  In this paper we offer a reconstruction of the evolution of Leibniz's thought
concerning the problem of the infinite divisibility of bodies, the tension
between actuality, unassignability and syncategorematicity, and the closely
related question of the possibility of infinitesimal quantities, both in
physics and in mathematics.
  Some scholars have argued that syncategorematicity is a mature acquisition,
to which Leibniz resorts to solve the question of his infinitesimals namely the
idea that infinitesimals are just signs for Archimedean exhaustions, and their
unassignability is a nominalist maneuver. On the contrary, we show that
sycategorematicity, as a traditional idea of classical scholasticism, is a
feature of young Leibniz's thinking, from which he moves away in order to solve
the same problem, as he gains mathematical knowledge.
  We have divided Leibniz's path toward his mature view of infinitesimals into
five phases, which are especially significant for reconstructing the entire
evolution. In our reconstruction, an important role is played by Leibniz's text
De Quadratura Arithmetica. Based on this and other texts we dispute the thesis
that fictionality coincides with syncategorematicity, and that unassignability
can be bypassed. On the contrary, we maintain that unassignability, as
incompatible with the principle of harmony, is the ultimate reason for the
fictionality of infinitesimals.
