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Dictionary of the History of Ideas | ||

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*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.

Dictionary of the History of Ideas | ||