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#### IV

This new transformation of the logico-mathematical
sciences started with the working out of the non-
Euclidean geometries (N. Lobatchevsky, 1826; 1855;
W. F. Bolyai 1828; and B. Riemann, 1854); since they
reflected on the Euclidean axiomatization itself it stood
in need of logical reinterpretation. It became clear that
Euclid's postulates are not necessary truths since it is
possible to construct perfectly consistent logical sys-
tems on the negation of some of the postulates. As a
result, the thought gradually arose that truth in pure
mathematics was no longer a property of isolated
statements or formulas but must refer solely to the
formal consistency of the whole system. Demonstration
ceased to be categorical, and no longer aimed to ad-
vance evidence but simply to establish a link from
principles as premisses to consequences as conclusions,
i.e., between primitive propositions and theorems.
Mathematics has thus become a hypothetico-deductive
science, to use M. Pieri's expression. Of the two func-
tions which mathematical principles served conjointly,
only one remains, namely, to serve as premisses of a
deductive system.

By the same token, demonstration also retains only
one of its former functions, but finds it necessary to
fulfill this function by meeting new formal require-
ments. So long as the material truth of propositions
was the chief concern, demonstration, in trying to
establish the latter, only played the role of a means;
one might eventually do without it, tolerate its gap
and its ambiguities, provided that intuition could fill
the gap by playing its persuasive role. Everything
changed when the logical organization of the system
came to the foreground. The aim of demonstration is
no longer a pedagogical or didactic one; it aims to
establish “objective relations” (B. Bolzano) which hold
between propositions. And logical rigor has to be im-
posed all the more, because in the generalized systems
of geometry the new propositions are often resistant
to our intuition and can therefore be supported only
by a logical apparatus that is faultless. These new
requirements reflect naturally on the Euclidean system
itself in which some inadequacies become more ap-
parent; namely, the concealment of links in the logi-
cal reasoning with appeal to diagrams as substitutes,
a confusion between the fruitful principles of theory
and the governing rules of reasoning, the inclusion of
definitions among the principles, etc.

In 1882 M. Pasch in his lectures on the new geome-
try (Vorlesungen über neuere Geometrie) formulated the
following conditions for a rigorous deductive exposi-
tion: (1) primitive terms and primitive propositions,
through which all other terms are defined and all other
propositions are demonstrated, must be stated ex-
plicitly with no omissions; (2) the relations among
the primitive terms formulated in the primitive
propositions must be purely logical relations without
the intrusion of geometrical intuitions, and the
demonstrations must appeal only to these logical
relations.

A theory axiomatized according to these require-
ments will then no longer contain at its base the three
kinds of propositions (definitions, axioms, postulates)
in the traditional geometrical demonstrations inspired
by Euclid, but will consist of a group of propositions
of a single kind; it will make no difference whether
they are called postulates or axioms since the axioms,
having lost their privileged self-evidence, have hence-
forth the same function as postulates. These primitive
propositions, like all the others belonging to the system,
are composed of two sorts of terms: those which belong
distinctly to the theory—in this case, the geometrical
terms, e.g., in Pasch: point, segment, plane, superposa-
ble on
... —and those which serve to state the logical
relations among these primitive terms, for example, all,
and, not, if..., then, is a
..., etc., eventually with
terms borrowed from presupposed theories, for exam-
ple, the terms of arithmetic in this case. Just as the
primitive propositions are simply postulated without
proof or even strictly asserted, so the primitive terms
are taken as indefinable for analogous reasons, since
definitions cannot be reduced indefinitely to others. But
how will their meaning be determined, if there is no
question of allowing one to refer to some prior intuitive
meaning? It will be determined, and exclusively so, by
the relations among them which the primitive propo-
sitions state within the relational framework set by the
axioms.

This last point is especially important in that its
effect is to subordinate terms to relations, that being
the direction already of all modern science in opposi-
tion to that of the ancients. Without going into its
philosophical implications this reversal has had a con-
siderable scientific bearing. It determined a turn in the
employment of axiomatization by making of it not only
a mode of exposition supremely satisfactory from a
logical point of view, but also a new scientific tool
whose importance soon became apparent. It has be-
come clear that this sort of “implicit definition” of the
primitive terms by the group of axioms, as J. Gergonne
already knew, only determines their meaning as a total
system in an equivocal manner which makes possible
a variety of interpretations, as, for example, in certain
systems of equations the values of the unknowns are
determinable by the whole group of the terms in their

168

mutual relations, not each one separately, thus allowing
very many interpretations. In other words, only the
relations are determined exactly and universally by the
axioms, but nothing prevents the same system of rela-
tions
from being able to support different systems of
specific interpretations. The object of an axiomatic
system is therefore, properly speaking, a certain ab-
stract structure. Such a structure, which undoubtedly
has been suggested by some concrete embodiment, is
nevertheless capable of being interpreted by many
other “models” which possess a structural identity
(isomorphism). However, the structure can also be stud-
ied by itself by going past the “threshold of abstrac-
tion” (F. Gonseth), without regard for the more
concrete interpretations. Far from being indigent or
destitute of meaning, this relative indeterminateness
accounts, on the contrary, for the chief value of axiom-
atic systems in that it enables one to disengage what
many diverse and apparently heterogeneous theories
have in common from a formal viewpoint, and thus
to think the many in the one (εἰς ἔν τὰ πολλά).

The truth is that it was not necessary to wait for
modern axiomatics or even non-Euclidean geometries
to become aware of the fact that the same system of
relations might handle different contents. Physicists and
mathematicians could not have failed to notice this
fact. Thus it was, for example, that the projective
geometry of J. V. Poncelet made use of the “principle
of duality” which enabled Gergonne to expound its
principles (1824) by writing them in two columns, in
which the terms point and plane were interchanged
when one passed from right to left, the relations of
these two terms to straight lines, as fixed by the axioms
of the theory, being identical. However, the generali-
zation of this procedure, which seemed so exceptional,
did not appear clearly until the systematic develop-
ment of axiomatics at the beginning of the twentieth
century. The idea came to be clearly perceived already
in the first axiomatic systems constructed in conformity
with the ideal conceived by Pasch. This occurred in
the system of arithmetic by G. Peano (Arithmetices
principia nova methodo exposito,
Turin, 1899) which
rests on five axioms containing three primitive terms,
and in the system of geometry by D. Hilbert who in
his Foundations of Geometry (Grundlagen der Geom-
etrie,
Leipzig, 1899) divides its twenty-one axioms into
five groups depending on whether they deal with con-
nection, order, congruence, parallelism, and continuity
respectively. These axiomatic systems have been fol-
lowed in this century by many others, and they tend
to be concentrated on the theory of sets as the basis
of the whole of mathematics (cf., Hao Wang and R.
McNaughton, Les systèmes axiomatiques de la théorie
des ensembles,
Paris, 1953).