Math and philosophy are inextricably linked. Unfortunately, students would never be able to know this from reading most mathematics textbooks. Elementary and high school mathematics textbooks are all geared around specific tools for specific types of problems. This is helpful to a –point—students are exposed to how to use mental tools to solve problems. However, we want to train our children to be able to think creatively and to be able to build new solutions to new problems, not just be stuck applying the old solutions to the old problems.
In order to do this, you have to learn how mathematics and philosophy interrelate. Mathematical principles and equations are primarily derived through what I will call ontological reflection. “Ontological” is an adjective which applies to questions of “being.” Ontological reflection, therefore, is taking the time to examine a question by looking at the fundamental meanings and definitions of mathematical terms. As an example, we will use ontological reflection to derive the equation of a circle.
So, let’s start with a unit circle (a circle with a radius of 1):
In order to find the equation, we need to reflect on the ontology of circles. In other words, we need to ask ourselves, “What does it mean for something to be a circle?”
Well, a circle is a set of points on a plane which are all equally distant from the center point. In other words, if we were to draw lines from any point on the circle to the center, they would all have the exact same length. We call this distance between the center point and the points on the circle the radius. Therefore, if we pick any arbitrary point on the circle, that point will be a radius length from the center, like this:
Since this is the unit circle, the distance from the center to the edge is exactly one. This is true of any line we might want to draw between the center and any point. Let’s call the point on the line that touches the circle P.
Since this is an arbitrary line, we will simply give the coordinates of P as (x0, y0). Now, take a close look at the line. Since it starts at the origin, it is starting to look like a right triangle. If we draw a line coming straight down from P to the X-axis, we would have a right triangle like this:
You will notice that the radius line is the hypotenuse of our newly-formed right triangle. This is true by definition, since we are drawing a line straight down from the touching-point and forming a ninety-degree angle with the X-axis.
So, if this forms a right triangle, then the lengths of each side are given by the formula A2 + B2 = C2, where C is the hypotenuse. So, in our example, what are the lengths of the A and B sides of the triangle? Well, since the point only goes out x0, then the length of A must be x0. Since it only goes up y0, the length of B must be y0. Since the hypotenuse of the triangle is, by definition, the length of the radius, that means that C is the radius of the circle.
Therefore, if you plug everything back in, you get x02 + y02 = radius2. Now, this is only for one point. Can we generalize it? Yes, we can, since our reasoning about x0 and y0 were not dependent on which point on the circle we drew our line to, this means that for every point on the circle the equation holds true. So how do we know if a point will exist on the circle? Well, let’s go back to the definition of a circle: a circle is all of the points which are of a given distance from the center. Therefore, every combination of x and y which satisfy the equation given will work. Therefore, our equation doesn’t just hold for specific points, it holds for all points. Therefore, we can rewrite our equation like this:
x2 + y2 = radius2
And that is the general equation for a circle with the center on the origin.
So, let’s look back on how we derived that equation. We started with reflecting on what it means to be a circle. Once we knew what it meant to be a circle, everything else fell into place. Conversely, if we had been incorrect on what it means to be a circle, everything else following would have likely been wrong, as well.
Therefore, philosophy allows us to build up mathematical principles by understanding the meaning of each term. In order to build equations, we must know what a thing really is at the deepest level. That is philosophy.
For my students, in nearly every example we do together, I make them answer the philosophical questions first: what are we trying to find out and what does that answer mean. By forcing students to tackle this problem explicitly, it trains them to ask better questions in the future. It keeps them from blindly following others by training them in the habit of asking the deep questions first, and then using the answers to those deep questions to find their way forward.