Archytas: The Philosopher Who Asked What Happens at the Edge of the Universe

Imagine you’re standing at the very edge of everything. Not the edge of the city, or the edge of the Earth, but the edge of the entire universe. You can go no farther. Now—what happens if you stretch out your hand?

You can probably guess that something weird is going on. Either your hand goes through into… something, in which case the universe wasn’t really the edge after all. Or it stops at an invisible wall. But then what’s on the other side of that wall? More wall? Nothing at all? The question seems silly at first, but the more you think about it, the harder it gets.

About 2,400 years ago, a Greek philosopher named Archytas asked exactly this question. And he used it to argue that the universe must be infinite—that there is no edge, no matter how far you go. People have argued about whether he was right ever since.

Archytas was unusual. He wasn’t just a philosopher sitting around thinking. He was also a leading mathematician, a music theorist, a general who won seven elections in a row to lead his city’s army, and the person who once rescued Plato from a tyrant. He lived in the Greek city of Tarentum, on the heel of Italy’s boot, during a time when it was one of the most powerful cities in the Greek world. And he believed that almost everything—music, politics, and even the shape of animals—could be understood through numbers and mathematical relationships.

The Edge-of-the-Universe Argument

Let’s start with that argument, because it’s one of the most famous thought experiments in history, and it still gets people thinking today.

Archytas said to imagine you’re at the outermost edge of the universe. Not just the edge of what we can see, but the very boundary of everything that exists. Now, can you extend your hand or a staff beyond that edge? If you can, then there’s something beyond the edge—which means it wasn’t really the edge. If you can’t, that seems strange too. Why would there be a wall? What would it be made of? What would be on the other side? Either way, Archytas said, you end up concluding that there’s always more space to extend into. And that means space is unlimited—it goes on forever.

Plato and Aristotle both rejected this argument. They believed the universe was a finite sphere, with nothing outside it—not empty space, not anything. But Archytas’ argument was extremely influential. The Roman poet Lucretius used a version of it to argue for infinite space. Much later, Isaac Newton and John Locke took it seriously. Even today, scientists aren’t sure whether the universe is finite or infinite. We just don’t know. The argument hasn’t been settled.

But there’s a catch. Modern physics allows for a universe that is finite but doesn’t have an edge—like the surface of a sphere, but in three dimensions. If you walk in a straight line on Earth, you never hit an edge; you just loop around. Archytas’ argument assumes that if the universe were finite, it would have to have a boundary you could reach. That might not be true. Philosophers still argue about this.

The Sciences: Four Sisters

Archytas is probably the earliest person we know of who grouped together four sciences as a family: geometry, arithmetic (he called it “logistic”), astronomy, and music. A thousand years later, this set of four became known as the quadrivium, and together with three other subjects they made up the seven liberal arts that students studied for centuries.

Why these four? Archytas thought they were all ways of finding the numbers and proportions that make sense of the world. Music works because the intervals we hear—the difference between a low note and a high note—correspond to simple ratios of string lengths. A string twice as long gives you an octave. A string in the ratio 3:2 gives you a fifth. The numbers aren’t just in our heads; they’re actually in the sounds. Astronomy works the same way: the planets move in patterns that can be described mathematically. Geometry and arithmetic are the tools for doing all of this.

Archytas began his book on music theory by praising his predecessors in these sciences. They had, he said, “discerned well about the nature of wholes,” and because of that, they could understand individual things correctly. This is a key idea: you start with the big, general concepts—what sound is, what number is—and then you work down to particular cases, like the exact ratios that musicians actually used in his day.

Plato agreed with some of this. In his famous book the Republic, he quotes Archytas’ phrase about the sciences being “akin” to each other. But Plato had a serious disagreement with Archytas about what the sciences were for. Plato thought the whole point of studying mathematics was to turn the mind away from the physical world toward a higher, invisible world of perfect Forms. Archytas, on the other hand, wanted to find the numbers in the physical world—in the sounds musicians actually played, in the shapes of real things. Plato criticized this approach. He wanted mathematicians to study numbers themselves, not “heard harmonies” or visible shapes.

This is a real philosophical disagreement that still matters. When you learn math in school, do you think of it as a way to understand the world around you, or as a way to think about abstract ideas that don’t depend on any particular physical thing? For Archytas, those two things weren’t separate. For Plato, they were.

Doubling the Cube: A Mathematical Triumph

One of the most famous mathematical problems in ancient Greece was the “Delian problem”—how to double the volume of a cube. The story goes that the island of Delos was suffering from a plague, and the oracle told them they needed to double the size of a certain altar that was shaped like a cube. Simple, right? Just build a cube with sides twice as long. But that doesn’t work—doubling the side of a cube makes the volume eight times larger, not twice. Building two identical cubes and stacking them gives you twice the volume, but the result isn’t a cube anymore.

The problem turned out to be much harder than anyone expected. The mathematician Hippocrates of Chios had already figured out that solving it required finding two special numbers (or lengths) that would relate to each other in a particular proportional way. But nobody had actually found a method for constructing those lengths.

Archytas was the first person to solve it. His solution is remarkable—it involves imagining two different curved surfaces rotating in three-dimensional space and intersecting with each other. The point where they meet gives you the answer. The mathematician Eudemus, who wrote a history of geometry about a generation after Archytas, listed him alongside Theaetetus and Leodamas as one of the three most important mathematicians of Plato’s generation.

Later tradition tells a story that Plato criticized Archytas for using “mechanical” methods, claiming that this ruined the purity of geometry. But the story doesn’t make much sense, because Archytas’ solution doesn’t actually use any machines or instruments. It’s pure geometry in three dimensions, requiring only imagination. Most scholars today think the story was invented later, probably to create a dramatic origin story for the science of mechanics.

Numbers in Politics

Archytas didn’t think numbers were just for music and geometry. He thought they were the foundation of a just society.

In a fragment that survives from his writings, he says: “Once calculation was discovered, it stopped discord and increased concord. For people do not want more than their share, and equality exists, once this has come into being.”

What does he mean? Think about a conflict between rich and poor. The rich have more than they need; the poor don’t have enough. How do you resolve this fairly? Archytas thought that rational calculation—logismos—allows both sides to see what’s fair. The poor can see that they’re receiving what they deserve, and the wealthy can see that they’re giving what’s fair. Everyone agrees because the numbers are clear.

This connects to the kind of government Tarentum had. It was a democracy—unusual for a city with ties to Sparta, which was famously not a democracy. Archytas himself was elected general seven years in a row, even though the law normally prevented anyone from serving in consecutive years. That’s how popular he was. He believed that the ability to calculate and reason was something shared by most people, not just a few experts. This made him more democratic than Plato, who thought only a tiny number of philosopher-kings should rule.

What Makes Something What It Is?

Aristotle—who wrote an entire three-volume work on Archytas (sadly lost)—praised him for his way of defining things. Archytas gave definitions that combined two elements: what something is made of and what form or arrangement it has.

For example, he defined “windlessness” as “stillness in a quantity of air.” The matter is air; the form is stillness. He defined “calm on the ocean” as “levelness of sea.” The matter is sea; the form is levelness.

These examples might seem oddly chosen. But they show that Archytas was thinking about how to give precise definitions of things that aren’t obviously mathematical. Even here, he saw the world in terms of structure and proportion.

Aristotle also reports that Archytas was good at seeing similarities between very different things. He once said that “an arbitrator and an altar are the same.” This sounds bizarre until you think about it: both are places of refuge. Someone being persecuted can flee to an altar and be safe. Someone in a dispute goes to an arbitrator to be safe from unfair treatment. They share the same function, even though the context is totally different.

Why This Still Matters

Archytas is not as famous as Plato or Aristotle, but he was a major figure in his own time and an important influence on both of them. He represents a different way of doing philosophy: one that doesn’t separate the physical world from the world of ideas, that finds meaning in numbers and proportions, and that tries to use reason to solve practical problems in politics and daily life.

His edge-of-the-universe argument is still taught in philosophy classes as a classic thought experiment. His work on harmonics laid the foundation for later music theory. His solution to doubling the cube is a landmark in the history of mathematics. And his idea that rational calculation can create political harmony is still relevant whenever people argue about fairness and redistribution.

But there’s also something appealing about Archytas as a person. He was a successful general who loved playing with children. He refused to punish his slaves when he was angry, because he didn’t want to act out of emotion. He argued against the idea that pleasure is the only thing worth pursuing in life. He believed that reason and calculation could make life better for everyone, not just for a few.

The questions he asked—about what lies beyond the edge of the universe, about how numbers relate to reality, about what makes a society fair—are still open. Nobody has completely answered them. That’s partly why we still read about him today.

Appendices

Key Terms

Term	What it does in this debate
Logistic	Archytas’ word for the science of number and calculation, which he thought was the most powerful of all sciences
Quadrivium	The set of four sciences (geometry, arithmetic, astronomy, music) that Archytas was among the first to group together
Mean proportional	A number that relates two others in a series of equal ratios; finding two of them was the key to doubling the cube
Thought experiment	An imagined scenario used to test an idea or argument, like Archytas’ “what if you’re at the edge of the universe?”

Key People

Archytas (approx. 435–350 BCE) — Greek philosopher, mathematician, general, and music theorist from Tarentum; the only ancient philosopher who was also a major political leader
Plato (approx. 428–348 BCE) — Famous Athenian philosopher who knew Archytas, quoted him, but also disagreed with him about whether numbers exist separately from physical things
Aristotle (384–322 BCE) — Plato’s student who wrote more about Archytas than about any other predecessor; admired his definitions but rejected his argument for infinite space
Eudemus — Aristotle’s student who wrote a history of geometry that preserved Archytas’ solution to doubling the cube and his edge-of-the-universe argument

Things to Think About

Archytas assumed that if you can’t extend your hand past the edge of the universe, that’s “paradoxical” and counts as a problem. But is it really? What if the universe just ends, with nothing outside—not even space? Is that any weirder than the idea that space goes on forever?
Archytas thought that rational calculation could end political conflict because everyone would see what’s fair. But do you think people actually agree when they see the numbers? Think of a real argument you’ve had—could it really have been solved just by showing someone a calculation?
When you learn math in school, are you learning about the world around you or learning to think about abstract ideas that don’t depend on anything physical? Or both? Does it matter which one you think you’re doing?
Archytas defined “windlessness” and “calm on the ocean” by combining matter and form. Try defining something like “silence” or “darkness” the same way. What would you say is the matter and what is the form?

Where This Shows Up

Physics and cosmology — The question of whether the universe is finite or infinite is still unresolved. The James Webb Space Telescope is gathering data that might eventually help decide.
Music theory — The discovery that musical intervals correspond to number ratios is still the basis for how we understand scales and chords. If you’ve ever played an instrument, you’ve been using ideas Archytas helped develop.
Politics and fairness — Arguments about fair distribution of resources—taxes, social programs, reparations—often turn on debates about whether rational calculation can resolve conflict or whether something else is needed.
Mathematics education — The debate between Plato and Archytas about whether math should be about abstract ideas or real-world applications is still alive in debates about how math should be taught in schools.