107 / 298
Vico’s Ring
107
170
Kleinert reviews Aristotelian and Kantian categories in
Categories in Phi-
losophy and Mathematics,
cit., pp. 242-249.
171
They encompass both physical as well as non-physical “objects”, such
as physical states, mental states, ideas, theoretical constructs, situations, states
of affair, ergo any type of identifiable complex. The term “objects” commonly
used in category-theoretic exposition out of convenience, and habit, does not
intend to imply reification.
172
E. Kleinert,
Categories in Philosophy and Mathematics
, cit., p. 249.
173
The example is more fully worked out in my essay
Vico’s Three Realms
,
cit., pp. 73-74.
174
Kleinert explains: «Die raison d’etre des Kategorienbegriffs ist der Vor-
rang der Morphismen vor den Objekten. Wendet man dieses Prinzip auf die
Kategorien selbst an, gelangt man zu den Morphismen zwischen Kategorien,
den
Funktoren
(The raison d’être of the category concept is the precedence of
morphisms over objects. By applying this principle to the categories them-
selves, one arrives at morphisms between categories, i.e.
functors
)» (italics origi-
nal) (Id.,
Mathematik für Philosophen
,
cit., p. 72). While the term
morphism
is a
synonym for all kinds of transformations, just for expository purposes, we are
reserving the term for those within a category, and the term
functor
for those
that involve passing from one category to another. The process of categorical
generalization does not stop here; the next higher level of morphisms leads to
natural transformations
(of functors themselves), but these do not play an explicit
role in the present exploration of Vico.
175
E. Kleinert,
Mathematik für Philosophen
, cit., p. 73.
176
«Broadly speaking, measurement theories attempt to specify the condi-
tions under which empirical objects can be represented with numbers or other
mathematical entities. This task is complicated by the fact that mathematical
relations among numbers do not always correspond to empirical relations
among measured objects. For example, 60 is twice 30, but one would be mis-
taken in thinking that an object measured at 60
⁰
C is twice as hot as an object
at 30
⁰
C»: E. Tal,
Old and New Problems in Philosophy of Measurement
, in «Philoso-
phy Compass», 8, 2013, 12, pp. 1159-1173, p. 1163.
177
In a mathematical context, it often involves constructions and other
complex operations. See E. Kleinert,
Mathematik für Philosophen
, cit., pp. 73-74.
178
Again, in the mathematical setting, this is exemplified by
equivalence clas-
ses
and
power sets
, to mention elementary cases. As Kleinert observed: «[E]s ist
heute selbstverständlich geworden, jede mathematische Konstruktion nach
ihrer Funktorialität zu befragen (Today it has become a matter of course to
examine every mathematical construction as to its functoriality)» (
ibid.
, pp. 73-
74).