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Dictionary of the History of Ideas | ||

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*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

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,

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-*

tionsamong the terms. It then treats each system of

tions

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.

Dictionary of the History of Ideas | ||