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Horst Steinke
56
ICS,
in R. W. Shahan and J. I. Biro
(ed. by),
Spinoza: New Perspectives
, Norman,
University of Oklahoma Press, 1978, pp. 71-111. Lachterman described his
approach as follows: «The contours of Spinoza’s project stand out more
sharply when set alongside those works of his most prominent contemporar-
ies or near-predecessors» (
ibid.
, p. 74). Our goal is therefore strictly limited,
namely, only to make the “contours” of Vico’s geometrical method in the
“Elements” stand out, rather than to do full justice to one of the major works
of philosophy in history.
70
Historically, geometry of a fairly sophisticated kind was practiced long
before Euclid, but it were Greek philosopher-mathematicians who started to
cast it in a logico-deductive system, using such concepts as definitions, postu-
lates, axioms, theorems, corollaries, conclusions, proofs, etc. (For ancient
mathematics, see, for example, V. Katz
(ed. by),
The Mathematics of Egypt, Meso-
potamia, China, India, and Islam: A Sourcebook
,
Princeton-Oxford, Princeton
University Press, 2007; on the Greek achievement, see C. A. Wilson,
On the
Discovery of Deductive Science
, in «The St. John’s Review»
,
XXXI, 1980, 2, pp. 21-
31; for example, comparison of Babylonian and Greek mathematics motivated
the conclusion: «While Greek geometry was
abstract and reasoning
, Babylonian
geometry was
concrete and numerical
”, (J. Friberg,
A Remarkable Collection of Baby-
lonian Mathematical Texts
, in «Notices of the AMS [American Mathematical So-
ciety]», 55, 2008, 9 pp. 1076-1086, p. 1079; italics original). Thus, when refer-
ring to the “geometric method”, it is not geometry
per se
that is in view but the
logico-deductive approach in general; see also the same distinction made in
Th. C. Mark, “
Ordine Geometrica Demonstrata”: Spinoza’s Use of the Axiomatic
Method,
in «The Review of Metaphysics», 29, 1975, 2, pp. 263-286, pp. 264-
265).
While this is not the place for a closer look at the indispensable role that
logic and deduction play in mathematics, it still deserves mention that mathe-
matics cannot be absolutely equated to, or exhausted by, the application of
logic and deduction alone. See E. N. Giovannini,
Intuición y Método Axiomático
en la Concepción de la Geometría de David Hilbert
, in «Revista Latinoamericana de
Filosofía», XXXVII, 2011, 1, pp. 35-65; D. Babbitt - J. Goodstein,
Guido
Castelnuovo and Francesco Severi: Two Personalities, Two Letters
, in «Notices of the
AMS», 56, 2009, 7, pp. 800-808; referring to two leading contributors of the
Italian school of algebraic geometry in the late 1800’s and early 1900’s, stating:
«Castelnuovo was an unabashed champion of the role of intuition in the suc-
cess of the Italian school» (p. 801). More recently, the mathematician I. Stew-
art wrote: «Proofs are discovered by people, and research in mathematics is
not just a matter of step-by-step logic» (Id.,
Visions of Infinity: The Great Mathe-
matical Problems,
New York, Basic Books, 2013, p. 10).