# Pythagoras and Early Pythagoreanism

Philosophy I · Ancient Philosophy: From the Milesians to Plato · https://tryals.app/learn/philosophy-i/pythagoras-and-early-pythagoreanism

## Number as the Nature of Things

The Pythagoreans made a claim quite unlike the Milesians': the archē is not a substance at all but **number**. Things are what they are because of the numerical ratios in them, and to understand something is to grasp its proportion.

The evidence was **music**, and it is genuinely striking. Divide a vibrating string in simple whole-number ratios and you get the consonances:

| Ratio | Interval |
|---|---|
| 2:1 | Octave |
| 3:2 | Perfect fifth |
| 4:3 | Perfect fourth |

Nothing in the sound of a note suggests arithmetic, yet the intervals that sound harmonious correspond exactly to the simplest ratios. Here was an audible quality with a hidden mathematical structure, and if music worked this way, why not everything? The **harmony of the spheres** extended the thought to the cosmos: the heavenly bodies move at proportional distances and produce a music we cannot hear only because we have never known its absence.

The Pythagoreans were a **school and a religious community** at once. They held **metempsychosis** — the transmigration of souls between bodies, including animal bodies — practised dietary restrictions, held property in common and imposed secrecy. Mathematics was pursued as *purification*: understanding proportion was a way of ordering the soul, not a career. This fusion of mathematics with a doctrine of salvation is genuinely alien to modern habits of thought and shaped Plato deeply.

Then their own methods refuted them. The diagonal of a unit square has length $\sqrt{2}$, and it can be **proved**, the proof survives essentially unchanged, that no ratio of whole numbers equals it. Assume $\sqrt{2} = p/q$ in lowest terms; then $p^2 = 2q^2$, so $p$ is even, so $q$ must be even too, contradicting lowest terms.

**Incommensurable** magnitudes exist. Geometry contains lengths that no whole-number ratio can express, so "all things are number" fails in exactly the domain that had made it plausible. The legend that the discoverer was drowned at sea is almost certainly false, but it registers something real about how unwelcome the result was.

> **Common pitfall:** treating the irrationality of $\sqrt{2}$ as a technical curiosity. For the Pythagoreans it was a refutation of their metaphysics by their own best method, the first time in recorded thought that a proof destroyed the position of the people who produced it.

## Practice questions

8 of this lesson's 11 practice questions, with answers. The full set is in the app.

### 1. A string of length 60 cm sounds a note. What length, in centimetres, sounds the octave above it?

**Answer:** 30

**Why:** **30 cm**: halving the string raises the pitch by an octave, the 2:1 ratio. This is directly demonstrable on any stringed instrument, which is part of why the Pythagorean claim was so persuasive.

Page: https://tryals.app/practice/philosophy-i/pythagoras-and-early-pythagoreanism/a-string-of-length-60-cm-sounds-a-note-what-length-in-centimetres

### 2. Why was the discovery of incommensurable magnitudes so damaging to Pythagoreanism?

A. It showed that some geometrical lengths cannot be expressed as any whole-number ratio
B. It proved that harmonious musical intervals cannot be expressed by simple ratios
C. It demonstrated that the physical cosmos is fundamentally chaotic rather than ordered
D. It contradicted their core religious doctrine regarding the transmigration of souls

**Answer:** A. It showed that some geometrical lengths cannot be expressed as any whole-number ratio

**Why:** The diagonal of a unit square is $\sqrt{2}$, and no ratio of whole numbers equals it. Since "all things are number" meant whole-number ratio, geometry itself contained a counterexample, produced by the school’s own methods.

Page: https://tryals.app/practice/philosophy-i/pythagoras-and-early-pythagoreanism/why-was-the-discovery-of-incommensurable-magnitudes-so-damaging-to

### 3. The Milesians identified the primary principle with a material substance, whereas the Pythagoreans identified it with number. What follows from this shift for what it means to explain a natural phenomenon?

A. Explanation becomes an enquiry into the origins and divine purposes of the cosmos
B. Explanation requires reducing sensible qualities directly to tangible physical elements
C. Explanation shifts from identifying structural laws to describing physical elements
D. Explanation shifts from identifying constituent matter to discovering formal ratios

**Answer:** D. Explanation shifts from identifying constituent matter to discovering formal ratios

**Why:** The Milesians sought a material substratum, but the Pythagoreans showed that properties like musical consonance depend on mathematical proportion rather than matter. Confusing this with elemental reduction or teleology misses that formal order itself became the explanatory principle.

Page: https://tryals.app/practice/philosophy-i/pythagoras-and-early-pythagoreanism/the-milesians-identified-the-primary-principle-with-a-material

### 4. Arrange the steps of the classical proof that the square root of two is irrational.

**Answer:**

1. Assume it equals a ratio of whole numbers in lowest terms
2. Square both sides to get p squared equals two q squared
3. Conclude that p must be even, and write it as twice some number
4. Substitute back and find that q must be even as well
5. Note that both being even contradicts the assumption of lowest terms

**Why:** This is reductio ad absurdum applied to arithmetic, and the proof survives essentially unchanged after 2500 years. The contradiction is precise: both numbers turn out even, which the lowest-terms assumption forbade.

Page: https://tryals.app/practice/philosophy-i/pythagoras-and-early-pythagoreanism/arrange-the-steps-of-the-classical-proof-that-the-square-root-of-two

### 5. Which were features of the Pythagorean community?

A. Rejection of mathematics as a distraction from religion
B. A requirement of secrecy about doctrines
C. Communal property and dietary restrictions
D. Belief in the transmigration of souls

**Answer:** B. A requirement of secrecy about doctrines; C. Communal property and dietary restrictions; D. Belief in the transmigration of souls

**Why:** Metempsychosis, communal life and secrecy are all attested. Mathematics was not a distraction from the religious project, it *was* the project, pursued as a way of ordering the soul through understanding proportion.

Page: https://tryals.app/practice/philosophy-i/pythagoras-and-early-pythagoreanism/which-were-features-of-the-pythagorean-community

### 6. The Pythagoreans kept their mathematics strictly separate from their religious practice.

**Answer:** False

**Why:** False, the two were a single project. Understanding proportion was pursued as *purification*, which is why the school looks to modern eyes like a research programme and a religious order at the same time.

Page: https://tryals.app/practice/philosophy-i/pythagoras-and-early-pythagoreanism/the-pythagoreans-kept-their-mathematics-strictly-separate-from-their

### 7. Match each musical interval to its string-length ratio.

**Answer:**

- Octave → Two to one
- Perfect fifth → Three to two
- Perfect fourth → Four to three
- Unison → One to one

**Why:** Simplicity of ratio tracks consonance exactly, which is a genuinely surprising empirical fact and not something anyone could have predicted from how notes sound.

Page: https://tryals.app/practice/philosophy-i/pythagoras-and-early-pythagoreanism/match-each-musical-interval-to-its-string-length-ratio

### 8. Sort each claim by whether the Pythagoreans could maintain it after the discovery of incommensurables.

**Answer:**

- Still maintainable: Halving a string raises its pitch by an octave, Mathematical structure underlies natural phenomena
- Refuted by the discovery: Every magnitude is expressible as a ratio of whole numbers, All things are number, in the sense of whole-number ratio

**Why:** The empirical discoveries survived; the metaphysical generalisation did not. Mathematical structure does underlie nature, but not every magnitude is a ratio of whole numbers, so the doctrine had to be weakened.

Page: https://tryals.app/practice/philosophy-i/pythagoras-and-early-pythagoreanism/sort-each-claim-by-whether-the-pythagoreans-could-maintain-it-after