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Horst Steinke



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”


. 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



. 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-




Unlike categories in earlier philosophical inquiry


, at least to

a relative degree, in the current incarnation categories consist not

only of “entities”, understood in the broadest of senses


, but

also of particular types of relationships. These special kinds of

relations take the form of




. 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, […]»


. An intuitive example of an elemen-

tary category – however only when viewed in isolation – is the

temperature scale on a thermometer: its


are numbers, its