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Dictionary of the History of Ideas

Studies of Selected Pivotal Ideas
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7 occurrences of Dictionary_of_the_History_of_Ideas
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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


164

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: ((pq) ∙ 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.