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5.1
The “mathematics of relationships”
The conceptual tool that would seem to offer itself in meeting
our needs of untangling the trichotomous nexus, at least in theo-
ry, can be found in the “mathematics of relationships”
167
. Alt-
hough, strictly speaking, it is not the only branch of mathematics
that describes and systematizes relationships, for our present
purposes the branch of
category theory
will play the most important
role
168
. This role, however, needs to be qualified and circum-
scribed right from the beginning: we are not interested in the
mathematical formalisms themselves but rather in their underly-
ing general conceptual insights that allow making distinctions be-
tween different types of relations. In a way, the epistemic move-
ment pursued here is thus in the opposite direction of the origi-
nal intent of category theory which put mathematical structure
around the informal notion of relations between entities of dif-
ferent kinds. We are, in effect, taking certain mathematical pro-
cesses that are part of the theory, strip them of their formalism,
and keep only their essential “logic”. As a result, what we are left
with are new tools to probe the problematic of Vico’s trichoto-
my, in order to supplement and complement metaphorical de-
scriptions
169
.
Unlike categories in earlier philosophical inquiry
170
, at least to
a relative degree, in the current incarnation categories consist not
only of “entities”, understood in the broadest of senses
171
, but
also of particular types of relationships. These special kinds of
relations take the form of
transformations
or
morphisms
. As the term
implies, the relations in view are not static or simply fixed in time
but amenable to variation and being acting upon, and acted with.
The standard definition of a category, therefore, typically begins
with a statement like: «A category consists of objects A, B, […]
and morphisms f, g, […]»
172
. An intuitive example of an elemen-
tary category – however only when viewed in isolation – is the
temperature scale on a thermometer: its
objects
are numbers, its