The Pythagoreans thought that numbers were the building blocks of things

Aristotle (384 - 322 BC), in the Metaphysics, reports to us that the Pythagoreans believed that the elements, or building blocks, of things were numbers. It is a difficult claim to understand, but the idea seems to be what it sounds like: the world that we experience is made of numbers.

In today’s world, the name ‘Pythagoras’ is associated with the Pythagorean theorem, but in antiquity, Pythagoras, who probably lived between around 570 BC and 495 BC, was associated with a school of philosophy, the details of which are very sketchy. Aristotle rarely but occasionally mentions a few prominent Pythagorean philosophers, such as Philolaus (ca. 470 - 385 BC), Eurytus (fl. 400 BC), and Archytas (410 - 350 BC), and he more typically refers to them as ‘the so-called Pythagoreans’. That’s the case, for instance, when he tells us that the Pythagoreans believed that the elements of things were numbers.

It isn’t clear why Aristotle inserts ‘so-called’. It is possible that he doubts that these figures really were associated with Pythagoras himself, even though people generally called them Pythagoreans. Perhaps he doubted that Pythagoras, who lived much earlier than they did, shared their thoughts.

At any rate, Aristotle tells us that the so-called Pythagoreans believed the elements of things were numbers. Why did they think this, and what exactly does this mean?

It is impossible to divorce this Pythagorean belief from the general context in early Greek philosophy, and Aristotle is reporting to us this general context when he reveals this fact about the Pythagoreans.

The context was that early Greek philosophers rejected mythological explanations of the world and preferred to give naturalistic explanations. This didn’t mean that they were atheists, but it meant that they gave early versions of what we would consider scientific explanations and generally preferred pictures of the world that sought the basic sources, elements, or building blocks of things.

We’ve talked about some of their views before. For instance, here is a post I wrote about Thales’ belief that water was the source of everything, and here is a post I wrote about Anaximenes’ belief that air was the source of all things. Anaximenes perfectly captures this intellectual trend: he thought that everything was made of air, and he identified physical, mechanical processes that transformed air into fire, oceans, school buses, rocks, etc.

When Anaximenes identified these processes, he most likely saw himself as out-competing Thales and his hypothesis about water. It is as if he were saying to Thales that his own theory had greater appeal than Thales’ because he could identify how air became all these disparate things, whereas Thales highlighted no such processes.

In this context, Pythagoreans believed that numbers were the elements of things.

Let’s look at Aristotle’s report of Pythagorean beliefs and see why. We have to rely on such reports because we don’t have any complete surviving Pythagorean works, so let’s jump into Aristotle’s text.

“Among these thinkers [i.e., those early Greek philosophers who tried to specify the fundamental building blocks of things] and before them, the so-called Pythagoreans were the first to latch on to mathematics. They both advanced these inquiries and, having been brought up in mathematics, thought that its starting-points were the starting-points of all beings. Since among these starting-points the numbers are by nature primary, and since they seemed to get a theoretical grasp on many similarities in the numbers to the things that exist, and to things that come to be, more so than in fire, earth, or water (for example, that such-and-such an attribute of numbers is justice, that such-and-such an attribute is soul and understanding, whereas another one is appropriate time, and—one might also say—each of the rest likewise), and, further, seeing in harmonies attributes and ratios that are found in numbers—since, then, the other things seemed to have been made like numbers in the whole of their nature, and numbers were primary in the whole of nature, they took the elements of numbers to be the elements of all beings, and the whole heaven to be harmony and number” (Metaphysics I.5 985b23-986a3).

This long passage confirms what I’ve been saying so far. But it also shines a light on some other important facts that help illuminate the Pythagorean reasoning.

Firstly, the Pythagoreans saw a deep similarity to numbers in the things that exist and come to be.

Secondly, they saw a greater similarity to numbers in those things than in fire, earth, air, etc.

In other words, they thought that their candidate for the ultimate building blocks of things was the correct candidate. Why? Because theirs had more similarities to the things whose composition and features they sought to explain.

Pythagorean philosophy was based on a desire to explain the world without reference to mythology, just as other early Greek philosophers were doing the same thing. They thought that their view was more plausible because of the greater similarities.

Consider, after all, how weird it seems to be to say that fire is made of water.

But we can understand music mathematically, and astronomy mathematically, and so many other parts of reality mathematically. Geometry, whose name literally means ‘earth-measurement’ in ancient Greek, was most likely a development out of early land-surveying methods.

Understanding the world around us mathematically is simple, intuitive, and very powerful in terms of making predictions and explanations. That might have been a real mark in favor of the Pythagorean hypothesis.

We also have some independent reasoning. For instance, Philolaus, a prominent Pythagorean, reportedly believed that numbers had to be in things, or else we wouldn’t be able to understand them.

Here’s the reported belief: “all things that are known have numbers. For it is not possible for anything to be known or thought of without this” (DK44B4).

The idea is that the presence of numbers in things explains their knowability. We can’t understand something without numbers.

Note that this is a little different from what Aristotle reported to us. For instance, Philolaus’ claim is consistent with there being things that don’t have numbers in them at all. It’s just that we couldn’t understand such things.

We don’t know whether Aristotle had Philolaus in mind when he said that so-called Pythagoreans believed that numbers were the elements of things, but looking at Philolaus’ reported belief helps us fill out the Pythagorean worldview for us. It gives us another reason why the Pythagoreans might have thought that numbers were in things: their presence makes the world understandable to us.

But what does this even mean, anyway? Numbers are in things, or numbers are the elements of things.

Well, it certainly does not mean that if you melted down or cut up an orange, you’d find numbers there. It isn’t obvious what it does mean, but it looks like the Pythagoreans had in mind at least two things:

1. Numbers fundamentally structure the things we encounter and experience.

2. Mathematics is directly applicable to everything we encounter and experience.

If you can’t quite understand these, recall what I pointed out earlier: everything in the world can be studied mathematically. If you study biology, physics, music theory, astronomy, physiology, cosmology, etc., enough, you’ll end up doing something mathematical at some point. Nothing we encounter and experience is “un-mathematical.”

This Pythagorean insight was an important part of the history of not just philosophy and mathematics but of ideas in general. It was an attempt to wrestle with, for the first time ever in the West, the deep mathematical structure of everything around us.