Mathematics and Plato’s Guardians

In Plato’s Republic, the great philosopher lays out his thoughts and prescriptions for the best way to organize society. Power, he insists, should lie with a class of intelligent and moral people called Guardians. But how does the state ensure that its best citizens are both intelligent and moral?

The School of Athens by Raphael. Plato and Aristotle in the center of the image.
Book VII of The Republic discusses the education of the guardians, the rulers and protectors of the perfectly just city. Education is paramount in Plato’s Republic, and the guardians are going to receive a very carefully crafted form of education. The principle subject that the guardians must study is that subject which affects their soul. Socrates is even more specific. He says, “The guardians must study a subject that draws the soul from becoming to being.” Becoming is a region, a category of reality. It expresses those kinds of things which come into being and pass out of being, finite, mortal, temporary, transient, fleeting things, the things of the world of our senses. Anything we can touch with our hands or see with our eyes is changing, and anything we can sense will eventually disappear. The other great region of reality is being. The permanent, the changeless, the purely intelligible, that which has no interaction with matter, that which must be thought, but cannot be seen. The guardians need a subject that will turn them around, from becoming to being.

Mathematics as the Key to Education

statue of head of Socrates, greek phlosopher
Socrates, the ancient Greek philosopher and the mentor of Plato

What is this subject? Socrates identifies this subject by describing it as the lowly business of distinguishing the one, the two, and the three—the number. The Greek word for number is arithmos, and it’s the root of our word arithmetic. The guardians that are undergoing this rigorous form of education do not study mathematics for practical purposes. Of course, this is the way mathematics is studied in most universities today. It was the way most people even would have studied mathematics in ancient Greece; we learn a little bit of math, and then we use it. Not the guardians. The guardians study mathematics in order to turn around. They study the nature of numbers themselves. They’re interested not in commerce, they’re not interested in technical applications of mathematics, they’re interested in the pure study of numbers. In modern language, this is described as number theory. After they study arithmetic, the guardians study plain geometry, solid geometry, theoretical astronomy, and harmonics.

detail of euclid from Raphael's "The School of Athens"
Education for the Guardian class in Plato’s Republic, was to include vigorous study of mathematics—including geometry.

What is the nature of mathematics, and why it was so important to Plato? It was important because mathematics is the best preparation for dialectic, the study of the purely formal structure of the whole of reality. It’s really not so obvious what the relationship is between mathematics and the forms. That’s what I want to have us think about. Let me put the point in the following way: think of the kinds of issues in which we have very real disagreement. You and I might disagree about the painting in the museum, and I say it’s beautiful and you say it’s ugly. You and I might disagree about a specific tax policy, you might say it’s unfair to tax rich people more than we tax poor people, and I might say no, it’s perfectly just to do that; we disagree. These are the issues, of course, that human beings have always intensely engaged in conflict over. Now, contrast that realm of disagreement with the realm of mathematics.

They study the nature of numbers themselves. They’re interested not in commerce, they’re not interested in technical applications of mathematics …

None of us would ever disagree that two plus two equals four. We take that to be a simple universal objective truth. We take it to be 100 percent clear that two plus two equals four. Take us back to the museum and imagine the discussion in which we’re disagreeing about the beauty of the painting. Well, that’s a hard discussion to have because it’s not clear what you mean by beauty or what I mean by beauty. Our disagreement about the tax policy, it’s not clear what you think justice is or what I think justice is, and that’s perhaps the reason why our disagreement goes on for such a long time.

Mathematics as a Platonic Ideal

I would suggest that the very best way to think of the relationship between mathematics and the forms—and in turn to understand Plato’s deep appreciation of mathematics and the prominent place he gives it in the education of the guardians, because after all, their education seems to be almost exclusively mathematical—is to think of the platonic forms as containing many of the same qualities that mathematics has, but operating in a different sphere. Another word that might be useful here: think of the forms as a projection of mathematical qualities onto issues like beauty and justice. Socrates believes that there is a form of beauty, a form of justice, beauty itself, justice itself.

Think of the forms as a projection of mathematical qualities onto issues like beauty and justice.

They would be the answer to the famous Socratic question, what is beauty, what is justice; they would be forms. They would have precisely the same sorts of qualities that mathematical truth, as we would all agree, already has. These forms would be clear and distinct and universal and objective. This is very hard to imagine. It’s very hard to imagine being in a museum and having an intense disagreement about a painting and thinking it could be resolved in the same way that an arithmetic problem can be resolved. If I ask you to multiply 75 times 152, I know we will all reach the answer if we do the steps properly or if we use a calculator, and we will end up with the same answer, and we don’t disagree. You and I will not come to blows over that mathematical problem. We may very well, however, come to blows about tax policy. We may disagree so vehemently that we can’t find a common ground.

The great platonic hope, the great platonic projection, is to project these kinds of mathematical attributes onto precisely those questions that right now seem to be so far from being resolvable. In Plato’s youth, in the 5th century, he witnessed tremendous turmoil. He witnessed his fellow citizens literally killing each other. This made, without a doubt, an enormous impression on him. Much of his thinking, I think, can be derived from this impulse. How do we resolve conflict? How do we come to harmony among ourselves? The platonic forms may be conceived, in fact, as a hopeful vision in which conflict about those most basic values, the values that people are willing to do die for, values like goodness and justice, can be resolved.

Mathematics as a Community

Statue head of Plato, ancient Greek philosopher
Plato, the ancient Greek philosopher and author of The Republic

Let me shift focus a little bit and look at mathematics from another perspective. I think Plato would say that mathematics is a wonderful example of community. Here’s what I mean by that apparently strange statement. Mathematics is the great equalizer. There’s only one answer to a problem and it doesn’t matter whether you are a man or a woman or young or old or from Greece or from Persia, from Athens, or from Sparta, the answer is the same. I think this gives, for Plato, a kind of inspiration about learning in general. He can imagine a common group of students who are working together towards the attainment of mathematical truth. They’re bonded precisely by the common objective that they have, and because the objective is mathematical, it’s there to be had by all.

If you’ve ever known a mathematician, it’s likely this person will have told you that mathematics is beautiful. The greatest mathematicians have long felt this.

Here’s a last way to put this point and to make a suggestion. If you’ve ever known a mathematician, it’s likely this person will have told you that mathematics is beautiful. The greatest mathematicians have long felt this. They study mathematics not because it’s practical, although it is, not because it’s useful, but because the sheer beauty of formal structure, the sheer beauty of literally perfection, shines through in mathematical truth. To take a ridiculously simple example, the one I’ve cited, two plus two equals four is a perfectly true sentence. That has, as trivial as it is, a beauty to it. I think this notion of beauty has long inspired mathematically minded thinkers. I think it inspired Plato. As a result, in Plato’s Academy, mathematics seems to have been a prerequisite. One had to study geometry in order to enter Plato’s Academy.

The Dialectic

The culmination of the education of the guardians is called dialectic. Dialectic is the study of forms and is inspired by the “what is it” question that Socrates is famous for asking. The first and, perhaps, the most interesting point that Socrates makes about dialectic is that it’s potentially very dangerous, and it’s especially dangerous for young people. Reading Book VII, you’ll see that the curriculum of the guardians is very rigidly regimented. Guardians, until they’re about 20 years old, do very little else but engage in physical exercise and training, called gymnastic. Between 20 and 30, these future rulers only study mathematics, but when they’re 30 and up to about the age of 35, they start to get their first introduction to dialectic. To complete the sequence, between the ages of 35 and 50, the guardians will be required to go down into the cave where they will rule the city. Then, at the age of 50, they return to the study of dialectic, and only at that very late stage of their education will they finally get a peek at the Idea of the Good, the pinnacle of their study.

The first and, perhaps, the most interesting point that Socrates makes about dialectic is that it’s potentially very dangerous, and it’s especially dangerous for young people.

Now, the dialectic is potentially quite dangerous for young people. I want to elaborate a little bit on that. Imagine that there is a young Athenian soldier, and his leaders tell him that he must go to war, and his leaders try to inspire him by telling him that this will be a just war. Perhaps, this was a soldier in the year 431 B.C.E. when the Peloponnesian War broke out. This soldier, in my hypothetical story, is on his way to serve in the army when he bumps into Socrates. What does Socrates do? He says, where you going, and the kid says, I’m going to war. Why are you going to war? Because the cause is just and I’m willing, even, to lose my life if my city requires me to do so. Socrates would then hit him with his question, what is justice?

Well, if you study The Republic, you know just how hard it is to answer this question. It’s very difficult to imagine that a 19-year-old boy would be able to make any real progress in answering this question. So he leaves the conversation with Socrates puzzled, confused, in a state of wonder, of bewilderment. What is justice? I thought I knew, I thought it was what my leaders told me was just, but this man Socrates has disrupted me. This man Socrates has taught me that I do not know what I thought I knew.

Well, what might happen? Maybe this boy will become a deserter, maybe he won’t serve in the army or maybe even worse. Maybe this boy will say I don’t know what justice is, maybe I’ll go over to the Spartan side. Maybe they’re just; maybe these Athenians who’ve been ordering me around aren’t telling me the truth. Socrates has taught me that I don’t know what justice is; the door is therefore open to me to do what ever it is I might want to do.

picture of Socrates teaching Alcibiades
Socrates and Alcibiades

Now this story corresponds to an actual event with an actual person. His name was Alcibiades, a very famous Athenian. He was famous for two things: he was an associate of Socrates and he was a traitor to Athens in the Peloponnesian War who went over to the Spartan side. This, by the way, is no doubt one of the real reasons Socrates was executed in 399 B.C.E. He was thought to be associated with the traitor Alcibiades. The point is that dialectical inquiry, the inquiry that begins with the question “what is it” and leads to an inquiry into the forms, is potentially subversive of the city. This is why in the educational program outlined in Book VII, Socrates does not allow young people to even be exposed to dialectic until they’re at least 30 years old.

From the lecture series Plato’s Republic
taught by Professor David Roochnik, Boston University