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Horst Steinke
62
themes, […] of contemplation
sub specie aeternitatis,
and
amor Dei intellectualis
)».
(Id.,
Vico e la tradizione “platonica”. “La filosofia dell’umanità e la storia universale delle
nazioni”
, in «BCSV», XXII-XXIII, 1992-1993, pp. 65-102, p. 102).
105
G. Vico,
Liber metaphysicus
, Chapter VII, § IV, according to
On the Most
Ancient Wisdom of the Italians
, trans. by J. Taylor, with an introduction by R.
Miner, New Haven-London, Yale University Press, 2010, p. 121.
106
Ibid.
, chapter III, p. 53. The reference is to Section IV of
De ratione
,
found in
On the Study Methods of Our Time
, cit., p. 23, which reads: «We are able
to demonstrate geometrical propositions because we create them; were it pos-
sible for us to supply demonstrations of propositions of physics, we would be
capable of creating them
ex nihilo
as well».
107
Liber metaphysicus
, Chapter I, § I,
On the Most Ancient Wisdom of the Ital-
ians
, cit., p. 25. The originality of defining the
point
as an entitity that is “sim-
ple” in that it has no parts can be seen by remembering that it was not con-
sidered the only definition possible. Vico himself referred to the competing
view that «even the smallest particles […] are infinitely divisible”, and of «a
geometry which defines the point as a minimal particle divided endlessly»
(
ibid.
, pp. 61, 63).
108
An important means of devising such constructions are so-called “aux-
iliary constructions”, such as bisecting an angle, bisecting a line, drawing a line
at a right angle from a given point, and drawing a straight line perpendicular
to another line, as outlined in Book I, Propositions 9-12, of Euclid’s
Elements
.
See Euclid,
The Thirteen Books of Euclid’s Elements
, trans. by T. L. Heath, Cam-
bridge, Cambridge University Press, 1908, vol. 1, pp. 264-275. A specific ex-
ample of an important auxiliary construction, to prove the Triangle Sum Con-
jecture, can be found in M. Serra,
Discovering Geometry: An Investigative Approach
,
Emeryville, Key Curriculum Press, 3
rd
edition, 2003, p. 200. On the crucial
heuristic role of auxiliary constructions, see also J. Hintikka - U. Remes,
The
Method of Analysis: Its Geometrical Origin and Its General Significance
, Dordrecht-
Boston, D. Reidel, 1974, pp. 41-48.
109
§ 349 is discussed in some detail by Remaud (
Vico lector de Espinosa
, cit.,
pp. 198-201), and Otto (
“Contextualidad” científica y “convertibilidad” filosófica
, cit.,
p. 173), but without drawing a distinction between geometrical method and
demonstration.
110
See T. Crilly with D. Johnson,
The Emergence of Topological Dimension The-
ory
, in
History of Topology
, ed. by I. M. James, Amsterdam, Elsevier, 1999, pp. 1-
24, pp. 1, 20-22. Actually, there are several “dimension theories” which under-
lines the “creative” rather than “constructive” nature of the process of
demon-
stration
. Each “dimension theory” has been “composed”, to use a Vichian
term, in a different way, according to its own logic.