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Horst Steinke
48
of geometry and arithmetic (mathematics) is at the center of the
debate (and polemic with Cartesianism). In referring to
geometry
,
Vico introduces a key distinction between two kinds, or levels, of
geometry: «Permit me to conclude with the following: not the
geometrical
method
, but the geometrical
demonstration
should be
imported into physics»
105
(italics added). Throughout
Liber meta-
physicus
, Vico, as well as his
Responses
to reviews, leaves no ambi-
guity about what he means by
demonstration
; it is not the inferen-
tial or deductive procedure terminologically denoted by the
geo-
metric method.
Immediately after the above programmatic state-
ment, Vico added a brief example of what he meant: «Galileo
[…] consider[s]
first principles
of physics in terms of mathematical
first principles
» (italics added). Geometrical
demonstration
thus has a
foundational role in geometry. It is, in fact, a paradigmatic exem-
plification of the very
verum-factum
principle, as Vico himself stat-
ed earlier in
Liber metaphysicus
: «In my treatise
On the Study Methods
of Our time
, I said this as well:
the reason that we demonstrate things in
geometry is that we make them; if we were able to demonstrate things in
physics, we would have to make them too
»
106
(italics in the original).
Among the entities thus brought about are the primitives of ge-
ometry: point, line, surface, and higher-dimensional elements:
[M]an […], like God, […] creates point, line, and surface out of no
substrate, as if out of nothing; by the name point, he understands
something which has no parts; by […] line, […] the extension of a
point or length without width or depth; by […] surface, […] the join-
ing of two separate lines at one point or length with width, but without
depth
107
.
Geometrical
demonstration
, therefore, refers to original, funda-
mental geometrical/mathematical thinking that precedes the
practice of geometry as commonly understood, and which –
notwithstanding required ingenuity – takes such primitives as
given, and turns them into geometrical constructions
108
. The
fundamental difference between geometry as
method
and
demon-