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Vico’s Ring
65
clidean geometry, and other comparable forms of mathematical reasoning.
Interestingly, Fletcher, in his review of J. R. Goetsch,
Vico’s Axioms,
brought a
modern result into the discussion, namely K. Gödel’s
incompleteness theorem
(1931) (A. Fletcher,
Book Reviews
, in «NVS», 14 (1996), pp. 86-90, p. 89). It
stunned the mathematical community by showing that the deductive system is
unable to produce every true statement, or viewed vice versa, that not every
true statement could be proved within the system (see D. R. Hofstadter,
Gödel,
Escher, Bach: an Eternal Golden Braid
, New York, Vintage Books, 1980, pp. 15-
24, 82-102). This observation is of course anachronistic, in a strict sense;
however, it is not without basis altogether as far as the early modern era is
concerned. As the philosopher of logic, J. Hintikka, remarked: «This kind of
reaction [divorcing logic from mathematics] to Gödel’s incompleteness result
is not a peculiarity of twentieth-century philosophers. It is in reality part and
parcel of a long tradition which goes back at least to Descartes» (Id.,
The Prin-
ciples of Mathematics Revisited
, Cambridge, Cambridge University Press, 1998, p.
89). As Spinoza studies have suggested, Spinoza presupposed
deductive
(self-)completeness,
as a result of which «it turns out that his
Ethics
exhibits the
defining properties of the universe it describes. That is to say, the
Ethics
is
«that the conception of which does not need the conception of another thing
from which it must be formed» (E-I, def. 3. Th. C. Mark,
Ordine Geometrica
Demonstrata,
cit., pp. 283-284; see also M. Hooker,
Deductive Character of Spino-
za’s Metaphysics
,
cit., pp. 30, 301, on “incompleteness” in
Ethics
). For further
philosophical implications of Gödel’s incompleteness theorem, see e.g. E.
Moriconi,
Il mito del sistema completo,
in «Teoria», 2005, 2, pp. 183-190.
In Vico’s case, the question of completeness/incompleteness is of another
sort. To follow Hintikka’s terminology, it has to do with so-called
descriptive
completeness
(Id.,
Principles of Mathematics
, cit., pp. 91-95), of which Hintikka said:
«This notion is definable completely independently of any axiomatization of
the underlying logic, and hence independently of all questions of deductive
completeness» (
ibid.
, p. 95). An echo of Vico’s aspiration of descriptive com-
pleteness (whether fully realized or not) is detectable in statements of readers
such as: «the penchant for generalizing and turning every insight into a “prin-
ciple”» (D. R. Kelley,
Vico’s Road
,
cit., p. 17), and: «Vico reverses Occam’s
sense of the economy of thought. We find not Occam’s razor, but Vico’s
magnet. Principles are multiplied and as many as possible are drawn into the
presentation of a given point» (D. Ph. Verene,
Vico’s Science of Imagination
, Itha-
ca-London, Cornell University Press, 1981, p. 106).
124
As V. Hösle remarked: «Die Frage nach dem Ursprung der Axiome hat
Vico nicht beantwortet (Vico did not answer the question of the origin of the
axioms)» (Id.,
Einleitung
, cit., p. CXIII).