1 The Birth of Science - MSRI

280 10. Lost Science
symbolic logic has not been fully clarified, 195 his mention of this particular
number shows that in his time remarkably complex combinatorial
problems were being solved. The philologists who studied Plutarch in the
mid nineteenth century and were unable to explain this passage probably
had no inkling that they could not possibly understand the numbers
therein simply because combinatorics at that time had not yet recovered
a concept known to Hipparchus. (Fabio Acerbi has made progress toward
reconstructing ancient combinatorics from the meager available testimonia.
195a
One important conclusion we can draw from this episode is that, at least
in some cases, Plutarch can be shown to have correctly recorded sophisticated
scientific results of Hipparchus that are not otherwise documented.
Plutarch also mentions simpler combinatorial problems; for example, he page 349
says that Xenocrates estimated the number of syllables that can be made
with the letters of the alphabet as 1002 billion. 196
10.14 The First Few Definitions in Euclid’s Elements 197
As already discussed, there is much to suggest that Euclid subscribed to
the nominalist and constructivist blueprint discussed in Sections 6.4 and
195 Probably the ten simple statements are to be combined using logical implications. The
Plutarchan term translated here literally as “intertwining” is , which in Stoic logics ordinarily
means what we call logical conjunction (“and”). But bracketing (grouping) an expression
that involves only conjunctions is an idle exercise, since conjunction is associative. On the other
hand, if by intertwining is meant a nonassociative operation (of which the most obvious example
is implication), different groupings lead to essentially distinct logical compound statements, and
counting such compound statements acquires interest. (More precisely, the conjecture is that the
problem posed was to count chains of implications under different groupings, and that — as is
perhaps natural when one uses everyday language instead of operator symbols — the ungrouped
“intertwining” of A, B, C meant the chain of implications now represented by (A ⇒ B)∧(B ⇒ C),
as distinct from (A ⇒ B) ⇒ C on the one hand and A ⇒ (B ⇒ C) on the other; compare our
a > b > c. The interest of ancient logicians (particularly Stoics) in long chains of logical implications
is documented in several loci; see in particular Alexander of Aphrodisias, In Aristotelis Analyticorum
priorum librum I commentarium, 283, 7ff. (ed. Wallies) = [SVF], II, 257, where this way of linking
propositions is implied.)
195a [Acerbi: SH]. Some of the sources are Pappus, Collectio VII, 11–12; Boethius, De hypotheticis
syllogismis, I, viii, 1–7; a scholium to the Data of Euclide, printed in [Euclid: OO], p. 290.
196 Plutarch, Quaestionum convivalium libri iii, 732F. This estimate is somewhat obscure in that we
do not know how many letters are to be combined nor the rules to be followed in combining them;
moreover the number “1002 billion” is certainly rounded off and so offers little in the way of clues
to the original computation.
197 The material in this section is drawn from [Russo: Elementi] and [Russo: Elements].
Revision: 1.11 Date: 2003/01/06 02:20:46