Ann Varela: Who Was Rene Descartes? And why do we continue to quote him?

Jun 13, 2017 by

An Interview with Ann Varela: Who Was Rene Descartes? And why do we continue to quote him?

Michael F. Shaughnessy –

  1. Ann, first of all, Rene Descartes is a very well-known name in both math and philosophical circles, so we have to address two realms today- but first of all, when and where was he born?

Back in 1596, Descartes made his entrance into the world.  His birthplace was La Haye en Touraine (now Descartes, Indre-et-Loire), France.

Rene was just over a year old when his mother died soon after delivering another child.  His father sent him to be raised by his grandmother in the Poitou district.  Descartes was a sickly child, and this caused him to start his schooling a bit later than other children.

2. What do we know about his early childhood?

By 1607, Descartes arrived at the Jesuit Collège Royal Henry-Le-Grand at La Flèche where he began his studies in mathematics and physics.  He studied Galileo’s work there.  Upon graduation in 1614, he spent two years (1615–16) studying at the University of Poitiers, where he earned a Baccalauréat and Licence in Canon and Civil Law.  This accomplishment was his father’s deepest wish.  He later moved to Paris.

3. I know that he is known as the father of analytical geometry, which uses algebra to describe geometry. While I have taken both College Algebra and College Geometry, I am not sure how he used algebra to describe or define geometry- other than perhaps labeling right angles and sides of triangles- Can you clarify?

The algebraic equations used to describe the size and position of geometric shapes on a coordinate system are what analytic geometry is all about.  Analytic geometry is also known as coordinate geometry, which uses the coordinate system.  This coordinate system is now named after Descartes’ name in its Latin form: Renatus Cartesius; hence, the Cartesian coordinate system.  We can use the Cartesian coordinate system to manipulate equations in two or three dimensions.  In geometry, we study planes (a flat, two-dimensional surface that continues infinitely) in two or three dimensions.  See Figure 1.

File:Intersecting planes.svg

Figure 1:  Two intersecting planes in three-dimensional space

By David Eppstein – Own work, Public Domain,

Analytic geometry is concerned with defining and representing geometrical shapes in a numerical way and obtaining numerical information from those shapes’ numerical definitions and images.  The numerical output, however, might also be a vector (an object that has length and direction) or a shape.

In analytic geometry, the plane is given a coordinate system, by which every point has a pair of real number coordinates, or ordered pairs described by an x-value and a y-value (x, y).  See Figure 2.

Figure 2: Cartesian coordinate system

Now, perhaps you can see how numbers are assigned to geometric objects with the use of an ordered pair.  For example, in Figure 2, one of the ordered pairs given is (2, 3), which represents a point located two units to the right and three units up from the origin (0, 0).

With regards to the circle in Figure 2, numerical values for x and y would need to satisfy, or produce true statements, when substituted into the equation of the circle provided.  Only certain points will satisfy the equation for that particular equation, thus each circle has its own unique equation.

Descartes’ founding of analytic geometry made it possible for Sir Isaac Newton and Gottfried Leibniz to develop calculus.

4. While in Bohemia in 1619, he apparently invented analytic geometry, which is a method of solving geometric problems algebraically and algebraic problems geometrically. Why is this important?

Analytic geometry is significant because it has important applications in a wide variety of fields such as agriculture, science, astronomy, physics, and business. There are numerous examples of how analytic geometry is used to solve problems found in real-life.  In business, financial information is illustrated in graphs.  Data gathered from experiments (from many different disciplines) may also take the form of a graph.  Points and curves are used in 2-D animation.  Communication and social network analysis rely on analytic geometry as well.  Bioinformatics is a more recent field that uses computer software for data visualization in science, which enables real-time data analysis through the creation of analytic applications and dashboards.  See Figures 3-6.

Figure 3.

Figure 4.

Figure 5.


Figure 6.

5. Apparently he also devised a way to depict known and unknown values along with “pioneering the standard notation” that uses superscripts to show the powers or exponents. Can you give us a few examples?

Descartes came up with the notion to use x, y, and z for unknown values.  We continue to use these variables in equations to this day, as well as a, b, and c to denote known numerical values.

A great example of Descartes’ use of known and unknowns is found in linear equations in two variables.  The equation of a straight line is Ax + By = C, where x and y represent an ordered pair on the line.  A, B, and C represent real numbers known as the coefficients and constant terms.

With regards to exponents, we use these in various formulas which help us model certain real-life events, like falling objects or projected objects.  Have you ever built a model rocket and launched it?  The path that the rocket follows may be modeled with the following function:  h(t) = -16t2 + vt + h0, where t is time in seconds, v is the initial velocity, and h0 is the initial height of the object.  See Figure 7.

Figure 7.

Other examples of exponents are found in formulas for chemistry (half-life), biology (growth and decay rates), business (compound interest formula), health (body mass index), construction (area and volume), statistics (population growth/decline), medicine (infectious disease rates and bacterial growth), engineering and electronics (angular frequency of oscillations in a certain type of circuit), and computer graphics, just to name a few.

6. Now venturing into philosophy, his best known philosophical statement is (or was): “Cogito ergo sum”; I think, therefore I am”. Why was this important at the time? And is it still important today?

After Descartes published his “Discourse on the Method”, he expanded upon the idea that thought exists.  He argued that the mere act of wondering whether or not one exists was proof that one did indeed exist.  When Descartes spoke of “thought,” he included all forms of mindfulness involving the senses, such as hearing, sight, touch, smell, taste, and emotions.  He later abandoned this idea of senses playing a role in existence, which he proved in his Wax Argument.  With the Wax Argument, it was clear to see that a wax entity and a melted wax entity were in fact both wax, but did not possess the same characteristics, therefore presenting themselves as two distinct entities.  He determined that the senses could be deceptive.

Descartes’ philosophy was important because it allowed 17th century science to organize a physical world explained by a mathematical nature and permitted mathematical concepts to explain that physical world.  It was because of this precise philosophy that a divine power, God, was no longer necessary to explain the nature of things, like how the universe works.

Even though we are no longer using Descartes’ statement to prove the existence of God, it may be argued that “I think; therefore I am” continues to cause one to ponder the meaning of their existence and why we have thought.  Do we think to exist or do we exist to think?

7. Some scientists get insight through dreams. Can you tell us about Descartes and his dreams?

Supposedly, after having three “visions” or dreams, which he thought to be influenced by divine intervention, he devised a universal method of deductive reasoning, based on mathematics that is applicable to all the sciences. This method, which he later described and elaborated on in his “Discourse on Method” (1637) and “Rules for the Direction of the Mind” (written by 1628 but not published until 1701), encompasses four rules:

(1) Accept nothing as true that is not indisputable.

(2) Split problems into their most basic parts.

(3) Solve problems by progressing from simple to complex thoughts.

(4) Recheck the analysis.

These rules are a clear application of mathematical procedures.  Additionally, Descartes insisted that all key notions and the scope of each problem must be plainly defined.

8. I have to say that I appreciate rule number two- but today we call it task analysis—Was Descartes also a type of pedagogy person? Teaching how to teach? It seems re-checking and reviewing one’s reasoning is certainly an appropriate methodology for learning!

I think Descartes’ focus was on thinking and not necessarily teaching.  His ideas seem circular, that is, one could argue his point forwards as well as backwards.  For example, “I think, therefor I am” could just as well be “I am, therefore I think.”  Likewise, as noted earlier, analytic geometry oscillates between numerical and graphical representations.

9. Apparently, Descartes was a part of Queen Christina’s court in Stockholm and advised her on philosophical issues. How did this come about? Why would Queen Christina be wanting a mathematician to advise her on philosophical issues? And what might she be investigating?

Queen Christina wanted Descartes to establish a new scientific conservatory and instruct her in his ideas about love.  She encouraged Descartes to publish the work he began with Princess Elisabeth entitled “Passions of the Soul”.

10. What was his most renowned work?

In his most renowned dissertation, the “Discourse on the Method of Reasoning Well” and “Seeking Truth in the Sciences” of 1637, he describes his own personal educational growth and quest for knowledge.  He believed that by initially doubting everything, he could reconfirm the original hypothesis by a process of rebuilding.  All previous knowledge relating to a given thought is excluded and a solitary definite fact from which to begin is sought.  This process of doubting and deductive reasoning gave way to contemporary science.  Basically, what Descartes is arguing for is a separation of the way we acquire knowledge of the truth.  One way encourages a creative, yet disciplined method of thinking, and the other thrives on logical procedures and methodology.

11. I have heard that ironically, it was in large part the mathematics of Descartes, that later made possible the defeat of Cartesian science. Now first of all what do we mean by Cartesian science and how did the math of Descartes impact science?

Cartesian science is based on Descartes’ idea that certain knowledge can be developed through reason originating in the brain from birth. Observation and experimentation were key to developing scientific theories.  Cartesians believed God’s supremacy was reason enough to accept uncertainty.

Descartes’ impact on science became more apparent in modern physics, as he provided the first markedly modern formation of laws of nature and a conservation principle of motion.  He came up with the concept of momentum of a moving body before Newton’s laws of motion.

He also made strides in optics and the study of the reflection and refraction of light, and he constructed a theory of planetary motion in the late 1600’s.  Descartes’ vortex theory postulated that these colliding molecular particles supplied the energy that pushed the planets towards the sun.  Nevertheless, Descartes’ vortex theory was refuted as supporting the model of orbiting planets.

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