so; nor should we regard it as a tendency of philosophy to become “scientific”
in its maturity. Mathematics is not identical with science; and as we have seen,
empirical-discovery science per se does not drive up the abstraction-reflexivity
sequence but tends to hold it constant on a given level, at least for long periods
of time, so that “lateral” exploration can take place. Mathematics has an
intimate effect on the structure of the intellectual community because it reveals
some of the innermost qualities of that community to itself.
A key social determinant of this influence is the connection between mathe-
matical and philosophical networks. Mathematics has existed in all world
civilizations. It reached some fairly elaborate developments in China (especially
the algebra of the Sung dynasty) and Japan (the calculus during the Tokugawa),
with bits of development in India, more in medieval Islam, and of course
ancient Greece. As we have seen, the network patterns have varied widely. In
China and India, mathematical and philosophical networks were totally sepa-
rate, whereas there was considerable network overlap in all four Western cases.
This is most notable in Greece and in modern Europe, where not only do the
networks overlap in regard to minor figures, but also the stars or network
centers in each are closely linked, and sometimes even coincide in the same
person.
In Greece, crucial developments in the content of philosophy and mathe-
matics were influenced by this connection. The Sophists, who catalyzed the
jump to higher abstraction in philosophy, were also the circle that popularized
mathematical puzzles. The Pythagoreans began as cosmologists based on the
primitive conceptions of mathematics; out of their ranks emerged the special-
ized mathematicians, as well as the key deep troubles such as the incommen-
surability problem of irrational numbers. It was through the combination of
the networks and concepts of the Sophists and the Pythagoreans that Plato
created the philosophy of abstract Forms. Philosophy also played back into
mathematics. Aristotle’s formalized logic became the basis for axiomatic sys-
tems which culminated in Euclid’s synthesis of geometry. In the Hellenistic
period, technical mathematics tended to become separate; the philosophy-
mathematics overlap continued, shifting into numerology, blending a classifica-
tory scholasticism with lay-oriented occultism and the cosmology of world
emanations. After the classic period, the mathematics-philosophy overlap no
longer contributed to the abstraction-reflexivity sequence; we might even say
that it was their overlap for a few generations which made the period “classic”
in both fields.
In modern Europe, the mathematical revolution of the 1500s produced both
a new form of mathematics and an impetus to modern philosophy, involving
the fusion for a while between the most creative mathematical and philosophi-
cal networks. Why should the network fusion have this effect? The answer
Sequence and Branch in the Social Production of Ideas^ •^847