Mixed Mathematics 153mathematics as a respectable way of generating real understanding of the
natural world, but also saw it as a means of accomplishing things of which
the natural philosophers were incapable—as shown by Aristotle’s math-
ematically demonstrable errors regarding falling bodies.^12 Both Benedetti
and Galileo, perhaps independently,^13 had made “thought- experiment”
arguments to show that Aristotle’s apparent position that the speed of
fall of heavy bodies was proportional to their absolute weights was logi-
cally incoherent; they preferred analyses drawing on the mathematical
mechanics of Archimedes.
The same sort of move was also made in the second half of the six-
teenth century in astronomy, starting with the signal attempt by Coper-
nicus to recast physical understanding on mathematical- astronomical
grounds, and pushed to perhaps its furthest extent by Johannes Kepler in
the early seventeenth century. Kepler’s Lutheran, but also mathematical,
understanding of natural philosophy was in its specifi cs unusual for math-
ematicians in this period. His astronomical and associated optical work
took mathematical entities and concepts as substantive constituents of an
astronomy that he approached as a part of physics, or natural philosophy,
as well as an enterprise of mathematical modeling. His astronomy was a
theological endeavor that sought God in the construction of the universe
and incorporated both qualitative causal explanations for celestial mo-
tions and formal mathematical accounts of celestial operations that put
mathematics at the center of physical knowledge. To that end, he wrote
that the astronomer “directs all his opinions, both by geometrical and by
physical arguments, so that truly he places before the eyes an authentic
form and disposition or furnishing of the whole universe.”^14 Kepler saw
himself, in effect, as a philosophical astronomer.
Galileo, who was not very interested in mathematical astronomy of
Kepler’s kind, began to call himself a “philosophical astronomer” when
he embarked on his telescopically based campaign against Ptolemy and
Aristotle; again, the idea was to show that the astronomer could tell the
natural philosopher what was what, rather than being subordinated to
him. It is telling that, when Galileo negotiated with the secretary to the
Grand Duke of Tuscany for a court position in 1610, he famously made
it clear that he was not interested in being just another court mathemati-
cian (a status held by Kepler). Instead, he would be the “Philosopher and
Mathematician” to the Grand Duke. In the correspondence, Galileo was
careful to point out that he had “studied a greater number of years in
philosophy than months in pure mathematics.”^15 His concern to stress
his “philosophical” credentials suggests that, while annoyed at the higher
status of his philosophical colleagues in the universities, Galileo had also