Routledge History of Philosophy Volume IV

RENAISSANCE AND SEVENTEENTH-CENTURY RATIONALISM 101
quantity or quantities with letters of the alphabet, as well as those that were
‘known’, and then to operate on both in the same way. This produced formulae
that looked far more like equations in the modern sense than anything that had
gone before, and this trend is even more accentuated in the work of Descartes,
where we often have no more than minor and accidental features of notation to
remind us that we are not in the twentieth century. All this may be seen as
reflecting a general psychological trend in mathematics to focus more on written
symbols than on what they are meant to symbolize.
It is this movement that has made it seem plausible to speak of Descartes as
the founder of ‘analytical geometry’, meaning thereby a geometry in which
curves may be substituted by the equations representing them. But this ignores
the extent to which Descartes still demanded explicitly geometrical constructions
for solving what we would regard as purely algebraical problems. Nevertheless it
does draw attention to how geometry and algebra could seem to proceed in
tandem. This trend was to continue into the work of Newton, and can be seen as
leading towards an eventual reduction of the former to the latter. With hindsight
it is possible to see ripening other fruits of mathematical method, such as those
associated with the infinitesimal calculus, but until well past the time of
Descartes these are decidedly muted in Comparison with the algebraic results.
As may be expected, philosophers were more explicit than mathematicians in
their discussions of method, although one head was often capable of wearing two
hats, as in the case of Descartes. In the Aristotelian tradition we frequently meet
with the methods of analysis and synthesis in the Latinate guise of resolution and
composition. As in mathematics, the terms are a little slippery, but resolution
was basically a form of argument from effects to causes, whereas composition
was demonstration of effects from their causes. The development of these ideas
by medieval and Renaissance commentators on Aristotle (and also on Galen’s
methodological writings) have led some scholars, for instance John Hermann
Randall, Alistair C.Crombie and William A. Wallace, to place strong, and
perhaps extravagant, emphasis on the positive role of Aristotelianism in
furthering the emergence of modern science.
One thinker who would not have been convinced by this was Francis Bacon.
Although Bacon had a grudging respect for aspects of Aristotle’s own thought,
he saw medieval scholasticism as the embodiment of sterility and futile
contentiousness. He therefore sought a more fruitful way of eliciting knowledge
from nature, of a kind that would eventually prove useful in practice. In this he
displayed some affinities with the magical and alchemical traditions, but their
secrecy and apparent obscurantism were antithetical to his programme for which
he sought more public, methodical and ‘democratic’ procedures. In this a central
role was played by his idea of induction, the careful collection and tabulation of
facts (although in practice often derived from nonetoo-reliable reports), and then
the cautious ascent to higher levels of generalization, with only a very wary use
of hypotheses, strictly controlled by the use of experiments. Although usefulness
was the remote aim, this could not properly be achieved without first seeking the