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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*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.

Dictionary of the History of Ideas | ||