Useless on purpose

The discovery of nature

Lesson 2 ended with the Greek self: something that belongs to itself and stays the same through change. The next step is subtle and important. The Greeks gave things a self too.

The Greek word was physis. A thing’s physis is its own inner way of being: where it comes from and what it tends toward. For Aristotle, a thing’s nature is a source of motion and rest inside the thing itself. A stone falls and an acorn becomes an oak because of what they are, not because anyone pushes them there. Historians call this the “discovery of nature,” and this course treats it as the starting point of Greek science.

Did ancient China have the same idea? This course says no, and points to language. The modern Chinese word for nature, 自然 zìrán, took on that meaning through nineteenth-century Japanese. In the Daodejing it meant something else: “so of itself,” spontaneous.

人法地，地法天，天法道，道法自然。

Humans follow earth, earth follows heaven, heaven follows the Dao, and the Dao follows what is so of itself.

Daodejing, chapter 25

Laozi isn’t placing the Dao beneath some realm called Nature. He means the Dao is simply what it is, with nothing above it.

If things have natures of their own, then understanding them means following their inner logic rather than imposing ours. That, in this reading, is what the Greeks meant by reason, logos: following the logic that belongs to things themselves.

Science was not for survival

Now we reach science itself. The common view says science exists to help us survive. In this reading, that has it backward. Humans survived for hundreds of thousands of years on technology: fire, farming, metalwork, medicine learned by trial and error. Science became an engine of economic and military power only in the nineteenth century. Before that, it made little practical difference.

So what was Greek science for? The course’s answer: it was a path to freedom. To become a free person, you studied science. Aristotle said so directly:

As the man is free, we say, who exists for his own sake and not for another’s, so we pursue this as the only free science, for it alone exists for its own sake.

Aristotle, Metaphysics I.2, translated by W. D. Ross

A free science, in this reading, has two features. It serves no outside purpose, and it supports itself.

Euclid’s three coins

An old story, preserved by the anthologist Stobaeus some seven centuries after Euclid, makes the first point. A student who had just learned his first theorem asked Euclid, “What shall I get from learning these things?” Euclid called his slave and said: “Give him three coins, since he must make a profit from what he learns.” (Livelier retellings have Euclid throw the student out.)

The insult is precise. Asking what knowledge will earn you is the question of someone who works for another’s purposes, so Euclid answers it the way you’d pay a servant. The Greeks, this course argues, made uselessness an explicit standard for the highest kind of knowledge.

Knowledge that needs nothing outside itself

If knowledge isn’t justified by results, what makes it true? The Greek answer was deduction: reasoning in which the conclusion follows necessarily from the starting points. The textbook example is the syllogism. All humans are mortal. Socrates is human. Therefore Socrates is mortal. If the premises are true, the conclusion cannot be false, and no experiment is needed to check it.

Euclid’s Elements builds all of geometry this way. It begins with statements so obvious they sound empty: “Things equal to the same thing are also equal to one another.” “If equals are added to equals, the wholes are equal.” Admittedly, they sound like empty talk. But chain them together, step by careful step, and they prove that the angles of every triangle add up to two right angles, 180 degrees, and that in every right triangle the square on the longest side equals the squares on the other two: the Pythagorean theorem.

The key word there is every. Measuring can show that a rule works for the triangles you measured. A proof shows it must hold for all of them.

The same theorem in China

It is sometimes said that ancient Chinese mathematicians knew the 3-4-5 right triangle but may not have grasped that one rule covers every right triangle. Historians of Chinese mathematics would push back here.

The general rule appears in the Zhoubi Suanjing, compiled around the first century BCE, and The Nine Chapters on the Mathematical Art devotes a whole chapter to right-triangle problems. In the third century CE, the commentator Zhao Shuang drew a figure called the 弦图 xiántú, the “hypotenuse diagram,” that shows why the rule holds for any right triangle. Try it:

So the difference isn’t that Chinese mathematicians couldn’t prove things. It’s how their knowledge was organized, and what it was for. Chinese mathematics grew around problems and step-by-step procedures: surveying fields, dividing grain, computing taxes, building dikes. It was knowledge for officials. Greek mathematics grew around a single chain of deduction from a few axioms, pursued as an end in itself.

China even had a school that loved definitions. The Mohists, followers of Mozi, defined a circle as “one center, equal lengths,” strikingly close to Euclid. But Mohist logic faded after the Han dynasty, while Euclid’s Elements stayed in use as a textbook for more than two thousand years. The deeper point survives the correction: what counts is what each culture valued and chose to keep.

What “mathematics” meant

Greek science stood on two pillars: mathematics and philosophy. The young studied mathematics first, then went on to philosophy.

But the Greek word mathēmata didn’t mean calculating. It meant “things that can be learned,” knowledge that can be taught and passed on. Four subjects qualified: arithmetic, geometry, music, and astronomy. This lesson covers the first two; music and astronomy come in Lesson 4.

Even “arithmetic” meant something different. The Greeks separated arithmētikē, the study of what numbers are and how they relate, from logistikē, the practical reckoning used for building, measuring earthworks, and trade. Plato said future leaders should study numbers “not for buying and selling, like merchants,” but to grasp their nature. Behind this stood Pythagoras, who taught that the order of the cosmos shows itself first of all in number.

Perfect numbers

Take the number 6. Its parts, the numbers that divide it evenly (not counting 6 itself), are 1, 2, and 3, and 1 + 2 + 3 = 6. A number that equals the sum of its own parts struck the Greeks as marvelous, and they called it perfect. It’s hard to resist a Chinese parallel, the lucky phrase 六六大顺 liù liù dà shùn, “sixes bring great smoothness.”

Here is a small lesson in the long life of “useless” questions. Euclid showed how to build even perfect numbers. About two thousand years later, Leonhard Euler proved that Euclid’s recipe produces all of them. But whether a single odd perfect number exists is still unknown. Mathematicians today are still working on a question the Greeks asked, one that has outlasted every empire since.

The crisis of √2

Pythagorean arithmetic soon hit a wall. Draw a square with sides of length 1. Its diagonal has length √2, a perfectly definite line you can draw with a straightedge. If “all things are number,” then √2 should be a ratio of two whole numbers. It isn’t.

The proof is short enough to follow in your head. Suppose √2 = p/q, a fraction in lowest terms. Squaring gives p² = 2q², so p² is even, which means p is even. Write p = 2k. Then 4k² = 2q², so q² = 2k², which means q is even too. But then the fraction wasn’t in lowest terms after all. The assumption contradicts itself.

Legend says the Pythagorean who revealed the secret, Hippasus, was drowned at sea. That’s almost certainly a story, but the shock was real, and later writers called it the first crisis in mathematics. The Greeks responded by moving their center of gravity from number to geometry. A diagonal may not be a ratio of whole numbers, but it can still be drawn, compared, and reasoned about. From then on, geometry sat at the heart of Greek science.