154 Dear
internalized many of the values that justifi ed his inferior position as a
mathematician: he apparently regarded being a “philosopher” as more
important, as well as more prestigious, than being a “mathematician.” At
the same time, in asserting his expertise in philosophy, he had contrasted
philosophy with pure mathematics, in effect explicitly leaving out mixed
mathematics. He seems to have seen his philosophical credentials as fol-
lowing from his expertise in mixed mathematics, taken as a legitimate
part of a truly philosophical science of nature. Pure mathematics could
not count as natural philosophy under any circumstances, of course, be-
cause it did not talk about the changeable natural things that defi ned
physics for Aristotle; it only dealt with the unchanging ideal entities of
pure quantity.^16
The ambiguity surrounding the philosophical and cultural status of
mathematical sciences in this period was therefore one that mathemati-
cally inclined scholars, whether academic or not (and, increasingly, they
were not), attempted to resolve as much by practical and rhetorical ac-
complishments as by philosophical innovations concerning such issues as
causal explanation. The Aristotelian conceptual categories in the terms of
which mathematics in the study of nature were usually debated remained
nonetheless common resources for all sides in the argument. Galileo’s fa-
mous work on free fall, parabolic trajectories, and related topics, published
in his Discorsi of 1638, drew on the legitimating resource of Archimedean
mechanics but was widely regarded among his mathematical peers as re-
taining methodological imperfections and uncertainties. Thus, Galileo’s
interlocutor and critic Giovanni Battista Baliani wrote in his De motu natu-
rali (On natural motion) of 1646 about the importance of discovering the
appropriate principles on which a true science could be based, and the dif-
fi cult role of experience in establishing or confi rming them. “I have begun
to search,” Baliani said of such principles; “others will decide whether I
have discovered anything.”^17 After a lengthy discussion of the behavior
of heavy bodies, Baliani then refl ects upon the status and character of the
knowledge that he has proposed.Hitherto, it seems to me that I have said as much as I can concerning the
science of the natural motion of solid weights, as from certain properties fa-
miliar to the senses many things unknown have been deduced and disclosed.
For according to Aristotle in this way alone is every science treated, as is seen
in practice with Euclid, and others, who treat true and simple sciences: from
whom [we see that] the geometer does not deal with the nature of quantity,
nor the musician with the nature of sound, nor the optician with the nature
of light, nor the mechanic with the nature of weight. But truly my mind