of Descartes’s system; issues of the cogito and of the relations among dual
substances became the deep troubles on which specifically philosophical crea-
tivity took off. A second key intrusion from mathematics came when Leibniz,
the most aggressive developer and purveyor of the calculus, used his command
of the new mathematical sophistication—now a quantum leap higher than in
Descartes’s generation—to solve these metaphysical problems. Leibniz’s ap-
proach to the two-substance conundrum was to reinstate a scholastic doctrine:
that essence contains all its predicates. His calculus provides a respectably
up-to-date analogy: each substance contains its past and future states, just like
a point moving along mathematical coordinates; since in geometrical repre-
sentation a moving point looks like a point at rest, the point must contain an
additional mathematical quality (later called “kinetic energy”) which generates
the motion from past to future. The infinitesimal calculus had opened the way
to conceiving of second-order (and higher) relations among first-order rela-
tions. Leibniz used this to solve an issue which had arisen over both Descartes’s
extended substance and the rival materialist doctrine of atomism, namely, that
all material substance is infinitely divisible, and thus ultimately fades away into
nothing. From the point of view inspired by Leibniz’s calculus, there is a
transcendent order which arises from examining the pattern of infinite subdi-
vision, which shows how infinitesimals nevertheless make up a solid contin-
uum. Leibniz’s monadology combines an infinite array of independent sub-
stances, each of whose essence logically contains all its predicates or qualities,
including its causal and time-space relations. It is a mathematician’s vision of
the universe.^26
The next big leap in metaphysics and epistemology comes with Kant. He
is not a star mathematician on the level with Descartes and Leibniz, but he
taught mathematics and science early in his career, and his first interests—and
his most prominent network contacts—were in science and mathematics. He
also inherited Leibniz’s position within philosophy, as part of the Wolffian
school which dominated the Prussian universities; Kant’s creativity was the
response of that network as it met the challenges posed both on the religious
side and by the Newtonians, who constituted a counter-wave to the Descartes-
Leibniz lineage throughout this period. Again with Kant, consideration of the
abstract properties of mathematics was the key to his philosophy. His key
discovery was the concept of the synthetic a priori. Up to this time the
conventional view was that mathematics epitomizes analytical knowledge, the
investigation of the implications of definitions and axioms. But such knowledge
is merely tautological; Kant wanted a version of knowledge which is synthetic,
non-tautological, amplifying what we know rather than merely clarifying what
we already knew.
Kant took his crucial step by arguing that mathematics is in fact not ana-
Sequence and Branch in the Social Production of Ideas^ •^851