Mathematics and Plato’s Guardians

From the lecture series: Plato’s Republic

By David Roochnik, PhD, Boston University

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?

statue of the Greek philosopher Plato, in front of the  main building of the nineteenth century neoclassical Academy of Arts in Athens, Greece
(Image: Alpha Media Production/Shutterstock)

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 will receive a very carefully crafted form of education. The principle subject that the guardians must study is that 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.

This is a transcript from the video series Plato’s Republic. Watch it now, on The Great Courses.

Mathematics as the Key to Education

Marble head of Socrates (1st century) displayed at the 
Louvre Museum.
Socrates, the ancient Greek philosopher and the mentor of Plato. (Image: Photographed by Sting/Public domain)

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, 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 to turn around. They study the nature of the numbers themselves. They’re interested not in commerce or the 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 plane geometry, solid geometry, theoretical astronomy, and harmonics.

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. The relationship is between mathematics and the forms is not obvious. Consider the point in the following way: Think of the kinds of issues in which we have a very real disagreement. You and I might disagree about the painting in the museum; 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 …

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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% 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. It’s a hard discussion to have because it’s not clear what you mean by beauty or what I mean by beauty. In 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

Sketch of the bust of Plato, the famous, classical Greek philosopher. By unidentified author, published on Magasin Pittoresque, Paris, 1842. Bust kept in Louvre museum.
Plato, the ancient Greek philosopher and author of The Republic.

The 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, as, 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 is to 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, distinct, universal, and objective. This is a difficult concept 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, we will all reach the answer if we do the steps properly or if we use a calculator; 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 projection is to project these kinds of mathematical attributes onto precisely those questions that currently 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 killing each other. This made, without a doubt, an enormous impression on him. Much of his thinking 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.

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Mathematics as a Community

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

Let’s shift focus a little bit and look at mathematics from another perspective. Plato would likely say that mathematics is a wonderful example of community. 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, young or old, from Greece, Persia, Athens, or 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.

Sketch of Plato in his academy
Plato in his academy. (Image: By After Carl Wahlbom/Public domain)

Here’s a final way to explain this point and to make a suggestion. If you’ve ever known a mathematician, this person will likely have told you that mathematics is beautiful. The greatest mathematicians have long felt this. They study mathematics not because it’s practical, not because it’s useful, but because the sheer beauty of a formal structure, the sheer beauty of literal perfection, shines through in mathematical truth. To take a ridiculously simple example, two plus two equals four is a perfectly true sentence. That has, as trivial as it is, a beauty to it. 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 to enter Plato’s Academy.

Learn more about the origins of one of the oldest branches of mathematics

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, 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 dangerous for young people. Imagine that there is a young Athenian soldier and his leaders tell him that he must go to war. 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 BCE when the Peloponnesian War broke out. This soldier is on his way to serve in the army when he bumps into Socrates. Socrates asks him his destination, 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?

If you study The Republic, you know 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. 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, 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 whatever it is I might want to do.

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Alcibades being taught by Socrates. Painting by François-André Vincent 1776
Alcibiades and Socrates (Image: By François-André Vincent – Musée Fabre (brochure)/Public domain)

Now this story corresponds to an actual event with an actual person. His name was Alcibiades, a 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 BCE. 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.

Common Questions About Plato’s Guardians

Q: How do Pythagoras and Plato coincide in their philosophy?

Pythagoras postulated that human nature resulted in appetite, reason, and courage. Plato conceded that the living state is simply a large version of the human body and exhibited all the traits Pythagoras had given to humans.

Q: How does Plato divide society?

Plato divides his version of a just society into three classes: the producers who make up the society and do most of the work, the guardians who make laws and decide what is best for the society, and the auxiliaries who are warriors that defend the society and enforce the laws.

Q: How does Plato envisage the guardians?

Plato develops a way of life for the guardians to be a wise, ascetic group of philosophers who essentially reject material possessions for the knowledge that they are internally made of divine gold and thus need nothing. They should have a fee only enough to survive the year and should not own excess property lest they become enemies of the state.

Q: What does Plato’s love mean?

Plato’s love was called platonic love and encouraged a rising through the layers of carnal, emotional love into love at the soul level and eventually love uniting with truth. This is platonic love.

This article was updated on October 15, 2020

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