scientist within two links for 16 of 25 major philosophers (64 percent) and for 15
of 61 secondaries (25 percent). If we start from the side of the mathematicians
and astronomers, of the 12 major figures, none has any direct contact with phi-
losophers important enough to be listed even as minor figures, and only 1 (the
state astronomer Shen Kua, during the Sung dynasty) is within two links of a
known philosopher. Of 9 mathematical scientists of secondary importance, only 1
has any philosophical contacts. By comparison, 63 percent of major Islamic scien-
tists (12 of 19) are within two links of a known philosopher. Sources on Chinese
mathematicians (Mikami, 1913; Needham, 1959; Sivin, 1969; Libbrecht, 1973;
Nakayama and Sivin, 1973; Swetz and Kao, 1977; Graham, 1978; CHC, 1979,
1986; DSB, 1981; Institute, 1983; Ho, 1985; Qian, 1985; Chen, 1987; Li and Du,
1987).- The exception which proves the rule is the Buddhist monk I-hsing, the only
important Buddhist mathematician and astronomer. He worked for the emperor
at the time of the abortive Buddhist near-theocracy in the early 700s, and in the
same temple where the great Hua-yen philosophy was created in the previous
generation. - The method was revived and extended in Tokugawa Japan, leading to indigenous
development of the calculus (Smith and Mikami, 1914). Here again we see a brief
confluence of networks. Mathematics was carried in a network of samurai, bu-
reaucratic officials of the shogun, and other high-ranking lords in positions con-
cerned with accounts, astronomy, and other practical administration. By itself this
was not conducive to intellectualized mathematics; but by the 1670s, these mathe-
matician-officials were also running private schools in Edo and Kyoto, similar to
the competitive expansion of Confucian schools which underpinned the philosophi-
cal outburst of the same period. Merely practical considerations were transcended
by social competition over prestige; much as in Italy during the time of Tartaglia
and Cardano, leading mathematicians attracted pupils by publishing problems and
challenging others to solve them. Topics went far beyond practical problems,
beginning with magic squares and circles, escalating to solving equations of very
high degree. The most important developments from circle measurement problems
to an integral calculus were made in the school of Seki Kowa from the 1670s
through the early 1700s. Methods were kept secret, although occasionally spilled
out in publications during the late 1700s. In the key period of innovation, the
mathematical networks connected with the philosophical centers. Seki worked for
the same Edo lord who employed the somewhat younger Ogyu Sorai; a Seki
grandpupil became head of the Mito school. The mathematical lineages continued
down through the early 1800s, but the connection with philosophy was not taken
further; mathematics did not become invoked as an epistemological ideal, and
mathematical methods were not generalized as abstract principles. Mathematical
innovation stagnated after the mid-1700s; most likely the increasing bureaucrati-
zation of education in the later Tokugawa had a restricting effect on mathematics
just as it did on philosophy. The wide expansion of low-status practical training
schools for commercial mathematics, stimulated by the growth of the commercial
996 •^ Notes to Pages 550–551