Willard Van Orman Quine

20 robert j. fogelin
with clarifying concepts by defining them, some in terms of others. The
doctrinal studies are concerned with establishing laws by proving them,
some on the basis of others. Ideally the obscurer concepts would be defined
in terms of the clearer ones so as to maximize clarity, and the less obvious
laws would be proved from the more obvious ones so as to maximize
certainty. Ideally the definitions would generate all the concepts from clear
and distinct ideas, and the proofs would generate all the theorems from selfevident
truths. (EN 69–70)
Specifically, the logicist program was an attempt, on the conceptual
side, to reduce the concepts of mathematics to concepts of logic, and
then, on the doctrinal side, to exhibit all the truths of mathematics
as truths of logic. Such a reduction, if carried through, would be a
triumph for epistemology, for the truths of logic seem epistemically
secure and the reduction of mathematics to logic would make the
edifice of mathematics epistemically secure as well. Unfortunately,
as Quine tells us, this goal has not been fully attained. He explains
why in these words:
This particular outcome is in fact denied us, however, since mathematics
reduces only to set theory and not to logic proper. Such reduction still enhances
clarity, but only because of the interrelations that emerge and not
because the end terms of the analysis are clearer than others. As for the end
truths, the axioms of set theory, these have less obviousness and certainty
to recommend them than do most of the mathematical theorems that we
would derive from them....Reduction in the foundations of mathematics
remains mathematically and philosophically fascinating, but it does not do
what the epistemologist would like of it: it does not reveal the ground of
mathematical knowledge, it does not show how mathematical certainty is
possible. (EN 70)
The moral to be drawn seems clear: Though progress has been made
on the conceptual side, advances in the foundations of mathematics
have left what Quine calls the doctrinal task essentially unfulfilled.
Quine expresses no hope that, with time, we might do better. Another,
more interesting, moral seems to lie in back of this: In order to
make progress on the conceptual issues, it may sometimes be necessary
to abandon the doctrinal goal, or at least to make the doctrinal
goal much more modest.
Quine holds that a striking parallel obtains between the epistemology
of mathematics and what he calls the “epistemology of natural
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