Willard Van Orman Quine

Quine and Logical Positivism 227
Carnap and the other members of the Vienna Circle were convinced,
following Frege, that Mill’s attempt to establish mathematics
as a species of empiricist knowledge, grounded by induction, was
untenable:
The conception of mathematics as tautological in character, which is based
on the investigations of Russell and Wittgenstein, is also held by the Vienna
Circle. It is to be noted that this conception is opposed not only to apriorism
and intuitionism, but also to the older empiricism (for instance of J. S. Mill),
which tried to derive mathematics and logic in an experimental-inductive
manner as it were. (Verein Ernst Mach 1973, 311)
That the viability of empiricism demanded a solution to this problem
was clearly recognized within the circle, as shown in the following
quotations. In 1930, Hahn asserted, “Only the elucidation of
the place of logic and mathematics...(which is of very recent origin)
made a consistent empiricism possible” (1980, 21). In the same
year, Carnap gave this account of the importance for empiricism of
establishing the analyticity of mathematics:
[E]mpiricism, the view that there is no synthetic a priori knowledge, has
always found the greatest difficulty in interpreting mathematics, a difficulty
which Mill did not succeed in overcoming. This difficulty is removed by the
fact that mathematical sentences are neither empirical nor synthetic a priori
but analytic. (1959, 143)
This view was echoed by Blumberg and Feigl in their 1931 paper:
Logic is a priori because it is analytic. Thus, the difficulties which the older
empiricism and positivism encountered in attempting to account for logic
and mathematics on an empirical basis disappear. Empiricism, the denial
of synthetic judgments a priori, is now in a position to develop a theory of
knowledge capable of doing full justice to logic and mathematics. (p. 285)
Carnap summed up this development in his “Intellectual Autobiography”
as follows:
Thus we arrived at the conception that all valid statements of mathematics
are analytic in the specific sense that they hold in all possible cases and
therefore do not have any factual content.
What was important in this conception from our point of view was the
fact that it became possible for the first time to combine the basic tenet
of empiricism with a satisfactory explanation of the nature of logic and
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