Willard Van Orman Quine

248 daniel isaacson
of Quine’s philosophy of mathematics would show in any case that
the analytic-synthetic distinction was unnecessary for empiricism,
thereby leaving those who would defend it with little to be gained
from doing so.
For Carnap, mathematics was the fundamental issue for his attempt
to uphold empiricism. Quine saw that this was so:
I think Carnap’s tenacity to analyticity was due largely to his philosophy of
mathematics. One problem for him was the lack of empirical content: how
could an empiricist accept mathematics as meaningful? Another problem
was the necessity of mathematical truth. Analyticity was his answer to
both. (TDR 269)
The case of mathematics was not central to Quine’s formulation
of empiricism, but he held that his empiricism could account for
mathematics, as of course he realized it must. He followed the preceding
characterization of Carnap’s philosophy with this account of
his own:
I answer both with my moderate holism. Take the first problem: lack of
content. Insofar as mathematics gets applied in natural sciences, I see it
as sharing empirical content. Sentences of pure arithmetic and differential
calculus contribute indispensably to the critical semantic mass of various
clusters of scientific hypotheses, and so partake of the empirical content
imbibed from the implied observation categoricals....
What then about the other problem, that of the necessity of mathematical
truth? This again is nicely cleared up by moderate holism, without the help
of analyticity. For let us recall that when a cluster of sentences with critical
semantic mass is refuted by an experiment, the crisis can be resolved
by revoking one or another sentence of the cluster. We hope to choose in
such a way as to optimize future progress. If one of the sentences is purely
mathematical, we will not choose to revoke it; such a move would reverberate
excessively through the rest of science. We are restrained by a maxim
of minimum multilation. It is simply in this, I hold, that the necessity of
mathematics lies: our determination to make revisions elsewhere instead. I
make no deeper sense of necessity anywhere. (TDR 269–70)
Quine’s philosophy of mathematics has to be gleaned from scattered
remarks, such as those just quoted. No section, let alone chapter,
of Word and Object is devoted to mathematics, and indeed there
is no entry for mathematics in the index of this most central text for
Quine’s philosophy. No one of Quine’s myriad papers is devoted to
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