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.