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minds, it contained no point which was not disputed and hence doubtful.” While the fate of many of Descartes’ own philosophical ideas was not going to be much different in that significant shortcomings in his propositions have been pointed out by later philosophers, his fresh skepticism of even the most basic concepts certainly makes him modern to the core. More important from the perspective of the present book, Descartes recognized that the methods and reasoning process of mathematics produced precisely the kind of certainty that the scholastic philosophy before his time lacked. He pronounced clearly:

Those long chains, composed of very simple and easy reasonings, which geometers customarily use to arrive at their most difficult demonstrations, gave me occasion to suppose that all the things which fall within the scope of human knowledge are interconnected in the same way [the emphasis is mine]. And I thought that, provided we refrain from accepting anything as true which is not, and always keep to the order required for deducing one thing from another, there can be nothing too remote to be reached in the end or too well hidden to be discovered.

This bold statement goes, in some sense, even beyond Galileo’s views. It is not only the physical universe that is written in the language of mathematics; all of human knowledge follows the logic of mathematics. In Descartes’ words: “It [the method of mathematics] is a more powerful instrument of knowledge than any other that has been bequeathed to us by human agency, as being the source of all others.” One of Descartes’ goals became, therefore, to demonstrate that the world of physics, which to him was a mathematically describable reality, could be depicted without having to rely on any of our often-misleading sensory perceptions. He advocated that the mind should filter what the eye sees and turn the perceptions into ideas. After all, Descartes argued, “there are no certain signs to distinguish between being awake and being asleep.” But, Descartes wondered, if everything we perceive as reality could in fact be only a dream, how are we to know that even the Earth and the sky are not some “delusions of dreams” installed in our senses by some “malicious demon of infinite power”? Or, as Woody Allen once put it: “What if everything is an illusion and nothing exists? In that case, I definitely overpaid for my carpet.”

For Descartes, this deluge of troubling doubts eventually produced what has become his most memorable argument: Cogito ergo sum (I am thinking, therefore I exist). In other words, behind the thoughts there must be a conscious mind. Paradoxically perhaps, the act of doubting cannot itself be doubted! Descartes attempted to use this seemingly slight beginning to construct a complete enterprise of reliable knowledge. Whether it was in philosophy, optics, mechanics, medicine, embryology, or meteorology, Descartes tried his hand at it all and achieved accomplishments of some significance in every one of these disciplines. Still, in spite of his insistence on the human capacity to reason, Descartes did not believe that logic alone could uncover fundamental truths. Reaching essentially the same conclusion as Galileo, he noted: “As for logic, its syllogisms and the majority of its other percepts are of avail rather in the communication of what we already know…than in the investigation of the unknown.” Instead, throughout his heroic endeavor to reinvent, or establish, the foundations of entire disciplines, Descartes attempted to use the principles that he had distilled from the mathematical method to ensure that he was proceeding on solid ground. He described these rigorous guidelines in his Rules for the Direction of the Mind. He would start with truths about which he had no doubt (similar to the axioms in Euclid’s geometry); he would attempt to break up difficult problems into more manageable ones; he would proceed from the rudimentary to the intricate; and he would double-check his entire procedure to satisfy himself that no potential solution has been ignored. Needless