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,
Breslau, 1884), and Fundamental Laws of Arithmetic,
derived by symbolic representation of
concepts (Grund-
gesetze der Arithmetik begriffsschriftlich abgeleitet,
Jena
1893-1903); A. N. Whitehead and B. Russell, Principia
Mathematica, 3 vols. (Cambridge, 1910-13,
reprinted
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
E . L. Post (1921) constructed the first three-valued 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
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.