1 The Birth of Science - MSRI

10.13 Combinatorics and Logic 279
many centuries after Philo of Byzantium. It was taken up again, in quite
another technical context, in modern theories of deterministic chaos.
10.13 Combinatorics and Logic
Cicero says that anyone who thinks that the ordered universe we know
might have arisen accidentally, through the casual concourse of material
particles, should also allow that, by shuffling and scattering on the ground
a bag of letters of the alphabet, one might get Ennius’ Annals, ready for
reading. 190 The curious thing is that, in choosing an event whose practical
occurrence can be excluded even though it is possible in theory — what
we now call an event of extremely low probability — Cicero should use an
example that seems to suggest an awareness of the enormous number of
ways in which a set of letters can be combined. Plutarch not only reports
a similar example, 191 but writes:
Disorder, like Pindar’s sand, “eludes numbering”. . . . The facts allow
only one true statement, but an unlimited number of falsehoods.
Rhythms and harmonies follow precise ratios, but no one can comprehend
all the musical slips that people make playing the lyre or
singing or dancing. 192
It has often been thought that combinatorial calculations were unknown
to ancient science, but twice in Plutarch’s dialogues we find this remark:
Chrysippus said that the number of intertwinings obtainable from
ten simple statements is over one million. Hipparchus contradicted
him, showing that affirmatively there are 103,049 intertwinings[.] 193
This passage stumped commentators until 1994, when David Hough,
then a graduate student in mathematics, noticed that 103,049 is the tenth
Schröder number, 194 representing the number of ways in which a sequence
of ten symbols can be bracketed (subdivided into hierarchically organized
groups). Although the meaning of Hipparchus’ calculation in terms of
190 Cicero, De natura deorum, II, xxxvii, 93.
191 Plutarch, De Pythiae oraculis, 398B–399E.
192 Plutarch, Quaestionum convivalium libri iii, 732E–F.
193 Plutarch, De Stoicorum repugnantiis, 1047C–E, and Quaestionum convivalium libri iii, 732F (in the
latter passage the number transmitted by the manuscripts, 101,049, was long ago emended to agree
with the other passage, the corruption from “three thousand” to “[a] thousand” being by far the
more likely one).
194 This notion was introduced in 1870 in [Schröder]. The link between it and Plutarch’s passage
was published in [Stanley]; further remarks can be found in [Habsieger et al.]
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