Willard Van Orman Quine

Quine and Logical Positivism 221
As to the nature of mathematics, Comte sees it as
divided into two great sciences, quite distinct from each other – Abstract
Mathematics, or the Calculus (taking the word in it most extended sense),
and Concrete Mathematics, which is composed of General Geometry and
of Rational Mechanics. The Concrete part is necessarily founded on the
Abstract, and it becomes in its turn the basis of all natural philosophy; all the
phenomena of the universe being regarded, as far as possible, as geometrical
or mechanical. (p. 50)
Abstract mathematics is based on “natural logic” (a notion Comte
never spells out) and is purely instrumental:
The Abstract portion [of mathematics] is the only one which is purely instrumental,
it being simply an immense extension of natural logic to a certain
order of deductions. (p. 50)
Concrete mathematics, that is to say, geometry and mechanics,
“must, on the contrary, be regarded as true natural sciences, founded,
like all others, on observation.” (p. 50)
Comte’s division of mathematics into abstract and concrete evidently
does not correspond to the usual distinction between pure
and applied (most branches of mathematics, and certainly the calculus
and geometry, are both pure and applied, depending on how
they are being used and developed). It might almost be said that for
those parts of mathematics that Comte labels as abstract, his position
follows Hume and looks forward to Carnap, and for those parts that
he sees as concrete, his views bear some affinity to the concept of
mathematics as empirical developed soon after by Mill – and even
more affinity to the views of Quine developed a century later.
Despite the centrality of mathematics for Comte’s positive philosophy
(“it is only through Mathematics that we can thoroughly
understand what true science is” [p. 55]), it must be borne in mind
that the driving force of Comte’s philosophy was social and political,
leading him to utopian schemes in which positive philosophy
would supplant established religion, a development he considered
to be well underway owing to the progress of eighteenth-century
Enlightenment.
Comte’s younger contemporary John Stuart Mill described himself
as “long an ardent admirer of Comte’s writings” ([1873] 1924, 178).
Admiration led to correspondence, but eventually Comte espoused
views that Mill abhorred, and they parted company. Comte and Mill
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