transcript
Speaker 1
Then there's democracy, but there's no name for a degraded democracy—at least, I don't think he gives one in The Politics. But to have that sort of informed citizenry, a demos that's educated enough to sustain a good democracy, requires some sort of moral grounding or foundation. I think Plato points to this, but in our modern idea of the liberal individual, there is no grounding; there is no direction. It's just, "do what you want as long as you're not breaking laws or interfering with the freedom of others."
Having no ideal, such as "the good," or even a moral foundation for how the populace should be educated is harmful and leads to the degradation of democracy.
I keep thinking of Nietzsche here. Nietzsche seems to be this great champion of the "will to be yourself." It doesn't matter, essentially, what you will to be, as long as you aspire to some sort of ideal, great version of yourself completely unbounded by any standard of goodness. There are nuances to how you read Nietzsche, but that's how a lot of people receive him. I think that kind of expresses our modern liberal ideal, which may be harmful to having a well-functioning democracy.
It also makes me think about what Simone Weil writes about education. She picks up on a Platonic theme regarding the necessity of a certain kind of education. For Weil, education is basically about the cultivation of attention. She actually posits that the best way to train your attention is to study things that you're not interested in. For example, studying math for the sake of math, but not necessarily out of any kind of love for it. She says that's actually better when it comes to training your attention. You have to dedicate yourself to the study, but your attention is best trained if you overcome your own inclinations.
In her view, the ultimate form of attention—what all training of attention prepares you for—is prayer. Prayer is the highest mode of attention in her view. But it's perhaps not prayer like we normally think about it; it's prayer as a kind of meditation, awaiting goodness and grace to enter into the void created through your kenosis—your self-emptying—when you're fully attentive to something.
I read about a priest who was a very good friend of Weil's. He said that when you talked with her, you had the distinct sense of a person who was fully attentive to everything you said and did. In that respect, she was walking the walk as well as talking the talk.
Pedro
It goes to the question of what education is actually for. Reminds me of being in academia now—it seems to become more and more about filling seats than anything else. It's a neoliberal model of education where we just fill as many seats as possible to make as much money as possible.
Because I wasn't alive at the time, I only hear stories about it, but there was a stronger sense in the United States that higher education possessed a distinct moral value. Its purpose was to educate "liberal individuals"—not liberal in the political left-or-right sense, but meaning free individuals who are capable of being responsible with their own freedom and educated to participate responsibly in the political process. That's the ideal of a liberal education as I understand it.
Today, that seems to be becoming more degraded in favor of an instrumental approach to education: "I got my degree, now I can just go get my job and do my own thing."
My worry with moving away from socially responsible forms of education and sliding down those standards is that we are sliding back toward a society like the one Plato warns about. If education just becomes another means to an end to get what I want and rise up the social ladder, it just develops more tyrannical personalities. Education becomes entirely about what it can do for me, or what I can take out of it.
Speaker 1
In my opinion, there's nothing more dehumanizing; all forms of dehumanization stem ultimately from that attitude toward education. If the soul is a process of formation itself, then education can't be a means to something else. Paideia, or Bildung—education—is the essence of human existence. It's how we become more fully human, and I don't think there is a final limit to that becoming.
Learning as a lifelong pursuit has to be the basis of any just society. The meaning of human life as such is education in the highest sense.
I think you've made quite an accurate diagnosis, Pedro, regarding the way education gets instrumentalized in our time as just a means to the end of getting a well-paying job. Even if education were understood as a means of creating a well-informed citizen who could participate in self-governance, that would still technically be instrumentalizing education to some degree—though it would be instrumentalizing it toward a much higher value than a purely economic one.
I think we need to reach even higher than that, toward something infinite like "the good." We'll never be done trying to become adequate to that transcendent good that Weil and Plato are trying to direct our attention toward. In the end, it definitely feels like it comes down to paideia.
I think one of the most common questions I get as a math teacher is, "What is this good for?" It's the one question that I don't want them to ask. A lot of my task is simply to make them stop asking that question, because if I try to answer it, it never helps. They don't get any more motivated.
Where I stir the most motivation in my students is actually when I start talking about some of the paradoxes surrounding the concept of infinity—things like Hilbert's Hotel or Cantor's analysis of transfinite numbers. They very often find that highly stimulating. Or, if I talk about the fact that they cannot actually see a triangle because of the way it's defined; it's not perceivable. They can think it, but they cannot see it.
When I put these kinds of paradoxes in front of them, I can feel that they are genuinely stimulated—much more so than if I try to explain to them what math is "good for" in the real world.
Matthew Segall, Pedro Brea and Karsten Jensen
Mathematically, a triangle is the simplest possible polygon (a closed, flat shape bounded by straight sides) in Euclidean geometry. Rather than defining it by how it looks, mathematics defines it strictly by its structural components, its coordinates, and the invariant laws it must obey.
Here is how a triangle is constructed and described mathematically.
1. Elements of a Triangle
A triangle is defined by a set of three distinct points in a two-dimensional plane, called vertices (plural of vertex), denoted as $A$, $B$, and $C$.
These vertices are connected by three straight line segments, called sides.
The sides are the intervals $[AB]$, $[BC]$, and $[CA]$.
Conventionally, the lengths of the sides opposite to vertices $A$, $B$, and $C$ are labeled with lowercase letters $a$, $b$, and $c$, respectively.
The space enclosed by these segments creates three interior angles, usually denoted by the Greek letters $\alpha$, $\beta$, and $\gamma$ (or simply $\angle A$, $\angle B$, $\angle C$).
2. The Conditions of Existence: Triangle Inequality
You cannot just pick any three random line lengths and form a triangle. To mathematically exist in a flat plane, the segments must satisfy the Triangle Inequality Theorem.
This law states that the sum of the lengths of any two sides of a triangle must always be strictly greater than the length of the remaining side:
If $a + b = c$, the three points would fall on a single straight line, creating what mathematicians call a "degenerate triangle" (which is visually just a line segment).
3. Invariant Properties (The Rules That Never Change)
No matter how you stretch, skew, or scale a triangle in flat (Euclidean) space, it is bound by rigid mathematical truths:
The Angle Sum Theorem: The sum of the interior angles is always exactly equal to a straight angle ($180^\circ$ or $\pi$ radians).
$$\alpha + \beta + \gamma = 180^\circ$$The Boundary and Space: The total distance around the shape is its perimeter ($P = a + b + c$). The amount of two-dimensional space it encloses is its area ($A$), which can be calculated fundamentally as:
$$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$
4. Classification Matrix
Mathematicians categorize triangles based on two distinct criteria: the relationships between the lengths of their sides, or the measures of their interior angles.
| By Sides | Definition | By Angles | Definition |
| Equilateral | All three sides are equal ($a=b=c$). All interior angles are exactly $60^\circ$. | Equiangular | All angles are equal (identical to equilateral). |
| Isosceles | At least two sides are equal ($a=b$). The angles opposite those sides are also equal. | Right-angled | One angle is exactly $90^\circ$. The sides obey the Pythagorean theorem: $a^2 + b^2 = c^2$. |
| Scalene | All three sides have different lengths, and all three angles are different. | Acute | All three interior angles are strictly less than $90^\circ$. |
| Oblique / Obtuse | One interior angle is strictly greater than $90^\circ$. |
5. Advanced Analytic Description
In coordinate geometry, a triangle can be defined as a convex hull of three coordinate points. If you have three points on a Cartesian plane—$A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$—the triangle is the set of all points that can be written as a "weighted average" (convex combination) of those three coordinates.
Using these coordinates, mathematicians can calculate the precise area of the triangle without knowing its height or angles, using a matrix determinant:
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