the Eleatics and Sophists, and the Mohists responded with formal geometrical
definitions. Here Chinese intellectuals seem to have been on a similar path as
the Greeks. But this development of mathematics from the philosophical side
was not followed up after the Mohist chain disappeared.
When bureaucracy consolidated in the Han dynasty, mathematicians, as-
tronomers, and philosophers alike compiled canonical texts. The mathematics
textbook Chou Pei was apparently assembled by the Confucian school and
endorsed the kaitian astronomical system; the Nine Sections textbook favored
the rival huntian cosmology. Later the leading skeptical and anti-occultist
Confucians of the Old Text school overlapped extensively with the astronomers
of the huntian system. The upsurge of mathematics and astronomy at this time
seems to have been connected to the struggle between rival branches of Con-
fucianism. The scholasticism of this period did not encourage abstract philo-
sophical treatment, however, as Confucians absorbed the divination texts into
their synthesis and emphasized numerological correspondences for magical
prognostication. The tail end of the Han intellectual network did produce one
last flurry of abstract mathematics in the governmental disintegration of the
late 200s c.e. The greatest of the early Chinese mathematicians, Liu Hui, was
apparently a scholar-official who recombined surveying techniques from a
low-status bureau with those of astronomical calculation used in a higher-
status bureau, treating higher-degree equations and introducing the first ex-
plicit proofs (Swetz, 1992). This combination of specialties happened at the
time of (and in some network links with) the philosophers of the “Dark
Learning” movement, which similarly reorganized official and oppositional
networks, revising their classic texts and thereby raising Chinese philosophy
to its high point of abstract metaphysics.
Thereafter the networks of mathematicians and philosophers were almost
totally separate. In subsequent dynasties the mathematicians were almost all
government officials, cut off from the administrative and historical bureaus
and literary circles in which the Confucian philosophers worked. The Bud-
dhists who dominated philosophy down to 900 c.e. were generally entirely
outside the realm of government administration and cut off from official
mathematics.^22
There was one more round of innovation in Chinese mathematics, the
generalized “celestial element method” for solving higher algebraic equations
in several unknowns, produced in the Sung dynasty. It is tempting to connect
this to the Neo-Confucian movement in philosophy, but there were very few
network connections between that movement and mathematics. Neo-Confu-
cianism began with a branch concerned with occult numerology, but its most
successful lineage was opposed both to numerology and to reformers who
wanted to elevate mathematics in the official examination system. Nor do the
550 • (^) Intellectual Communities: Western Paths