*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).