Dictionary of the History of Ideas Studies of Selected Pivotal Ideas |

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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

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.

Dictionary of the History of Ideas | ||