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
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).