Descartes for Future Mathematicians and Philosophers

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Ricky Andrade, Walker White, Jared Gardner Ms. Miller December 14th, 2018 Geometry Honors Rene Descartes Reality is constantly being questioned by philosophers, people who wonder who we are and why we exist. Of these philosophers, one stands out among the crowd for not only beating the political climate of his time, but for his additions to philosophy as a whole as well as to geometry and mathematics. His name was Ren© Descartes, and he created one of the most basic elements of geometry, alongside finding the relationship between geometry and algebra. He faced a society that was deeply devoted to religion, and his ideas to break away from so-called Scholasticism were always a risk he had to take.

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R©ne Descartes was born on born March 31, 1596, in La Haye, Touraine, France. He immediately inherited a noble rank, due to his father, Joachim, being a councillor in the Parlement of Brittany in Rennes. This did not last long because his mother died one year after he was born. His father remarried soon after, leaving him in La Haye for his first maternal grandmother to raise him. When he was just ten years old, he was sent to a Jesuit college at La Fl??che, which was established in 1604 by Henry IV. In this Jesuit college over one thousand young men and boys were trained to pursue careers such as military engineering, the judiciary, and government administration. He studied at the Jesuit school until 1614 or 1615. He then enrolled at the University of Poitiers, where he went on to earn his Baccalaureate and License in Canon & Civil Law. At the age of twenty-two he enlisted in the army of Prince Maurice of Nassau. It is not known what his duties were exactly, although it is most likely that he was part of the what is now known as the Corps of Engineers. That division would have applied their knowledge in mathematics to design various structures and machines that aimed to protect and assist the soldiers in fighting the opposing army. While in the army, Descartes was stationed in Breda, where he met Isaac Beeckman. In notes Descartes wrote, he seemed to describe their relationship as a student teacher relationship. This new relationship Descartes had developed with Beekman would open Descartes’ eyes and he developed an interest in the sciences. In Discussions about a wide variety of natural science with Beekman would lead Descartes to write Compendium Musicae, which is based of specific questions Beekman posed. As said by a Cambridge scholar, one of the main ideas in Compendium Musicae is to identify the affections or properties of sound, as far as they may be known mathematically, which are capable of pleasing and of arousing the passions. It ended up being one of the most widely read books on mathematics in the second half of the seventeenth century. The book was not published during Descartes’ life time. He then left the army in 1619 and went to Germany where he would begin to work on his first published project ever. Discourse on the Method, published in 1637, shows how Descartes went about solving problems. It is said that The Discourse was meant to serve as an introduction to three essays Descartes had been laboring over??”Optics, Meteorology, and Geometry??”which contain science now regarded obsolete. The Discourse, however, remains one of the world’s most influential works of philosophy.

In the upcoming years, he jumped around europe, and finally landed in Paris. The notes had been writing told us that he had been in contact with Father Marin Mersenne ,who was a member of the Order of Minims. His relationship with Mersenne would turn out to shape his life in a major way. Mersenne prompted Descartes to makes his thoughts on natural philosophy public. It was also by the works of Mersenne that Descartes’ work would make its way into some of the smartest people in Europe’s hands. After 1628, when Descartes left Paris, he began to work on projects which are what he is known for today. Descartes is most well known for his contributions to philosophy, and geometry and mathematics. However, to understand the origin of these contributions we have to look back at his philosophies and how he differed from his contemporaries. Descartes’ entire world view was defined by his belief that nothing was certain unless it could be proven without a shadow of a doubt. This is where his quote I think, therefore I am comes from, as he can prove he exists due to his ability to think. He thought that Scholastic science and philosophy was worthless, as it simply stated things were the way they were for the sake of it. He looked for mechanisms and patterns in what occurred around him, seeing deduction as being capable of providing an explanation to everything. To him, the idea that all must be understood through sensation was ignorant, as senses could always deceive, looking instead to analytical thinking for answers. This led him to discover his passion for geometry, which had clear ideas and distinctions, as opposed to the wild and chaotic nature of relying on the senses. According to Philip J. Davis, author of Descartes’ Dream, Descartes was first and foremost a geometer; he claimed he was in a habit of turning all problems to geometry. He could start with simple facts and use deduction to answer a bigger question. This found its way into all parts of Descartes’ life, as he would search for facts that were true beyond a shadow of a doubt to base his works on, showing the extreme skepticism characteristic of philosophers today.

Descartes had a clear passion in mathematics, stating himself that I took pleasure, above all, in mathematics, because of the certainty and the absoluteness of its reasons; but I had not yet discovered its true use I was astonished that with such solid foundations nothing more eminent had ever been built upon them. With the revelation that mathematics had not changed in a significant way, he set out to build up on these foundations. When working on proofs, he developed a habit of doubting everything and needing to cut a problem into as many parts as needed before continuing to work, starting with simple facts and building from there. Eventually, Descartes made the connection between geometry and algebra, blending the two together to solve some of the tougher problems he faced, which led him to develop the framework for the Cartesian coordinate system. He would plot a curve on a graph with an x- and y-axis, and then as algebraic equations, and would use this new method to solve problems that had stumped some of the greatest mathematicians of yesteryear. Ever since Descartes had created the Cartesian coordinate system, we have been able to implement it in quite the number of versatile ways. In class, we have learned to exploit the distance formula to determine and prove different shapes (i.e. triangles, parallelograms, etc.) and, of course, find the measure of a line segment between two points in two-dimensional space. The midpoint formula is another formula able to be used in Cartesian coordinates. A common occurrence in our textbook had us drawing the problems themselves, whether it was to solve a mystery of a disappearing proof or draw a missing shape given written information, or editing a shape by translating it or reflecting it over an axis. Descartes’ coordinate system, along with his other mathematical discoveries, has revolutionized the way we learn math, even to this day.

We have sporadically used other types of Cartesian inventions in math. The most common example of this is the superscript used in exponents (i.e 24=2222). Along with that, with the creation of the Cartesian coordinates came the formula for a circle, ellipse, parabola, and hyperbola. This Cartesian geometry, or analytic geometry, unlocked the ability for mathematicians and others to solve equations both algebraically and geometrically, and helped further development in calculus. Descartes did not only contribute to geometry, however; he was responsible for creating the rule of signs for polynomial equations, making a method which could determine the number of positive or negative real roots of a polynomial. Descartes re-discovered the formula for amicable numbers (two integers for which the sum of proper divisors of one number equals the other), originally founded by Thabit ibn Qurra. He also devised a universal method of deductive reasoning, based on mathematics, that is applicable in a variety of ways, such as in sciences or other types of math (Watson). He later applied this method to create Discourse on Method and Rules for the Direction of the Mind. Descartes’ groundbreaking work even set the foundation to visualize geometries of dimensions higher than the third; something that could not have been done physically. Descartes’ life in the fields of philosophy and geometry was not as clear as it may seem. Religion, or more specifically Christianity, was accepted as fact during his time, and people who went against the views of the Church faced persecution and had their works banned. Galileo, a contemporary of Descartes, was arrested after his radical theory that the Sun, instead of the Earth, was at the center of the solar system. His works were blacklisted, and Descartes feared the same judgement. Despite his views of Scholasticism and his need to scientifically prove everything, he wanted to be a devout Christian and be remembered as such, so he never published one of his biggest works, The World. According to Kurt Smith of Stanford, The World would have contained . . . a treatise on physics, a treatise on mechanics (machines), a treatise on animals, and a treatise on man,. While some parts of The World has been found in Discours de la Methode, another of his works, and others published after his death, the majority of The World would never see the light of day. His attempts to try and stay in the Church’s good graces did not fair him very well, however, as after his death many of his works were found on the Church’s list of banned books until 1966.

Descartes laid the groundwork for future mathematicians and philosophers, breaking away from Scholasticism and connecting geometry and algebra. There is a reason we always bring up his famous quote, I think therefore I am. It broke away from conventions held in his time, and he managed to lay a foundation for modern-day skepticism. While some of his beliefs may not have aged well, the contributions he made to math will always be remembered for as long as we use graphs.

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