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 self-evident 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.