|||Descartes, René. Rules for the Direction of our Native Intelligence in The Philosophical Writings of Descartes. Cambridge. Cambridge University Press, 1984.|||
It is not possible for us ever to understand anything beyond those simple natures and a certain mixture or compounding of one with another. Indeed, it is often easier to attend at once to several mutually conjoined natures than to separate one of them from the others. For example, I can have knowledge of a triangle, even though it has never occurred to me that this knowledge involves knowledge also of the angle, the line, the number three, shape, extension, etc. But that does not preclude our saying that the nature of a triangle is composed of these other natures and that they are better known than the triangle, for it is just these natures that we understand to be present in it. Perhaps there are many additional natures implicitly contained in the triangle which escape our notice, such as the size of the angles being equal to two right angles, the innumerable relations between the sides and the angles, the size of its surface area, etc. pp. 15-6.
Rene Descartes, sometimes referred to as the “Father of Algebra”, gives to us an entirely new level of abstraction. When he separates the nature of a triangle into its components, one in particular stands out, “the number three”. We take it for granted today, because we have been performing this type of abstraction our entire lives.
The ancient Greeks, and even the Medievals, would not know what is going on if I handed them an algebraic expression, “3x + 4y = 36”. For thinkers before Descartes, numbers were always attached to things. You possess ten fingers, two ears, etc. When mathematical abstraction is performed, you abstract the shape of a triangle from some particular triangular thing. You imagine the triangle in your mind. You have abstracted it from the particular matter and accidents that the real triangular object has.
With numbers, however, Descartes takes abstraction to an entirely new level. He abstracts number from the particular measured thing. No longer does he have to think about five dogs, or five fingers, but he can think of five itself, or “five-ness”. It does not ultimately have to be grounded by five objects. We now have an intellectual concept, or ens rationis (being of reason, or logical being). He has also opened up the possibility for other beings of this kind. Zero, negative numbers, the square root of negative one, these are all beings of reason. The Greeks had no conception of these beings.
Again, we take this for granted today. When students begin algebra and are handed a bunch of practice problems, they don’t need any context to begin solving for x. Letters replace numbers; we treat them as numbers. We actually go backwards when a student is having trouble. We ground the problem with objects. How would you represent to him the division of a negative number by a negative number?
Descartes’ new level of abstraction also paved the way for calculus. Have you ever paused to think about the different representations for the three orders of equations that we use to express what happens when an object is dropped and accelerates due to gravity? I am standing on a second floor balcony. I drop a tennis ball straight down. It accelerates downward until it hits the ground. Let us imagine the graphs of the three equations used to express respectively acceleration versus time, velocity versus time, and distance versus time. If I find the area under the line in the velocity versus time graph, I find the total distance traveled by the ball. The area of this solid triangle represents a one-dimensional value, distance traveled. We have now fully removed numbers from the original figures that they represent (one dimension, distance) and placed them in a different figure (two dimensions, area). In the same way, picture the graph of acceleration versus time. It is a horizontal line. We now have a graph with a horizontal line depicting movement that was completely vertical.
A professor of mine, Dr. Richard Hassing, once posed an interesting question to a class: “In the graph of velocity versus time, what is represented by the hypotenuse of the triangle?” The individual points on the line represent the velocity at a given time, but what does the line as a whole represent? The line itself does not correspond directly to anything. Jacob Klein writes on this topic, “This reconciliation is achieved by means of symbolic figural representation. Everything depends on understanding that the ‘figures’ with which the ‘mathesis universalis‘ deals, namely ‘rectilinear and rectangular planes’ as well as ‘straight lines’, have, as far as their mode of being is concerned, no longer anything to do with the ‘figures’ of what had up till then been the ordinary ‘geometry’.”1 Descartes gave the practical sciences a new tool, which, most of the time, we do not fully appreciate.
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Jacob Klein, Greek Mathematical Thought and the Origin of Algebra. trans. Eva Brann (New York, NY: Dover Publications, 1992), 203. ↩