Vico’s Ring
89
morphisms
are the
changes
or
transitions
in temperature readings
173
.
Such complexes of entities/objects with their capacity or affinity
of being altered, metamorphosed in some way function thus as
the “primitives” of category theory. In this language, the three
spheres of Vico’s grand explanatory project, the trichotomy of
“philosophy”, “philology”, and the actual historical world, are
epistemologically equivalent to three categories
sui generis,
and any
of the descriptions made regarding them so far pertained only to
their internal structure, considered individually
.
However, this
aspect of category theory obviously does not go far enough to
throw light on the crucial question of the relationships
between
them, separately from the morphisms
within
them.
Conceptually, this necessitates moving the category concept
to the next higher level: taking individual categories themselves
as the objects, and allowing for morphisms between them also.
These higher-level morphisms have been given the special no-
menclature of
functors
174
.
This is, in fact, the level at which Vico’s
trichotomy is being considered. However, merely exchanging
nomenclature, that is,
functors
for different metaphors, accom-
plishes little, if anything. The value of functors stems from the
fact that they come, and act, in multiple forms and ways, and
thus enable bringing to the surface features that would not come
into view otherwise. For our present purposes, two types of
functors are particularly relevant: (1) the “forgetful functor”, and
(2) the “contravariant functor”.
In mathematics, the somewhat quaint term “forgetful func-
tor” describes a specific process of abstraction, as when a “set
with structure” of group theory is subsequently viewed without
the “structural” properties, the formalized group-theoretic opera-
tions and relationships, resulting in a mere underlying un-
structured “set”
175
. Usually, in mathematics, “forgetting” certain
important properties of complex entities has far-reaching impli-
cations, such as turning geometry into topology by factoring out
metrics (distance, angle). The example of a temperature scale,