II
At the start of the modern period, the instrumental
and exemplary nature of
mathematics recognized by
the new science led to extending the
mathematical
mode of exposition to various disciplines. This occurred
first in the extension of the work begun by the Greeks
to the science of
nature and, more exactly, to that part
which is generally regarded as its
foundation, namely,
Mechanics. Galileo was inspired by the method of
Archimedes, and tried to do for Dynamics what Archi-
medes had done for Statics. Descartes, in his Principles
of Philosophy (Principia
philosophiae, 1644), postulated
three “laws of
nature” dealing with motion, justifying
them a
priori through God's perfection, and claiming
that he could
demonstrate all of physics by means of
these three laws. Finally, and above
all, Newton in his
Mathematical Principles of Natural Philosophy (Philo-
sophiae naturalis
principia mathematica, 1687), orga-
nized Mechanics in the form of a logical system which
has
remained classical. It was taught often best, espe-
cially in France, as a mathematical discipline. Newton's
work
opens with the statement of eight definitions and
three axioms or laws of
motion, starting from which
Books I and II demonstrate a great many theorems.
However, the prestige of the Euclidean axiomatic
model was such that after
going beyond mathematics,
it won over disciplines which are outside of
science
properly speaking. Descartes, while maintaining his
preference
for the analytical order of his Meditations,
had
already agreed, to satisfy the authors of the Second
Objections, to expound in synthetic order the “reasons
which prove the existence of God and the distinction
between the mind and
the human body, the reasons
arranged in a geometric manner,”
demonstrating his
propositions through definitions, postulates, and
axioms. His example was followed by Spinoza, with
a breadth and rigor which
fascinated many minds, in
his Ethics, demonstrated in a
geometric order (Ethica
ordine geometrica
demonstrata, 1677); Spinoza's work
was expounded by subjecting it,
from one end to the
other with no exceptions, to the requirements of Eu-
clidean standards with definitions,
postulates, and
axioms followed by propositions, demonstrations,
corollaries, lemmas, and scholia.
Jurisprudence, along with metaphysics and ethics,
also entered upon the road
of axiomatization. When-
ever Leibniz wished to
give examples of disciplines
containing rigorous reasoning he mentioned the
works
of the Roman jurisconsults as well as of the Greek
mathematicians. He offered an example himself of a
juridical exposition by
definitions and theorems in his
sample of legal persuasion or demonstration
(Specimen
certitudinis seu demonstrationum in
jure, 1669) in which
he refers to “those ancients who
arranged their rebut-
tals by means of very
certain and quasi-mathematical
demonstrations.” Not long before,
Samuel von Pufen-
dorf had published his Elementa jurisprudentia uni-
versalis (1660), written under the double inspiration
of Grotius and his own teacher Weigel who taught both
law and mathematics.
Pufendorf wished to show that
law, rising above historical contingencies,
contains a
body of propositions which are perfectly certain and
universally valid, and capable of being made the con-
clusions of a demonstrative science. As a matter of
fact,
here, as in Leibniz, axiomatization was still only
making a
start. Instead of producing the propositions
and their proofs as logical
consequences of principles,
Pufendorf presented them substantially in
extensive
commentaries which follow each one of his twenty-one
definitions in order to avoid, he said, “a certain aridity
which
might have run the risk of distorting this disci-
pline if we had presented it by cutting it up into small
parts,
as is the manner of mathematics.” In the wake
of Pufendorf the
so-called school of “natural law and
human right”
elaborated for more than a century theo-
ries
in which “one deduces through a continuous chain
leading from
the very nature of man to all his obliga
tions and all his rights,” restating the subtitle of
one
of Christian Wolff's works. Wolff, as a disciple of
Leibniz,
boasted of accomplishing what others had only
proposed to do, namely, to
deal with the theory of
human actions according to the demonstrative
method
of the mathematicians (Philosophia practica
universalis,
methodo scientifica pertracta, Frankfurt and
Leipzig,
1738-39). Nevertheless, here also, we are quite far from
the
logical rigor and even the mode of presentation
of Euclidean geometry.