Mixed Mathematics 165some communicative function, if only by virtue of its connotations and
the disciplinary alliances that it suggested. In time, its slipperiness (which
had evidently proved functional), like that of “physico- mathematics,”
resulted in its general abandonment among the professionalizing scien-
tists of the nineteenth century, with the notable exception of the British.
The remaining bastion of mixed mathematics throughout the nineteenth
century was the University of Cambridge, where it had persisted in the
mathematical tripos through the reforms of William Whewell around
midcentury, taking on new life at the heart of a British physics that came
to be associated with Maxwell’s Cavendish Laboratory in the 1870s and
thereafter.^52
If, as John Heilbron has suggested, mixed mathematics in its associa-
tion with scientifi c instruments contributed centrally to the quantifi ca-
tion of the physical sciences (often, “experimental natural philosophy”)
in the decades around 1800,^53 and pointed to an ever- growing absorption
into physics of concepts and techniques that had once counted strictly as
“mathematics,” one might conveniently see mixed mathematics as hav-
ing always been modern physics in disguise. However, on that view, mod-
ern physics would have to be seen in its common late- nineteenth- century
guise as a positivistic discipline concerned with appearances (what Heil-
bron himself has dubbed “descriptionism”) and eschewing deep under-
standing of “things in themselves.”^54
In his chapter in this volume, Heilbron sees the mid- eighteenth cen-
tury as representing the narrow point of a meta- temporal hourglass from
which a much- restricted natural philosophy began to expand again as it
absorbed Lavoisier’s new quantifi ed chemistry and adopted the mixed-
mathematical traditions into itself. What “natural philosophy” under-
stood in this way had lost as a consequence of this process was precisely
its claim to provide causal understanding. In practice, the “mathematiza-
tion” of the physical sciences in that period meant the growth of a natural
philosophy eviscerated of its causal pretentions. Heilbron’s chapter sums
up this view as its story proceeds down through the nineteenth century:
“An instrumentalist attitude toward theory often accompanies quantifi ca-
tion. Mixed mathematics may be precise, but, as Aristotle knew, there is
no truth in it.”^55
The old Aristotelian view of mixed mathematics as a body of disci-
plines that did not speak of the causes of the things with which they dealt
had never itself been unchallenged, as Kepler bears powerful witness; a
Platonizing tendency associated with the mathematical sciences had al-
ways had the potential to tinge the philosophical understanding of math-
ematical aspects of the physical sciences. One of the issues that emerges