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Dictionary of the History of Ideas | ||

#### 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

to be found in recent editions of the

*Encyclopedia*

Britannica(1962). It is mentioned in the

Britannica

*Enciclopedia*

Italiana(1949); there it is defined as the “name adopted

Italiana

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

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-

sophiae naturalis principia mathematica,

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

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

ordine geometrica demonstrata,

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

versalis

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,

methodo scientifica pertracta,

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

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(

chanics

*Mécanique analytique,*1788), Joseph Fourier

in his thermodynamics (

*Théorie analytique de la cha-*

leur,1822), and J. C. Maxwell in his

leur,

*Treatise on Elec-*

tricity and Magnetism(1873), it does not postulate its

tricity and Magnetism

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

ophie des als ob,

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

and, not, if..., then, is a

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

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-*

tionsfrom being able to support different systems of

tions

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

etrie,

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).

des ensembles,

####
*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,

mathematische Untersuchung über den Begriff der Zahl,

Breslau, 1884), and

*Fundamental Laws of Arithmetic,*

derived by symbolic representation of concepts(

derived by symbolic representation of concepts

*Grund-*

gesetze der Arithmetik begriffsschriftlich abgeleitet,Jena

gesetze der Arithmetik begriffsschriftlich abgeleitet,

1893-1903); A. N. Whitehead and B. Russell,

*Principia*

Mathematica,3 vols. (Cambridge, 1910-13, reprinted

Mathematica,

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

*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

philosophicus

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-

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

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.

##
*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,

formalisme

*Pensée formelle et*

sciences de l'homme(Paris, 1960), esp. Ch. VI; W. and M.

sciences de l'homme

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;

tive Sciences,

R. L. Wilder,

*Introduction to the Foundations of Mathe-*

matics(New York, 1952); F. Enriques,

matics

*Historic Development*

of Logic,trans. J. Rosenthal (New York, 1933).

of Logic,

ROBERT BLANCHÉ

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

Dictionary of the History of Ideas | ||