96 Shank
Not least, the deductive ideal of demonstration raised important related
questions about certainty in specifi c scientiae: Where did the foundation of
certainty lie? And how could one get from there to the various sciences?FROM THE QUADRIVIUM TO THE INTERMEDIATE SCIENCES
(SCIENTIAE MEDIAE)
Aristotle had already raised such disciplinary questions by calling astron-
omy, optics, and harmonics “the more physical of the mathematical sci-
ences” (Physics 2.2). As one of the fi rst Latin commentators on Aristotle’s
Physics and Posterior Analytics, Robert Grosseteste (d. 1253) explored the
gray areas in these works, notably the relations of mathematics and natu-
ral philosophy. He articulated a framework for their interaction that would
resonate into the later Middle Ages. He held that, for disembodied intel-
lects, theology offered greater certainty than mathematics and physica. For
us here below, however, mathematics yielded greater certainty, an echo of
Ptolemy’s Almagest. But Grosseteste had also read Aristotle. He therefore
treated disciplines like astronomy, music / harmonics, and perspectiva not
simply as mathematics, but as composite subjects: “purely mathematical
subjects to which are added natural ‘accidents’” (nonessential properties).
He used the expression “subalternate sciences” (scientiae subalternatae) to
convey Aristotle’s view: astronomy and optics / perspectiva were sciences
subalternated to geometry, harmonics / music to arithmetic.^61
What does this mean? A science is subalternate to another when it bor-
rows some principles from that other (the “subalternating” science). Take
a planetary sphere (from astronomy) or a light source (from perspectiva).
Although each is a physical entity, aspects of each—the sphere and the
rectilinear ray, respectively—can be abstracted for geometrical treatment.
Since the principles that govern spheres and straight lines come from ge-
ometry (not astronomy or optics), geometry is the subalternating science
in each case.
It is crucial to emphasize that Grosseteste gave the mathematical sci-
ences a signifi cant causal role. Echoing Aristotle (Posterior Analytics 1.13),
he argued that when the natural philosopher (physicus) and the “per-
spectivist” (perspectivus) each considers the rainbow, the former knows
only the fact (quia—that it is) while the latter knows its cause (propter
quid—why it is).^62 The practitioner of a subalternate science that draws
on mathematics thus possesses causal knowledge (the most valued kind);
the practitioner of physica (natural philosophy) in this case does not. The
combination of mathematics and natural philosophy thus gave optics a