Willard Van Orman Quine

220 daniel isaacson
“[M]athematics gives us a shining example of how far, independently
of experience, we can progress in a priori knowledge” (A4/B8), that
is, knowledge obtained independently of experience. Thus far, Kant’s
account is compatible with Hume’s, but Kant powerfully challenged
empiricism by his further doctrine that while mathematical knowledge
is a priori, it is synthetic, that is, not merely determined by
relations between concepts: “All mathematical judgments, without
exception, are synthetic” (A10/B14).
It was Auguste Comte who introduced the terms ‘positive philosophy’
and ‘positivism’ 9 as labels for an empiricist philosophy based
on a conception of science founded strictly on observation:
[T]he first characteristic of the Positive Philosophy is that it regards all phenomena
as subjected to invariable natural Laws. Our business is, – seeing
how vain is any research into what are called Causes, whether first or final, –
to pursue an accurate discovery of these Laws,...to analyse accurately the
circumstances of phenomena, and to connect them by the natural relations
of succession and resemblance. (Comte [1853] 1974, 28)
(Compare this statement of Comte’s with this declaration of Quine’s:
“As an empiricist I continue to think of the conceptual scheme of
science as a tool, ultimately, for predicting future experience in the
light of past experience” [TDEa 44].)
Comte’s positive philosophy is structured by his “hierarchy of
the positive sciences” (chap. 2 of the introduction to his Cours de
Philosophie Positive). Comte held that there are six “fundamental
sciences” (p. 43): mathematics, astronomy, physics, chemistry, physiology,
and social physics (p. 50). (The last of these he later called
sociology, and he is generally credited as the originator of this discipline.)
The hierarchical order between these fundamental sciences is
determined by their “successive dependence” (p. 44). Comte’s Cours
de Philosophie Positive consists of six books devoted successively to
each of the fundamental sciences in their hierarchical order.
For our purposes, the significant point in Comte’s ordering of the
sciences is the place of mathematics in that ordering, and its nature.
Mathematics is the most fundamental of all sciences, so fundamental
as to make it more than just one among the sciences:
In the present state of our knowledge we must regard Mathematics less as a
constituent part of natural philosophy than as having been, since the time
of Descartes and Newton, the true basis of the whole of natural philosophy;
though it is, exactly speaking, both the one and the other. (p. 49)
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