = 1

A

n = 2

A+B, AB

n = 3

A+B+C, A+BC, (A+B)C, AB+C, A(B+C), ABC

n = 4

A+B+C+D, A+B+CD, A+(B+C)D, (A+B+C)D, A+BC+D, (A+B)(C+D), (A+B)C+D, A+B(C+D), A+BCD, (A+B)CD, (A+BC)D,
AB+C+D, A(B+C)+D, A(B+C+D)
, AB+CD, A(B+D)D, A(B+CD), (AB+C)D, ABC+D, AB(C+D), A(BC+D), ABCD.

For n = 1,2,3,4 there are 1,2,6,22 compound statements respectively and for n = 10 elements there are 206098 combinations, which is exactly twice Hipparchus's number. The expressions are symmetrical in the sense that the number of distinct parenthetization of (+ * * +) is the same as (* + + *) by the duality of AND and OR and Hipparchus might have had something like these compound Boolean statements in mind.

The number of 103049 that appears in various combinatorial problems suggests that Hipparchus knew some non-trivial combinatorics. The exact problem which results in 310952 combinations is unknown. Habsieger et al consider this number to represent (s10+s11)/2 which is 310952. As considered by others it would be strange if these results were produced in isolation without other knowledge of combinatorics. This suggests that probably a significant body of combinatorics knowledge of the ancient Greeks was lost.

References

[Sch1870] Schröder, E. "Vier kombinatorische Probleme." Z. Math. Phys. 15, 361-376, 1870.

[ST97] Stanley, R. P. "Hipparchus, Plutarch, Schröder, Hough." Amer. Math. Monthly 104, 344-350, 1997. also available from:

http://www-math.mit.edu/~rstan/papers.html (and as a local copy in case the original is not available here)

[Hab98] Habsieger, L.; Kazarian, M.; and Lando, S. "On the Second Number of Plutarch." Amer. Math. Monthly 105, 446, 1998

Eric W. Weisstein. "Plutarch Numbers." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PlutarchNumbers.html