University of Virginia Library

Search this document 
Dictionary of the History of Ideas

Studies of Selected Pivotal Ideas
7 occurrences of Dictionary_of_the_History_of_Ideas
[Clear Hits]
  
  
expand section 
  
expand section 
  
  

expand sectionVI. 
expand sectionV. 
expand sectionVI. 
expand sectionI. 
expand sectionVI. 
expand sectionV. 
expand sectionIII. 
expand sectionIII. 
expand sectionVI. 
expand sectionVI. 
expand sectionV. 
expand sectionV. 
expand sectionIII. 
expand sectionVII. 
expand sectionVI. 
expand sectionVI. 
expand sectionIII. 
expand sectionIII. 
expand sectionII. 
expand sectionI. 
expand sectionI. 
expand sectionI. 
expand sectionV. 
expand sectionVII. 
expand sectionVI. 
expand sectionV. 
expand sectionIII. 
expand sectionIII. 
expand sectionIII. 
expand sectionII. 
expand sectionI. 
expand sectionI. 
expand sectionI. 
expand sectionVI. 
expand sectionVII. 
expand sectionIII. 
expand sectionVII. 
expand sectionVII. 
expand sectionVII. 
expand sectionV. 
expand sectionVI. 
collapse sectionVI. 
  
  
  
expand sectionVI. 
expand sectionVI. 
expand sectionVI. 
expand sectionVII. 
expand sectionIII. 
expand sectionIV. 
expand sectionVI. 
expand sectionVI. 
expand sectionVI. 
expand sectionV. 
expand sectionV. 
expand sectionV. 
expand sectionIII. 
expand sectionIII. 
expand sectionVII. 
expand sectionIII. 
expand sectionI. 
expand sectionV. 
expand sectionV. 
expand sectionVII. 
expand sectionVI. 
expand sectionI. 
expand sectionI. 
expand sectionI. 
expand sectionI. 
expand sectionVI. 
expand sectionIII. 
expand sectionIV. 
expand sectionIII. 
expand sectionIV. 
expand sectionIV. 
expand sectionIV. 
expand sectionVI. 
expand sectionVI. 
expand sectionVI. 
expand sectionV. 
expand sectionIII. 
expand sectionVI. 

7 occurrences of Dictionary_of_the_History_of_Ideas
[Clear Hits]

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


171

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


172

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.