Descartes reasoning received its major impetus from the same
mathematics as used by Galileo but extended further by a new and
original approach to mathematics. This was his own personal
contribution to science, analytic geometry. Even considering all
this, to an extent many would rather deny, modern science and
modern philosophy are both indebted to Descartes for his unique
method.
For those who do not have a strong mathematical background a few
words on this fascinating branch of mathematics might be in
order. Geometry, as we are all aware, deals with rules
concerning the forms of geometric figures. For example given a
triangle we immediately know certain things about it because of
its form. We know that it has three sides, and that the sum of
its interior angles total two right angles. We know these things
to be true because they are true of all triangles because of
their form. Analytic geometry goes a step farther. Through
analytic geometry any mathematical function can be pictured on a
graph using "Cartesian coordinates" as a geometric form and any
geometric form, once placed on a coordinate graph, can be
described by a mathematical function. These mathematical
functions can be manipulated according to standard mathematical
rules and the resultant can be shown as another geometric form.
The conclusion that must have been drawn by Descartes, though he
never mentioned it explicitly, was that all knowledge is thereby
interconnected. It is wrong, he said, to make the various
sciences different in the same way that we make the arts
different. By this he meant that flute playing and agriculture
require very different physical talents but the sciences of
physics, chemistry and optics do not require different
intellectual talents since they all are interconnected through
mathematics. In fact he believed that all knowledge was
interconnected through mathematical relationships and the entire
physical universe was interconnected through mechanical
relationships including the new laws of physical science being
developed by Galileo. But through his analytic geometry these
mechanical relationships, particularly when they can be expressed
mathematically through natural law, are translatable into
geometric forms which are then translatable back into
mathematical functions. Thus it was possible to interconnect all
knowledge.
If all this is true then it is obvious that in order for a person
to deduce all true knowledge, he need only find one indubitable
truth that could be known to be true on its own, not deduced from
some other truth. Once that truth was learned it was merely a
matter of logical and mathematical manipulation to prove out all
of the true knowledge that exists anywhere in the world. Thus
Descartes problem could be stated simply, he must locate that one
indubitable truth, one that cannot be denied regardless of the
circumstances. Once having obtained that truth one would have
the key that would unlock all knowledge. Remember that, like
Aquinas, he did not believe that reason could contradict
revelation thus the result of his efforts would be the sum total
of all religious as well as scientific truths.
The most famous scholastic philosopher of the late middle ages
was the Spanish Jesuit Francis Suarez. Descartes attended the
Jesuit academy and it has been said that he never went anywhere
without his Suarez tucked under his arm. Thus he was very
familiar with the system he was about to dismantle. Remember
that the Scholastic method was to begin with truths from the
bible, then move to the teachings of the church, and finally to
reason. In his system he was attempting to develop a route to
pure truth bypassing the first two steps.
Descartes' approach to the solution to this problem was to
develop a method, a procedure for directing the mind such that it
could lead eventually to an indubitable truth if such existed. As
you may recall, Aristotle said that there were two uses for
logic, dialectic and demonstration. Demonstration was arguing
from first principles and led to truth. Dialectic meant arguing
from opinions and led to persuasion. Descartes' personal revolt
was against the dialectical methods of the scholastics. It was
mathematics, he said in his discourse on method, that led to
demonstration and truth. In order to impress you on the
Aristotelian nature of this I would like you to remember that it
was a particular truth, an indubitable truth about something
existing, that he was searching for and not the kind of platonic
generalizations that the scholastics argued over. Only if it
dealt with some existing thing could he make that jump that later
Hume was to claim was impossible, from matters of fact to
relations of ideas, from the facts of existence to the form of
existence.
It is a matter of fact that no one would deny that there are
truths that, in our daily interactions with life we assume to be
true. But though we don't normally require indubitibility, we
nevertheless treat them as though they were true. It solves a
problem that Hume was to find in his own skepticism, once we
doubt of everything there is no return to truth. There must be
some criterion by which doubting could be rejected. This
criteria for Descartes became the concept of clear and distinct
ideas. Mathematical relationships were a good example. However,
it is also clear that mathematical truths, though indubitable,
will not secure the purpose Descartes was searching for, since
there exists no evidence that mathematical relationships refer
directly to existing things. What he needed for his fundamental
truth was a truth that dealt directly with existence.
To facilitate this search for truth Descartes set up a series of
"Rules for the Direction of the Mind." Originally there was to
be thirty six such rules. Three sets of twelve. However, he
completed only twenty one. We are primarily interested in only a
few which will demonstrated the change in thinking that he was
introducing into the world. The first we will examine is .
rule V.
Rule V
Method consists entirely in the order and disposition of the
objects towards which our mental vision must be directed if
we would find out any truth. We shall comply with it
exactly if we reduce involved and obscure propositions step
by step to those that are simpler, and then starting with
the intuitive apprehension of all those that are absolutely
simple, attempt to ascend to the knowledge of all others by
precisely similar steps.
In this alone lies the sum of all human endeavor, and he who
would approach the investigation of truth must hold to this
rule as closely as he who enters the labyrinth must follow
the thread that guided Theseus.

These rules were meant to replace the rules of scholastic method
and the recourse to revelation and Church laws is conspicuously
missing. This does not mean, at least not for.
Descartes, a retreat from the truths
of religion and revelation. If there can be no conflict between
reasoning and revelation then there need be no reason to concern
oneself with one when applying the other.
Rule VI
In order to separate out what is quite simple from what is
complex, and to arrange these matters methodically, we
ought, in the case of every series in which we have deduced
certain facts the one from the other, to notice which fact
is simple, and to mark the interval, greater, less, or
equal, which separates all the others from this.
Although this proposition seems to teach nothing very new,
it contains, nevertheless, the chief secret of method, and
none in this whole treatise is of greater utility. for it
tells us that all facts can be arranged in certain series,
not indeed in the sense of some ontological genus such as
the categories employed by philosophers in their
classification, but in so far as certain truths can be known
from others; and thus, whenever a difficulty occurs we are
able at once to perceive whether it will be profitable to
examine certain others first, and which, and in what order.

Thus, he is trying to say, if all knowledge is interconnected,
and we can see the trail of that interconnection, then we can
examine it step by step until we arrive at a simple and
indubitable truth. This assures us that we have obtained true
knowledge. It also means that the justification of any later
derived truth lies precisely in that trail of indubitability.
Rule VII
If we wish our sciences to be complete, these matters which
promote the end we have in view must one and all be
scrutinized by a movement of thought which is continuous and
nowhere interrupted; they must also be included in an
enumeration which is both adequate and methodical.
It is necessary to obey the injunctions of this rule if we
hope to gain admission among the certain truths for those,
which we have declared above, are not immediate deductions
from primary and selfevident principles.

These three rules will suffice to show the gist of his method.
He said that science is "indubitable cognition," which puts it
into the realm of absolute truth. His method requires two basic
facts that like Aristotelian first principles, no one could
doubt. The first is an indubitable truth, one that cannot
possibly be denied. The second is a series of deductive
arguments that lead from one simpler truth to any truth more
complex. By following these rules one can eventually determine
all of the truths of both science and religion. Of all the
Cartesian concepts this one has held most firm right up to our
modern scientific age. Without this idea. Without the
capability of accepting new developments not because they have
been determined to be necessary truths, as the Schoolmen would
have us do. But simply because over the years the scientific
community has built up a vast set of scientific concepts that
have been developed using Descartes' method. Thus scientists
have the faith that were it necessary the trail of scientifically
acceptable proofs could be followed back if not to an indubitable
source at least to something acceptable without further proof.
