VII
Modern axiomatic theory, in the beginning, did not
seem to come into science
except as a rigorous proce-
dure of exposition
and as a refinement of the deductive
presentation of a theory. Axiomatics
is now an integral
part of science, as a new discipline having its own field
of
studies. But it becomes at the same time a general
scientific tool; used
once simply as a means of expres-
sion it
assumes now the role of a method of research.
In this third aspect,
axiomatic theory is intimately
associated with the modern theory of groups.
An axiom-
atic system may itself be regarded
as the repre-
sentation of a group,
namely, the group of operational
transformations which it permits its terms
to undergo.
Both axiomatics and group theory are devoted to dis-
engaging formal structures, and thereby
succeed in
exposing the unexpected relationships among appar-
ently heterogeneous theories. Since the
beginning of
this century, therefore, not only have all branches of
mathematics, from set theory to the calculus of proba-
bilities, been axiomatized in many ways, but this
work
of axiomatization has in addition had the effect of
reorganizing
the division of various mathematical dis-
ciplines; and redistribution of such disciplines is no
longer based
on the nature of the objects studied but
on the common or different
features of their funda-
mental structures.
Axiomatization has proceeded from logic and math-
ematics, from which it arose, to become progressively
extended
to the whole gamut of the sciences. An idea
which is intimately related to
it, the idea of a model,
explains this extension. This can be seen, for
example,
by the place occupied by the idea of model (or inter-
pretation of a formal system of
axioms) in the Proceed-
ings of the 1960 International Congress of Logic,
Methodology,
and Philosophy of Science (Stanford,
1962) and by the generally
acknowledged importance
of models in contemporary scientific work. The
term
“model of a deductive theory” is applied to
another
deductive theory which has the same logical structure;
that is
to say, all of the terms and propositions of the
model are in a
“biunique” relation to the first theory;
the first
theory can then, of course, be regarded recip-
rocally as a model of the second. So it is possible for
two or
more concrete or semi-concrete theories, even
when they bear on totally
different objects, to be ex-
pressed by one
and the same abstract calculus, or in
other words, they may be derived from
the same axiom-
atic system of which they are
simply different inter-
pretations. We
can thus understand how axiomatics was
able to become a universal
scientific tool; the axiom-
atized systems
of logic and mathematics were only
particular applications of this
intellectual instrument
to a privileged but in no sense exclusive domain.
It was natural, nevertheless, for Mechanics and
Physics, the most
mathematized sciences, to have
soonest and best appropriated the axiomatic
method.
Theoretical physics had for a long time been ex-
pounded in deductive form. It had, when occasion
called for it, transposed one formal structure to another
(e.g., electromagnetic theory to the theory of light
waves), and it was
from the language of physics that
axiomatics borrowed the very term
“model” (e.g.,
Kelvin's mechanical model of
electricity). Theoretical
physics now gradually lends itself to the growing
needs
of axiomatization, not only for the presentation of
classical
theories, but also for introducing new theories:
e.g., the special theory
of relativity (H. Reichenbach,
Axiomatik der relativistischen Raum-Zeit
Lehre,
Vieweg, Braunschweig, 1924) and quantum-theory (H.
Weyl, Gruppentheorie und Quantenmechanik,
Leipzig,
1923). Then, axiomatization has been applied to scien-
tific domains scarcely mathematized;
because of the
very fact that a formal axiomatic system eliminated
the
memory of the intuitive ideas which had gone into
it and had thus ceased to
remain attached to strictly
mathematical notions, it became aptly
disengaged from
them in such domains. Thus, it was possible to extend
axiomatization to Biology (J. H. Woodger, The Axio-
matic Method in Biology,
Cambridge, 1937) and to
Psychology (C. L. Hull, Mathematico-Deductive Theory
of Rote Learning: a Study in Scientific
Methodology,
New Haven and London, 1940). These theories do
not
then reach the complete formalization which the theo-
ries of Logic and Mathematics have attained by being
reduced to pure calculi on signs; nevertheless, they are
on the road to
formalization.
We venture even to say that today axiomatization,
if not in all its rigor at
least in spirit, inspires the
present refurbishing of methods in the human
sciences.
In the nineteenth century the economists of the so-
called “classical”
school had frequently proceeded in
a deductive manner (D. Ricardo), and
certain ones had
even introduced the use of algebraic formulas (A.
Cournot, L. Walras). But in our own time we witness
a wide movement,
bearing on the totality of the sci-
ences of
man under the impetus and example of lin-
guistics, to modify profoundly their style of inquiry;
instead of
limiting research by the Baconian precept
of mounting gradually by
prescribed steps (per gradus
debitos) from
the experienced facts to more and more
general laws, some scientists,
without giving up such
an empirical investigation of new materials, try to
go
immediately from observation of the facts to the con-
struction of a formal theory conceived
as a system of
relations and performing the role of an axiomatic sys-
tem for these facts. Confronted with social,
economic,
linguistic, ethnological facts, contemporary struc-
turalism tries, as Claude
Lévi-Strauss said to an inter-
viewer (Le Nouvel Observateur, 25-31 Jan.,
1967), “to
represent these facts in the form of models
taking
always into consideration not the terms but the
rela-
tions
among the terms. It then treats each system of
relations as a particular case of other systems, real or
merely
possible, and seeks to explain them as a whole
on the level of the rules of
transformation which permit
one to go from one
system to another.” The exemplary
status of axiomatic procedures
is well shown when, for
instance, we see some linguists today aiming at
the
construction of a “formalized grammar” apt to
elimi-
nate intuition in the learning of a
foreign language,
thus bypassing the traditional inductive procedure
by
a radical reversal, in the name of a “Cartesian linguis-
tics” (N. Chomsky,
Carresian Linguistics, New York,
1967).
Axiomatic formalization is tried also in other do-
mains, which this time transcend the boundaries of
science, properly
speaking, viz., in cases where the
evaluation of a statement is no longer
in terms of what
is true or false, but according to what is just and
unjust.
Confined for a long time to deductive statements, logic
has
for the last few decades been extended to norma-
tive, evaluative, and imperative sentences. Many works
have been
devoted to the constitution of a “deontic
logic,”
following G. H. Wright (An Essay in Modal
Logic,
Amsterdam, 1951). Now such a logic is exactly
adapted to the language of
the law, and many efforts
are being made today to construct a
“logic of law”
(V. Klug, furistische Logik, Berlin, 1951); or if this
expression
appears disputable, in order to try to give
to the language of the law a
logically rigorous form,
what today can only mean a formalized axiomatic.
It
is true that attention has been concentrated on the
applications of
law, i.e., to the analysis of legal argu-
ments rather than to the axiomatizing of the legal
doctrines
themselves; but the idea is on the way. One
can well judge what the value
of the success of such
ventures would be not only as a speculative but
also
as a practical matter. The editors of legal codes, of
constitutions, international treaties, and even of con-
tracts only, are haunted by the two preoccupations of
avoiding contradictions and loopholes. These are pre-
cisely the problems of consistency and completeness
in the
theory of axiomatic systems. It would be obvi-
ously advantageous to be able to solve these problems
whenever the
system reaches a certain degree of com-
plexity, by substituting a formalized demonstration for
an intuition
which is always likely to be fallible and
incommunicable.