The turning point in Descartes's intellectual development occurred on November 10, 1619. He had attended the coronation of Ferdinand II in Frankfurt, and was returning to serve in the army of Maximilian of Bavaria. Due to the onset of winter, he holed himself up for a day, alone in a stove-heated room. With nothing else to occupy him, he set about thinking.

He first mused that accomplishments of single individuals are usually more perfect than group efforts. Cities and buildings are more beautiful when they are made according to a single plan than when they are patched together piecemeal. Similarly, laws are better when they come from a single mind than when they evolve gradually over time. Descartes cites God's law as an instance of this perfection. These musings suggest to him that a person is best served by following the guidance of his reason alone, and not letting his judgments be clouded by his appetites and by the opinions of others.

While it would be impossible to resolve the imperfections of a state or a body of sciences by tearing it all down and starting again from scratch, Descartes suggests that such a method is not quite as unreasonable on the individual level. He decided to let go of all his former opinions at once, and re-build them anew according the exacting standards of his own reason.

Descartes is very careful, first of all, to point out that this method is meant only on an individual level, and he strongly opposes those who would try to topple a public institution and rebuild it from the ground up. Second, he reminds us that he only wants to discuss his method with us; he is not telling us to imitate him. In particular, he notes that there are two types of people for whom this method would be unsuited: those who think they know more than they do and who lack the patience for such careful work, and those who are modest enough to think that they are more capable of finding out the truth if they follow a teacher. Descartes would count himself among this second group if he hadn't had such a number of teachers and embarked on so many travels as to realize that the opinions of even learned men vary greatly.

Before abandoning his former opinions entirely, Descartes formulates four laws that will direct his inquiry: First, not to accept anything as true unless it is evident; this will prevent hasty conclusions. Second, to divide any given problem into the greatest possible number of parts to make for a simpler analysis. Third, to start with the simplest of objects and to slowly progress toward increasingly difficult objects of study. Fourth, to be circumspect and constantly review the progress made in order to be sure that nothing has been left out.

An obvious starting place was in the mathematical sciences, where a great deal of progress and certain knowledge had been achieved by means of demonstration. Descartes found his work made considerably easier if, on the one hand, he considered every quantity as a line, and, on the other hand, developed a system of symbols that could express these quantities as concisely as possible. Taking the best elements of algebra and geometry, he had tremendous success in both these fields.

Before applying this method to the other sciences, Descartes thought it well to find some philosophical foundations for his method.

If we were to identify a starting point for modern philosophy, November 10, 1619 would be as good a date as any. We might pinpoint precisely the moment that Descartes resolved to cast all his former opinions into doubt. This process of methodological doubt is central to Descartes, and indeed to most of modern philosophy. The results Descartes achieves by employing this method of doubt are discussed in Part Four of the *Discourse,* so we will comment on his method in greater detail there.

It is important, of course, that Descartes does not simply scrap everything he knows, or else he would have no guidance in rebuilding his knowledge. The four rules he lays out are meant as guidelines, so that he will be able to rely on them, and not on unnoticed prejudices. Descartes had initially collected twenty-one rules entitled *Rules for the Direction of Our Native Intelligence* in 1628, but left the manuscript unpublished. The four rules we find here can be read as a major abbreviation of that effort. Essentially, they demand that an inquiry proceed slowly and carefully, starting with basic, simple, self-evident truths, building toward more complex and less evident propositions.

Descartes assumes a certain kind of theory of knowledge that was pretty much unquestioned in his day. In modern philosophical language, we call this a foundationalist epistemology. It sees knowledge as built up from simple, self-evident propositions, to higher and more complex knowledge. The theory states that if we were to analyze any complex proposition, we could break it down into increasingly smaller, simpler pieces until we were left with simple, non-analyzable propositions. These basic propositions would be either self-evidently true or self-evidently false. If they were all true, then we would know that the original complex proposition was also true. Of course, there are different variations of foundationalist epistemology; for example, the epistemology will shift depending on how the analysis is supposed to take place or on what the basic propositions are supposed to look like. But the general idea can be applied to Descartes easily. Knowledge is built up like a skyscraper, with the higher, complex knowledge built on simple, sturdy foundations.

This is just one of a number of theories of knowledge that are batted about these days. Another theory that will come into play later in the *Discourse* is a coherentist epistemology, one that states that knowledge is more like a circle than a skyscraper. According to this theory, there is no foundational knowledge that is more basic than other knowledge. All knowledge fits together in such a way that it is internally coherent, but there is no fundamental self-evident proposition that is itself beyond doubt and that justifies all the other propositions. A statement is true because it is consistent with everything else we know to be true, not because it can be analyzed into simple parts.

The reason that a foundationalist epistemology seems natural to Descartes at this point is that this is the epistemology that philosophy had inherited from Aristotle. As we have noted already in other sections of this SparkNote, Aristotelian scientific method works according to a system of syllogism and demonstration, where complex truths are logically deduced from simpler ones. This method implies a theory of knowledge according to which complex truths are built upon simpler ones that serve as an unquestioned bedrock of knowledge.

It is significant that Descartes should choose mathematics to study according to this method. Mathematics has had far more success than any other field (except logic) with deductive reasoning. Math is built upon simple, self-evident axioms that are then used, along with some rules of inference, to derive proofs of more complex propositions.

Descartes is not only one of the greatest philosophers of the modern world, he is also one of its greatest mathematicians. His discussion of algebra and geometry alludes to his discovery of analytic geometry that brought those two fields together. Until Descartes, algebra and geometry were two totally separate fields of study. He invented the Cartesian co-ordinate system that every math student knows and loves. That's the co-ordinate system with the x-axis and the y-axis that allows you to plot lines and curves and whatever other shapes you please. Geometrical figures could be plotted onto the co-ordinate grid, and since every line and curve on the grid corresponds to an equation, geometrical figures can be expressed as equations. Geometrical figures become algebraic equations, and algebraic equations can be graphed as geometrical figures. This all seems pretty commonplace to us today, but if you try to imagine solving math problems without graphing anything you'll begin to understand the colossal contribution Descartes made to mathematics.

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