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AXIOMATIZATION

Axiomatization as the organization of a deductive
system in a strictly axiomatic form dates from the last
part of the last century. The very use of the term

“Axiomatics” as a noun is even more recent; it is not
to be found in recent editions of the Encyclopedia
Britannica (1962). It is mentioned in the Enciclopedia
Italiana (1949); there it is defined as the “name adopted
recently to signify that branch of mathematical science
which deals with the ordering of principles” (F. En-
riques). If we adhered rigidly to this narrow definition,
the history of “axiomatics” and of axiomatization
would be a brief one, and its domain would be confined
to mathematics alone. Here we must adopt the broader
interpretation in which these terms are often under-
stood and in which the very word “axiom” is included:
an axiomatic system is one composed of propositions
deducible from a small number of initial propositions
posited as axioms. But what then is an “axiom”?

“There is,” Leibniz says, “a class of propositions
which, under the name of maxims or axioms, pass as
the principles of the sciences.... The scholastic phi-
losophers said that these propositions were self-evident
ex terminis, that is, as soon as the terms in them are
understood” (New Essays IV, vii, 1). And Bossuet de-
clares: “Those propositions which are clear and intelli-
gible by themselves are called axioms or first princi-
ples” (Connaissance de Dieu I, 13). Thus, in its classical
usage—with various modifications which we shall see
later—an axiom is characterized as combining two
features: as a principle it is the beginning or the basis
of a group of propositions which it serves to demon-
strate; as a self-evident truth known immediately as
such, it compels conviction without the aid of any
proof. It is, therefore, at one and the same time a
certainty by itself and the basis of our certainty with
respect to the propositions following from it.

Axiomatization will then consist in organizing a body
of propositions into a deductive system such that the
principles of this system appear indubitable by virtue
of their own self-evidence; the result is that the deduc-
tive apparatus performs the functions of communi-
cating or transmitting to the group of propositions of
the system the evidence and consequently the certainty
of the axioms; this produces what has been called a
“transfer of evidence.” The deduction is in such a case
categorical; it is demonstrative in the sense in which
Aristotle defines demonstration as the “syllogism of the
necessary,” the necessity residing both in the connec-
tion of the propositions and in the very positing of
the initial propositions. Such should be the ideal form
of scientific exposition, according to Aristotle: “it is
necessary that scientific demonstration start from
premisses which are true, primitive, immediate and
more evident than the conclusions, being prior to them
as their cause” (Posterior Analytic I, 2). This ideal was
to be perpetuated, with few exceptions, until the be-
ginning of the modern era.

I

The typical example, which has been invoked for
more than twenty centuries as an unsurpassable model,
is the method with which Euclid (ca. 300 B.C.) ex-
pounded geometry in his Elements. Most of the subject
matter had already been acquired; Euclid's merit is
due to the manner in which he organized it. “Euclid,”
Proclus says, “assembled the elements, arranged in
order many truths discovered by Eudoxus, completed
what had been begun by Theaetetus, and proved more
rigorously what had also been too loosely shown before
him.”

It is well known how Euclid's system is presented.
At the beginning of Book I appear statements of defini-
tions (ὅροι), postulates (αἰτήματα), and common notions
(κοιναὶ ἔννοιαι). Each of the succeeding books opens
with additional definitions intended to introduce the
geometrical entities belonging to each book. However,
the common notions and postulates given in the first
book suffice to demonstrate all the propositions and
solve all the problems constituting the whole work
with the sole exception (in the middle of Book I) of two
supplementary postulates about perpendiculars and
parallels. The “common notions” correspond to what
later generations called “axioms”; for example, things
equal to the same thing are equal to each other. Thus
Euclid accomplished the transformation of geometry
from an empirical science to a rational science after
its initiation by Pythagoras. No longer are merely
isolated problems treated “abstractly and by pure in-
telligence” (Eudemus), but the whole of geometry is
organized in a close network in which all the proposi-
tions are linked to each other by logical relations, so
that each proposition is made absolutely clear to the
mind, either through its own self-evidence or through
its logical dependence on the primary data.

This accomplishment was henceforth looked upon
as a model for all the sciences which, beginning with
Physics, were going in turn to be expounded in the
geometrical manner (more geometrico) even if they do
not attain the same level of systematic order as Euclid's
Elements. In antiquity, as a case in point, Euclid's
Optics was constructed on a few initial principles such
as the one which postulates that light rays are trans-
mitted in a straight line; then also Archimedes' On the
equilibrium of planes (third century B.C.) demonstrated
its propositions by starting with a few postulates such
as: equal weights suspended from a lever at equal
distances from the fulcrum are in equilibrium.

We can therefore understand why Euclid has been
regarded as the initiator of axiomatization. Although
not inaccurate, this view must be tempered, however,
by a few reservations. First, the fact is that Euclid's
Elements is not as logically perfect a work as had been

thought for a long time, and also it falls far short of
satisfying all the requirements of modern axiomatics.
It is also a fact that Euclid's Elements did not emerge
suddenly as an absolute novelty. Hippocrates of Chios
(fifth century B.C.) had also written an Elements, a work
unfortunately lost; but we know that he had attempted
in this work a systematic organization of mathematics.
Between the work of Hippocrates and that of Euclid
other efforts had been made by the mathematicians
Leo, Eudoxus, and Theudius.

It would be unjust to overlook certain works before
Euclid's, including even nonmathematical works in
which a very clear approach is made to an axiomatic
treatment. First of all, we find it in Aristotle, not in
his Physics, which, though attempting to be demon-
strative, is still far from axiomatic in form or rigor,
but in his logic or more precisely in his syllogistic
theory as it appears in the Prior Analytics. Of course,
Aristotle does not proceed explicitly through initial
axioms and demonstration of theorems. But from the
standpoint of modern formal logic, as has been shown
by J. Łukasiewicz (Aristotle's Syllogistic from the
Standpoint of Modern Logic, Oxford [1951]; 2nd ed.
enlarged [1957]), Aristotle's text allows one to read it
as an axiomatic work. To do that, it must be remem-
bered that Aristotle formulates his syllogisms not as
schema of inference as the later philosophers did, be-
ginning with Alexander of Aphrodisias and Boethius,
but as logical theses; for example, for the syllogism,
later called Barbara: if A is predicated of all B and
B of all C, then A is predicated of all C. This point
rectified, it appears that the four moods of the first
figure—the so-called “perfect” moods being self-
evident without demonstration—play exactly the same
role as axioms do, on which the moods of the other
figures depend as theorems; the “reduction” of these
moods to those of the first figure is really the same
as demonstrating them from axioms. And then we must
regard as primitive terms of the syllogistic theory the
four operators which function in the axioms to connect
the variables A, B, C, in the elementary propositions:
“belongs to all...,” “belongs to none...,” etc.
Aristotle advanced even further his reduction of the
implicit axiomatic base of his syllogistic theory when
he went on to demonstrate the third and fourth moods
(AII and EIO) of the first figure, by means of only the
universal moods (AAA and EAE), which may thus be
counted as only two axioms.

The Megaric-Stoic logic, contemporary with Aris-
totle's, also offers an example of progress towards axiom-
atization. As a point of departure, five undemon-
strated (ἀναπόδεικτοι) propositions are postulated,
which can easily be translated into the symbolism of
modern logistics; for example, the first proposition
would read: ((p ∃ q) ∙ p) ∃ q. They obviously involve
propositional variables connected by a few logical
operators taken as primitive terms. Not only did they
draw from these primitive propositions, as Cicero as-
sures us, “innumerable conclusions,” but they boasted
being able to reduce every logically important type
of reasoning to these primitive propositions. This was
accomplished by means of four rules of inference ex-
plicitly detached and formulated. What marks this
Megaric-Stoic logic as an advance over Aristotle's are
the following three features: the clear distinction be-
tween axioms and explicitly formulated rules of infer-
ence; the line drawn expressly between concrete rea-
soning (λόγοσ) and its formal schema (τρόπωσ); and the
claim—not challenged by their adversaries so far as
we know, but our inadequate information prevents our
checking this—to have erected a system which would
today be called “complete.” This logic is in fact a move
in the direction of modern axiomatics, anticipating our
modern calculus of propositions (see Benson Mates,
Stoic Logic, Berkeley and Los Angeles [1953]).

Later, and until we reach the rigorous axiomatic
systems of modern symbolic logic, several attempts at
a logic demonstrated in geometric fashion were ex-
pressly made, from Galen (second century A.D.) with
his proposed Logica ordine geometrica demonstrata to
the Logica demonstrativa of Saccheri (1692).

Thus towards the end of Greek antiquity, mathe-
matics, logic, and certain parts of physics had shown
in various degrees the beginnings of axiomatization.

II

At the start of the modern period, the instrumental
and exemplary nature of mathematics recognized by
the new science led to extending the mathematical
mode of exposition to various disciplines. This occurred
first in the extension of the work begun by the Greeks
to the science of nature and, more exactly, to that part
which is generally regarded as its foundation, namely,
Mechanics. Galileo was inspired by the method of
Archimedes, and tried to do for Dynamics what Archi-
medes had done for Statics. Descartes, in his Principles
of Philosophy (Principia philosophiae, 1644), postulated
three “laws of nature” dealing with motion, justifying
them a priori through God's perfection, and claiming
that he could demonstrate all of physics by means of
these three laws. Finally, and above all, Newton in his
Mathematical Principles of Natural Philosophy (Philo-
sophiae naturalis principia mathematica, 1687), orga-
nized Mechanics in the form of a logical system which
has remained classical. It was taught often best, espe-
cially in France, as a mathematical discipline. Newton's
work opens with the statement of eight definitions and
three axioms or laws of motion, starting from which

Books I and II demonstrate a great many theorems.

However, the prestige of the Euclidean axiomatic
model was such that after going beyond mathematics,
it won over disciplines which are outside of science
properly speaking. Descartes, while maintaining his
preference for the analytical order of his Meditations,
had already agreed, to satisfy the authors of the Second
Objections, to expound in synthetic order the “reasons
which prove the existence of God and the distinction
between the mind and the human body, the reasons
arranged in a geometric manner,” demonstrating his
propositions through definitions, postulates, and
axioms. His example was followed by Spinoza, with
a breadth and rigor which fascinated many minds, in
his Ethics, demonstrated in a geometric order (Ethica
ordine geometrica demonstrata, 1677); Spinoza's work
was expounded by subjecting it, from one end to the
other with no exceptions, to the requirements of Eu-
clidean standards with definitions, postulates, and
axioms followed by propositions, demonstrations,
corollaries, lemmas, and scholia.

Jurisprudence, along with metaphysics and ethics,
also entered upon the road of axiomatization. When-
ever Leibniz wished to give examples of disciplines
containing rigorous reasoning he mentioned the works
of the Roman jurisconsults as well as of the Greek
mathematicians. He offered an example himself of a
juridical exposition by definitions and theorems in his
sample of legal persuasion or demonstration (Specimen
certitudinis seu demonstrationum in jure, 1669) in which
he refers to “those ancients who arranged their rebut-
tals by means of very certain and quasi-mathematical
demonstrations.” Not long before, Samuel von Pufen-
dorf had published his Elementa jurisprudentia uni-
versalis (1660), written under the double inspiration
of Grotius and his own teacher Weigel who taught both
law and mathematics. Pufendorf wished to show that
law, rising above historical contingencies, contains a
body of propositions which are perfectly certain and
universally valid, and capable of being made the con-
clusions of a demonstrative science. As a matter of fact,
here, as in Leibniz, axiomatization was still only
making a start. Instead of producing the propositions
and their proofs as logical consequences of principles,
Pufendorf presented them substantially in extensive
commentaries which follow each one of his twenty-one
definitions in order to avoid, he said, “a certain aridity
which might have run the risk of distorting this disci-
pline if we had presented it by cutting it up into small
parts, as is the manner of mathematics.” In the wake
of Pufendorf the so-called school of “natural law and
human right” elaborated for more than a century theo-
ries in which “one deduces through a continuous chain
leading from the very nature of man to all his obliga
tions and all his rights,” restating the subtitle of one
of Christian Wolff's works. Wolff, as a disciple of
Leibniz, boasted of accomplishing what others had only
proposed to do, namely, to deal with the theory of
human actions according to the demonstrative method
of the mathematicians (Philosophia practica universalis,
methodo scientifica pertracta, Frankfurt and Leipzig,
1738-39). Nevertheless, here also, we are quite far from
the logical rigor and even the mode of presentation
of Euclidean geometry.

III

The systems we have discussed—logical, mathe-
matical, physical, metaphysical, ethical, or legal—all
have in common a dogmatic character. Axioms were
supposed to compel assent through their inherent self-
evidence transmitted to later propositions by means
of demonstrations. In the modern period we see this
conception gradually disintegrating, until we reach by
degrees our present conception of axiomatics. This
conception was attained by a progressive dissociation
of the two hitherto intimately related components of
the idea of axiom (self-evident and primary prop-
osition). The transformation was accomplished in
two stages: the first, in the seventeenth century in
connection with the advent of experimental physics;
the second, the beginning of which can be dated in
the early nineteenth century, with the construction of
non-Euclidean geometries.

Descartes still required that the principles of philos-
ophy (including natural philosophy or physics) satisfy
two conditions at the same time: “... one, that they
be so clear and so self-evident that the human mind
cannot doubt their truth when it concentrates on judg-
ing them; second, that the knowledge of other things
depends on the principles which can be known without
these other things but not conversely” (Letter, preface
to the French version of the Principles of Philosophy,
1647). Nevertheless, he admitted, and practiced him-
self, when he needed to, another mode of exposition,
although he regarded it as less perfect than the deduc-
tive mode. This other mode consisted in regarding basic
propositions (general principles) not as principles of
demonstration but as requiring, on the contrary, proof
by the empirical verification of their consequences. In
this he was in agreement, albeit accidentally, with the
practice of the new seventeenth-century physicists
who, following Galileo and Torricelli, were active
around him and Mersenne: Pascal, Roberval, Gassendi.
These principles, that is to say, the propositions from
which deduction starts, are only “suppositions” or
“hypotheses” in two senses of these words: premisses
and conjectures. For, as Pascal maintained, in physics
the experiments furnish the true principles, that is, the

foundations of our knowledge. When in the order of
exposition, though not in the order of discovery, the
order of the propositions is always the same, viz., that
of a deductive synthesis, the meaning of the proof is
reversed: instead of extending beyond the premisses
to the consequences, the truth rebounds from the con-
sequences to support the premisses.

The operational distinction in Physics between the
propositions which it states as its principles and those
that it invokes to establish them, is nowhere better
illustrated than in the Newtonian theory of gravitation,
expounded in Book III of the Principia; it instigated
the battle between Cartesians and Newtonians that
lasted for half a century. One of the chief reasons for
the antagonism of the Cartesians was the idea of at-
traction or action at a distance, which, far from being
a clear idea demanding assent as self-evident, was on
the contrary, unintelligible to them. To which the
Newtonians replied: clear or obscure, self-evident or
not, the principle of gravitation is still a compelling
truth because experience confirms it in very many
precise ways.

Thus classical science was faced with the institution
of a sharp separation between the experimental method
of the physicists and the demonstrative method of the
mathematicians. The result was the uncertain status
of Mechanics, halfway between geometry and physics.
In the middle of the eighteenth century, the Academy
of Berlin offered a prize for the best answer to the
question whether the laws of nature are necessary or
contingent truths, that is to say, whether they are
directly or indirectly purely rational statements or, on
the contrary, simply experimental findings. D'Alem-
bert, a declared Newtonian, replied in the way a
Cartesian would, and presented Dynamics as a demon-
strative science. And in the next century, William
Whewell still wondered about the “paradox of neces-
sary truths acquired by experience,” which suggested
to him his theory about the “progress of evidence.”

A nondogmatic use of deductive method was, in a
manner of speaking, quite ancient. Without discussing
the accidental employment of this method by the
mathematician in his indirect proofs, or by the dialec-
tician in his refutations through reduction of his adver-
sary's arguments to absurdity, we find in antiquity a
systematic use of the hypothetico-deductive method
in expounding one of the sciences of nature which had
already reached a high level of development. Alongside
of physical astronomy there had actually developed,
after Plato, a so-called formal or mathematical astron-
omy which was connected especially with Ptolemy's
work, and which continued to thrive in the Middle
Ages; it did not require that its principles should be
true but only that they allow one to calculate exactly
the empirical data (“to save the phenomena,” σώζειν
τὰ φαινόμενα). This was the notion to which Osiander
also referred when, in his introduction to Copernicus'
De revolutionibus orbium coelestium (1543), he asked
the astronomer “to imagine and invent any hypotheses
whatsoever,” adding: “it is not necessary for these
hypotheses to be true or even probable, the only suffi-
cient condition is that they must lend themselves to
a calculation which agrees with the observations.”
However, the new physics was just as opposed to these
hypotheses which are neither true nor false, as it was
to dogmatic theses. If it no longer posits its principles
as self-evident, it nevertheless proposes them as truths.
As Newton expressly says in the first of his Rules for
Philosophy (Regulae philosophandi) it does not suffice
that the causes invoked by the scientist should be fit
to explain the phenomena, but they must be true in
addition. This new way of employing the hypothetico-
deductive method would discredit in time and relegate
to oblivion the ancient method of “saving the phe-
nomena.”

To summarize, the deductive order, or axiomatic
order in the wide sense, may function in three different
ways, depending on the identifiable nature of its basic
propositions: (1) a categorical deduction which dem-
onstrates the truth of the consequences by the truth
of the principles laid down dogmatically; (2) a hypo-
thetical deduction which proves a posteriori the truth
of its provisional hypotheses by the truth of their con-
sequences; finally, (3) a pure hypothetical deduction
whose principles are introduced as fictions removed
from the domain of what is true or false, so that truth
comes into play only on the level of logical conse-
quences. Whereas Mechanics tried for a while to per-
severe still in the first conception (categorical deduc-
tion), classical physics opted clearly for the second (a
posteriori hypothetical inference), rejecting the two
past legacies of demonstrative physics and fictive
physics. Even when Physics seeks to combine a body
of experimental laws into a unifying theory and even
when it reduces this theory to a certain mathematical
structure, as did J. Lagrange in his Analytical Me-
chanics (Mécanique analytique, 1788), Joseph Fourier
in his thermodynamics (Théorie analytique de la cha-
leur, 1822), and J. C. Maxwell in his Treatise on Elec-
tricity and Magnetism (1873), it does not postulate its
axioms either as necessary truths or as arbitrary fictions,
but as a system of hypotheses; the truth of these hy-
potheses being tested finally by the precise and un-
erring agreement of the many consequences of the
hypotheses with the experiential data. It was only
around 1900 that this radically hypothetical conception
of Physics was revived, on the one hand with the
critique of scientific dogmatism by H. Poincaré, P.
Duhem, and H. Vaihinger's Philosophy of As If (Philos-
ophie des als ob, 1911); on the other hand, and above

all, the hypothetical conception came to the fore when
the logico-mathematical sciences were put into axiom-
atic form, in the strict and most rigorous sense which
the term “axiomatic” has assumed today.

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).

V

We see then that what had happened in antiquity
recurred, on a higher level of development, about 1900:
the axiomatization of mathematics duplicated the axi-
omatization of logic. In 1879, Frege had offered the
first truly axiomatic formulation of the calculus of
propositions. However, the two disciplines of logic and
mathematics are now intimately tied together, as the
very titles of fundamental works show: G. Frege's The
Foundations of Arithmetic, a Logico-mathematical En-
quiry into the Concept of Number (New York, 1950,
trans. of Die Grundlagen der Arithmetik, eine logisch-
mathematische Untersuchung über den Begriff der Zahl,
Breslau, 1884), and Fundamental Laws of Arithmetic,
derived by symbolic representation of concepts (Grund-
gesetze der Arithmetik begriffsschriftlich abgeleitet, Jena
1893-1903); A. N. Whitehead and B. Russell, Principia
Mathematica, 3 vols. (Cambridge, 1910-13, reprinted
1925-27), the classic of the new symbolic logic or
“logistics.” The nineteenth century had arithmetized
mathematics; the proposal of Frege and Russell was
to logicize arithmetic, that is, to construct its primitive
terms and primitive propositions with the aid of purely
logical terms and purely logical propositions. The latter
would themselves be reduced to a small number of
initial statements; thus the Principia Mathematica
makes the whole calculus of propositions rest on two
undefined terms, negation (∼) and disjunction (∨, the
nonexclusive “or”) and on five axioms. By thus de-
ducing arithmetic from logic, the relative indetermi-
nateness of the foundations of arithmetic was to be
corrected; such an indeterminateness was making of
arithmetic, and with it of the whole of mathematics,
a purely formal science in which, as Russell said in
a well-known sally, “one never knows what one is
saying or whether what is said is true.” However, such
an attempt made sense only on condition that it would
not invest the principles of logic with a similar indeter-
minateness, that is to say, that we accord to the funda-
mental ideas of logic an exact meaning and to its laws
an absolute truth. As completely axiomatic as it might
be, logic, in order to offer a solid basis for the whole
mathematical structure, should itself rest on evidence
and assert its principles categorically as mathematics
also used to do until recently.

But this logical dogmatism soon had to yield, as, not
long before, it was mathematical dogmatism which had
to yield in the wake of physical dogmatism. Just as
geometry in the nineteenth century had proliferated
into a multitude of non-Euclidean geometries, and had
besides, by its axiomatization, eliminated intuitive rep-
resentations, so, around 1920, logic all at once in its
turn went on to diversify itself and empty itself of its
substance. On the one hand, J. Łukasiewicz (1920) and

E. L. Post (1921) constructed the first three-valued and
n-valued logics respectively; these were soon followed
by a proliferation of non-Russellian systems. On the
other hand, L. Wittgenstein in his Tractatus logico-
philosophicus (London, 1922) characterized logical
laws, axioms, or theorems as simple tautologies, under-
standing by that term that they are devoid of all con-
tent: “all propositions of logic say the same thing, that
is, nothing”; they are pure forms which remain valid
whatever material contents are poured into them. The
primitive terms no longer retain anything of their
intuitive and pre-axiomatic meaning, which was prop-
erly a logical one; they retain only what the group
of axioms as a whole determines in its systematic
ambiguity; and although many systems are still con-
cerned with maintaining a very close correspondence
between these two meanings, there is no longer any
obligation to do so.

The axiomatization of logic is thus allied to that of
mathematics: its terms become rid of their semantic
burden; its axioms lose their self-evidence and fall into
the rank of postulates which are set up in a more or
less arbitrary manner, either to recover as consequences
a body of formulas previously given or simply to see
what set of formulas might be derived from them. Or
better it may be necessary to say that because of its
complete vacuity a system of logical axioms cannot
be distinguished from a system of mathematical axioms
or even, more generally, from any system of axioms
whatsoever. The distinction would only reappear if,
in descending from a pure science to applications, we
recognize that some system lends itself better to an
interpretation in logical notions and propositions, and
some others to interpretation by mathematical ideas
and propositions, taking the words “logical” and
“mathematical” here in their intuitive and pre-axiom-
atic meaning. Strictly speaking it is only on this level
of concrete or nearly concrete interpretations that we
rediscover the idea of truth. In a purely axiomatic
system, the axioms are no longer genuine propositions,
but simply “propositional functions,” i.e., empty for-
mulas which become genuine propositions for every
interpretation of the primitive terms, and become true
propositions if this interpretation satisfies all of the
axioms. In this last case, every theorem of the system,
i.e., every propositional formula deducible, directly or
indirectly from the interpreted axioms, becomes truly
a proposition and also a proposition which is true.

VI

Compared to the axiomatic efforts made at the end
of the nineteenth and beginning of the twentieth cen-
turies, which can be regarded retrospectively as naive
or semi-concrete, contemporary axiomatization is
characterized by three features anticipated, of course,
by what preceded it, but now sharply asserted and
inseparably united: symbolization, formalization, and
appeal to meta-theories.

Symbolization consists in substituting for the spoken
natural languages, with their national differences and
especially their imperfections in logical respects, a
system of written signs, a “characteristic,” which is an
immediate ideographic representation not exactly of
the ideas belonging to the theory axiomatized, if the
ideas happen to be already represented by signs, but
of the logical articulations of the discourse in which
the theory is developed. It is also not yet a question,
therefore, of “the universal characteristic” dreamed of
by Leibniz, but only of a logical characteristic which
allows one to express in an entirely artificial symbolism
the doctrines which, like Arithmetic, already make use
of a symbolism which is appropriated for their ideas
and operations. Frege in his Begriffsschrift (Halle, 1879)
and Peano in his Notations de logique mathématique
(Turin, 1894) proposed such systems of symbolic logic.
Frege's symbolism was quite cumbersome and has not
survived, whereas Peano's notation, essentially what
Russell adopted, has passed into the current usage of
symbolic logic.

The chief value of this symbolic notation is to make
possible a formal treatment of the sort of reasoning
about ideas, which is still tainted more or less with
subjectivity or with appeals to intuition in judging the
correctness of logical inferences, by replacing such
reasoning by a calculus of signs. Here Leibniz' ideal
of a calculus of reasoning (calculus ratiocinator) comes
to the fore again. Now in order to avoid any dispute
in the practice of such a calculus, it is first necessary,
as in a well-regulated game, that the rules governing
the calculus be explicitly formulated, and in such a
manner that they admit no ambiguity about their mode
of application. That is why formal axiomatic systems
state the rules according to which calculation may take
place besides stating the axioms serving as a basis for
the calculus. In that way the confusion was cleared
up which had prevailed for a long time with respect
to the distinction, on the level of logical principles,
between premisses and rules of inference. The rules
of inference are now made explicit and are expressly
distinguished from the system of premisses on which
the calculus operates governed by the rules. These rules
are generally divided into two groups, depending on
whether they govern the formation or the trans-
formation of expressions. Demonstration then amounts
to transforming progressively, without omitting any
step, one or more formulas correctly formed (the ab-
breviation “w.f.f.” is used for “well formed formulas”)
and already admitted as axioms or theorems, by indi-

cating at each step the number of the rule authorizing
this transformation, until step by step the formula to
be demonstrated is finally reached. Such a task has
become performable, in theory and in fact for rela-
tively simple cases, by a suitably constructed and
programmed machine; the computing machine can
with extreme rapidity try the various combinations
authorized by the rules of inference and retain only
those combinations which yield the result sought.

But how can one be sure, in the unrolling of the
theorems derivable from the axioms according to these
rules, that one will never run into a contradiction, that
is to say, into the possibility of proving both a formula
and the same formula preceded by the sign of negation?
Such a question was hardly a problem for the first
axiomatic systems which started from a system of
propositions practically certified, such as the body of
Euclidean geometry or that of classical arithmetic, and
simply proposed to make the system rest on a minimal
basis, entirely explicit. However, the problem of the
consistency of a system arises as soon as there is a doubt
about it, and furthermore, the problem of consistency
arises also in the reverse direction, when a certain
number of axioms are arbitrarily posited in order to
see what consequences flow from these axioms. In order
to be sure that the very axioms of a system are indeed
compatible, we must rise to a new level and take this
system as itself an object of study. In his Foundations
of Geometry of 1899, Hilbert had already raised ques-
tions about his axioms when he investigated their mu-
tual independence, their subdivision into five groups,
and the limitations which each had to impose on its
own respective domain. Taking very clearly into con-
sideration the specificity of this class of problems, he
proposed in 1917 the institution of a new science,
“Metamathematics,” which takes as its object of study
the language of mathematics already symbolized and
formalized, and in abstraction from its meaning pro-
ceeds entirely on its own in a mathematical manner
to create rigorous proofs. In this new science the prob-
lem of the proof of the noncontradictoriness of an
axiomatized mathematical system naturally occupied
an important place. In truth, however, the difficulty
had only been pushed back, for it was then necessary
to guarantee the validity of the metamathematical
procedures themselves. Whence arose the attempts to
find a means of proving the noncontradictoriness of
a system by means of the very axioms and rules of
inference within the system itself.

The halting of these attempts and their futility were
explained and sanctioned by the famous proof by K.
Gödel (1931); the proof itself was drawn by the rigor-
ous procedures of metamathematics and established
that the proposition which states the noncontradictori
ness of a system in which arithmetic can be developed
is not decidable within this system. In other words,
in order to prove that a formal system is not contra-
dictory, it is necessary to appeal to stronger means of
demonstration than those used by the system itself, and
by means of which the question of noncontradiction
is consequently carried over. Hence the theory about
a calculus cannot be constructed by means of the
resources alone of this calculus, nor can one speak
about a language without employing a metalanguage,
which would yield the same uncomfortable situation.
In short, formalism is not self-sufficient; its closure on
itself is impossible. The ideal of the calculus ratiocina-
tor and that of the caracteristica universalis are in the
end incompatible. One can postpone indefinitely but
cannot eliminate altogether the appeal to logical
intuitions.

This check on one of the objectives of metamathe-
matics is, in other respects, an important result to
credit to it. Besides, metamathematical logic poses
many other problems concerning completeness, decida-
bility, categoricity, isomorphism, etc.; the very analysis
of these ideas leads to their further diversification by
greater refinements and nuances. As an example, the
idea of noncontradiction appears as a special case of
the more general idea of consistency which is itself
presented in various forms.

Because of the close relationship between logic and
mathematics, which is highlighted by the formalizing
of their axiomatic systems, logic itself has experienced
analogous developments. By analogy with Hilbert's
metamathematics, Tarski constructed metalogic as a
distinct discipline. Beside the questions of syntax which
had at first been the main concern of metatheories,
Tarski emphasized the importance of the semantic
point of view. Through this new approach he estab-
lished limiting conditions under which the semantic
notion of truth replaces the syntactical idea of deriva-
bility; this he showed in a theorem (1935) which, con-
joined to other results, came at the same time in various
forms to converge on Gödel's result. But here also, the
field of metatheory was extended to many other prob-
lems. Metalogic today occupies in the activities of
logicians a place equal at least in importance to that
of logic properly so-called, since beyond the studies
specifically assigned to metalogic there is scarcely any
work in logic not accompanied by a critical examina-
tion on the metalogical level.

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.

BIBLIOGRAPHY

The works which mark the principal historical stages of
axiomatization have been indicated in the course of the
article. Among contemporary works which deal with axio-
matics, without being themselves axiomatic, are the follow-
ing: H. Scholz, Die Axiomatik der Alten (1930-31), reprinted
in Mathesis universalis (Basel-Stuttgart, 1961); F. Gonseth,
Les Mathématiques et la réalité, essai sur la méthode axio-
matique (Paris, 1936); J. Cavaillès, Méthode axiomatique et
formalisme (Paris, 1938); G. G. Granger, Pensée formelle et
sciences de l'homme (Paris, 1960), esp. Ch. VI; W. and M.
Kneale, The Development of Logic (Oxford, 1962). For an
introductory exposition: R. Blanché, L'axiomatique (Paris,
1955), trans. as Axiomatics (London, 1962). Also A. Tarski,
Introduction to Logic and to the Methodology of the Deduc-
tive Sciences, 3rd ed. rev. (New York, 1965), pp. 140, 234ff;
R. L. Wilder, Introduction to the Foundations of Mathe-
matics (New York, 1952); F. Enriques, Historic Development
of Logic, trans. J. Rosenthal (New York, 1933).

ROBERT BLANCHÉ

[See also Abstraction in the Formation of Concepts; Mathe-
matical Rigor; Number; Structuralism.]