Pythagoras: 'All Things Are Number' — The Philosopher Who Heard the Order of the Cosmos

Key Takeaways

• "All Things Are Number": From the discovery that musical harmony can be expressed in integer ratios, Pythagoras argued that the order of the entire cosmos consists of numerical relationships. This is the prototype of the scientific impulse to understand nature mathematically.
• Harmonia (Harmony): Just as numerical ratios produce beautiful order in music, so too do numerical harmonies pervade the cosmos, the soul, and society. The idea of the "Harmony of the Spheres" influenced astronomy from Kepler onward.
• Purification of the Soul and the Intellectual Life: The soul transmigrates and ascends through purification. Pythagoras regarded intellectual inquiry itself as a means of purifying the soul, establishing the prototype of "the philosophical life."

Life and Historical Context

Legend clings thickly to Pythagoras' life, and reliable evidence is exceedingly scarce. None of his own writings survive; the earliest testimonies are fragmentary references by contemporaries — Xenophanes, Heraclitus, and Empedocles. Detailed biographies were not written until more than seven hundred years after his death (Iamblichus' On the Pythagorean Life; Porphyry's Life of Pythagoras), and separating deified tradition from historical fact remains a frontier of current scholarship.

He is said to have been born around 570 BCE on the island of Samos in the Aegean Sea. His father Mnesarchus was reportedly a gem-engraver (or merchant). Samos at that time prospered under the tyrant Polycrates, but Pythagoras — according to tradition — detested autocratic rule and emigrated around 530 BCE to the Greek colony of Croton in southern Italy.

At Croton, Pythagoras founded a community and gathered several hundred disciples. The community was no mere academic circle; it was a religious, political, and residential commune. Communal property, dietary restrictions (especially abstinence from meat and beans), a period of silence as training, and secret oaths — these rules bore the unmistakable character of a religious brotherhood. The community wielded political influence as well, eventually provoking a backlash. Rioting broke out, and Pythagoras was driven from Croton to Metapontum, where he is said to have died around 490 BCE.

The community eventually split into two factions. The "Listeners" (akousmatikoi) memorised the master's teachings as maxims (akousmata) and upheld a religiously strict way of life centred on dietary rules and silence. The "Learners" (mathēmatikoi) pursued mathematical proof and theoretical inquiry (Iamblichus, On the Pythagorean Life, 81–87). The polarity between rational mathematics and mystical religion was already inscribed within the community itself.

Pythagoras lived in the generation immediately following the Milesian school (Thales, Anaximander, Anaximenes), which had laid the foundations of natural philosophy. To the question "What is the archē (origin) of all things?" Pythagoras gave an answer that was neither water nor air but "number" — therein lay his originality.

Mini-Timeline

• c. 570 BCE: Born on the island of Samos
• c. 550 BCE: Tradition (unverified) holds he travelled to Egypt and Babylonia
• c. 530 BCE: Emigrates to Croton in southern Italy; founds his community
• c. 510 BCE: Croton destroys neighbouring Sybaris; the community's political influence peaks
• c. 509 BCE: Anti-Pythagorean riots; the community's meeting-house is set ablaze
• c. 490 BCE: Dies at Metapontum
• 5th–4th c. BCE: Disciples Philolaus and Archytas carry on and develop Pythagorean thought

What Did This Philosopher Ask?

The Milesian school asked "What is the origin of all things?" and proposed water, air, or the Indefinite (to apeiron). Pythagoras' question operated on a different plane. He asked not "What is everything made of?" but "What structure does everything possess?"

From matter (what things are made of) to form (what order they exhibit) — this shift of perspective was decisive. Aristotle reports in Metaphysics Book A (985b23–986a3) that "the Pythagoreans devoted themselves to mathematics and were the first to advance it; steeped in that study, they came to believe that its principles are the principles of all things." From a natural philosophy that asked about material origins to a mathematical philosophy that asked about structure and relation — this leap was Pythagoras' greatest achievement.

Moreover, the inquiry did not stop at nature. Numerical order pervades the soul and society as well. Music restores order to the soul because the soul itself is a kind of numerical harmony. As for the question "What is justice?", a tradition preserved in the Magna Moralia (1182a14 — a work usually attributed to a pseudo-Aristotelian author) reports that the Pythagoreans defined it as "a number multiplied by itself" (a square number). A grand vision that sought to grasp nature, the soul, and ethics in a single framework — that was Pythagorean philosophy.

Core Theories

1. The Ontology of Number — What "All Things Are Number" Means

"All things are number" — what does this mean? We moderns think of numbers as abstract concepts, but for the Pythagoreans number was reality itself. This expression is better understood not as a verbatim quotation from ancient sources but as a summary formulation of Pythagorean doctrine by Aristotle (Metaphysics 985b23–986a3, 986a15–21).

The Pythagoreans arranged numbers spatially as points. One is a point, two a line, three a plane (triangle), four a solid (tetrahedron) — number is geometrical reality and the constructive principle of bodies. This may look naïve. But when we recall that modern physics describes elementary particles in terms of mathematical structures (group theory, symmetry), the intuition that "mathematical structure lies at the foundation of matter" proves remarkably prescient.

A more concrete illustration of this idea is the theory of figurate numbers. The Pythagoreans visualised numbers by arranging pebbles (psēphoi). Triangular numbers (1, 3, 6, 10 …) are numbers whose pebbles form a triangle; the tetraktys, 10, is simply the fourth triangular number. Square numbers (1, 4, 9, 16 …) are pebbles arranged in a square, generated by summing consecutive odd numbers (1+3=4, 1+3+5=9 …) — a vivid demonstration that number and figure are one. Number is not abstraction but shape; shape is directly number.

A caveat is in order. Distinguishing the teachings of Pythagoras himself from those of later Pythagoreans (especially Philolaus and Archytas) is difficult, and modern scholarship generally treats them collectively as "early Pythagoreanism."

2. Music and Number — The Discovery of Harmonia

The firmest ground in Pythagorean thought is the discovery of the relationship between music and number. According to tradition, Pythagoras was passing a smithy when he noticed that hammers of different weights produced consonant sounds. The smithy anecdote is physically inaccurate (the relationship between hammer weight and pitch does not yield simple integer ratios), but the discovery concerning string length and pitch is experimentally verifiable.

A string-length ratio of 2 : 1 gives an octave (consonance), 3 : 2 a perfect fifth, 4 : 3 a perfect fourth. All these ratios are composed from the first four natural numbers (1, 2, 3, 4). The Pythagoreans called their sum, 10, the "tetraktys" and revered it as a sacred number. 1 + 2 + 3 + 4 = 10 — this perfect number 10 was held to symbolise cosmic order itself, and disciples reportedly swore their oaths by it.

The impact of this discovery is incalculable. Behind the sounds we find beautiful lurk mathematical laws. Here was a concrete demonstration that sensory experience can be explained by number. It is the earliest forerunner of the modern scientific conviction that "the book of nature is written in the language of mathematics" (Galileo).

There is, however, a catch. Stacking twelve perfect fifths (3 : 2) does not return exactly to seven octaves (2 : 1) — (3/2)¹² ≠ 2⁷ — a tiny discrepancy later called the "Pythagorean comma." The harmony of integer ratios is not perfect, and this small mismatch harboured the same structural problem as the later crisis of irrational numbers: the world cannot be completely closed within ratios of whole numbers.

3. The Harmony of the Spheres — Cosmic Harmonia

The Pythagoreans extended musical harmony into cosmology. The heavenly bodies revolve on great spheres, each at a different speed; since moving bodies produce sound, the heavenly bodies must produce sound too. Yet we cannot hear it — because we have heard it since birth and have no silence against which to contrast it (Aristotle, De Caelo 290b12–291a6).

This is the idea of the "Harmony of the Spheres" (harmonia tōn sphairōn). As a scientific hypothesis it is, of course, untenable. What matters is the conviction that "the cosmos contains mathematical order yet to be discovered." When Kepler discovered the laws of planetary motion (1619, Harmonices Mundi), the tradition he consciously inherited was precisely this Pythagorean intuition.

Notably, the later Pythagorean Philolaus placed at the centre of the universe neither the sun nor the earth but a "Central Fire" (Hestia), with the earth and other heavenly bodies revolving around it (DK 44A16). This bold idea of removing the earth from the cosmic centre was cited by Copernicus himself in the preface to De Revolutionibus, demonstrating that Pythagorean cosmology was no mere fantasy.

The word "cosmos" (kosmos) — the universe as order — is traditionally said to have been first used by Pythagoras (Aëtius, though the attribution remains debated). As the antonym of "chaos," the term encapsulates the Pythagorean worldview.

4. The Table of Opposites — The Binary Structure of the Cosmos

Aristotle reports that the Pythagoreans posited "ten pairs of opposites" as fundamental principles (Metaphysics 986a22–b8): Limit and Unlimited, Odd and Even, One and Many, Right and Left, Male and Female, Rest and Motion, Straight and Curved, Light and Darkness, Good and Evil, Square and Oblong.

The table may look miscellaneous, but its foundation is the dualism of "Limit" (peras) and "the Unlimited" (apeiron). Odd numbers belong to the side of Limit: when an odd-numbered gnomon (an L-shaped row of pebbles) is added around a unit, the figure always remains a square (1, 1+3=4, 1+3+5=9 …). Even-numbered gnomons, by contrast, produce rectangles of ever-changing proportions, never settling into a fixed shape (Aristotle, Physics 203a10–15). The cosmos comes into being when Limit imposes order on the Unlimited — an idea that feeds directly into the ontology of "mixture of Limit and the Unlimited" developed in Plato's Philebus.

5. Transmigration and Purification of the Soul — Metempsychōsis

Pythagoras believed in the transmigration of the soul (metempsychōsis). Xenophanes mocked the belief: "They say that once, when a puppy was being beaten, he passed by and, taking pity, said: 'Stop — don't beat it. It is the soul of a friend; I recognised it by its voice'" (DK 21B7).

For Pythagoras, however, transmigration was not mere faith but the foundation of an ethical system. The soul is imprisoned in the body. By living rightly, the soul is purified and transmigrates into a higher form of being; an unjust life causes the soul to descend into an animal. The prohibition of meat-eating springs from the conviction that a human soul may inhabit the animal one is about to eat.

Crucially, Pythagoras regarded intellectual inquiry itself as a means of purifying the soul. According to tradition, he compared life to a festival. Among those who come to a festival are competitors, traders, and spectators (theōroi). The best life is that of the spectator — not seeking fame or profit but contemplating truth (bios theōrētikos). This idea, via Plato, culminated in Aristotle's ideal of the "contemplative life" (Nicomachean Ethics, Book X).

6. The Pythagorean Theorem — The Eternity of Mathematics

The square of the hypotenuse of a right triangle equals the sum of the squares of the other two sides — a² + b² = c². Whether the theorem was actually proved by Pythagoras himself is uncertain. The same relationship already appears on Babylonian clay tablets (Plimpton 322, c. 1800 BCE) and was known in ancient India and China.

Yet knowing something empirically and proving it as a universal proposition are quite different things. The Pythagorean contribution was arguably the advance from individual numerical examples (3-4-5, etc.) to a general proof — that is, to deductive mathematics.

Moreover, the theorem confronted the Pythagoreans with an inconvenient truth. The diagonal of a square with side 1 is √2, and the Pythagorean disciple Hippasus is credited with discovering that √2 cannot be expressed as a ratio of whole numbers — it is irrational. The very foundation of the doctrine "all things are ratios of integers" was shaken. According to legend, Hippasus was drowned for revealing the secret. Whatever the truth, the discovery of irrational numbers was the greatest crisis — and simultaneously the catalyst for the greatest leap — in the history of ancient Greek mathematics.

7. The Philosophical Life — The Birth of Philosophia

A tradition holds that Pythagoras was the first to use the word "philosophia" (φιλοσοφία, love of wisdom) — reported in Cicero's Tusculan Disputations V.3.8–9. He called himself not a "wise man" (sophos) but a "lover of wisdom" (philosophos).

The historical reliability of this anecdote is doubtful (it is likely a creation of Heraclides Ponticus). But the attitude it embodies — treating knowledge not as a finished possession but as an ongoing pursuit — harmonises deeply with the spirit of the Pythagorean tradition. As the festival metaphor shows, the contemplative life is the best life and the path of purifying the soul. Philosophy is not the accumulation of knowledge but a way of living.

Guide to Key Texts

• No writings by Pythagoras himself survive. The following are the principal sources for knowing Pythagorean thought.
• Philolaus, Fragments — The oldest primary source transmitting Pythagorean cosmology and number theory. The standard critical edition is Huffman (1993).
• Aristotle, Metaphysics, Book A — The most important ancient testimony on the Pythagorean philosophy of number.
• Iamblichus, On the Pythagorean Life (De vita Pythagorica) — A biography from the Neoplatonic era (3rd c. CE). Rich in legendary material, but an invaluable source for the community's way of life and rules.
• Porphyry, Life of Pythagoras — Another late-antique biography, alongside Iamblichus.
• Diogenes Laërtius, Lives of the Eminent Philosophers, Book VIII — A concise overview of Pythagoras' life and doctrines.

Major Criticisms and Controversies

1. Contemporary criticism: Heraclitus attacked Pythagoras by name — "Much learning (polymathiē) does not teach understanding (nous)" (DK 22B40). The charge that erudition is not true wisdom was directed at Pythagoras' encyclopaedic appetite for knowledge. Xenophanes, as noted above, ridiculed the doctrine of transmigration.

2. Aristotle's criticism: Aristotle objected that "in what sense number is the cause of things" was left unclear (Metaphysics 990a). Is number the material of things, their form, or their efficient cause? The Pythagoreans posited number as the principle of all things yet failed to develop a sufficient causal account.

3. The crisis of irrational numbers: The discovery that √2 cannot be expressed as a ratio of integers undermined the doctrine "all things are ratios of integers" from within. How the Pythagoreans responded to this crisis is unclear, but as a consequence Greek mathematics shifted its centre of gravity from arithmetic to geometry.

4. Modern source criticism: Walter Burkert's Lore and Science in Ancient Pythagoreanism (1962; English ed. 1972) argued that Pythagoras himself was primarily a religious leader and that most of the mathematical achievements attributed to him belong to later disciples. This view remains dominant, and the distinction between "the historical Pythagoras" and "the legendary Pythagoras" is a presupposition of current research.

Influence and Legacy

Influence on Plato: The Pythagorean influence on Platonic philosophy is substantial. The immortality and transmigration of the soul (Phaedo, Republic), the privileged status of mathematical entities (Republic VII, the educational curriculum), and the mathematical structure of the cosmos (Timaeus and the cosmology of regular solids) — all can be read as developments of Pythagorean themes. The tradition that "Let no one ignorant of geometry enter" was inscribed above the gate of the Academy symbolises the place of mathematics for Plato (though the historicity of the inscription itself is unverified).

Influence on modern science: In Harmonices Mundi (1619), Kepler explicitly pursued, as Pythagoras' successor, the correspondence between planetary orbits and musical ratios. Galileo's declaration that "nature is written in the language of mathematics" rests on a Pythagorean conviction at its core. Einstein and Heisenberg, too, invoked Pythagoras when speaking of their trust in the mathematical beauty of physical laws.

Music theory: The foundation of Western music theory lies in Pythagorean tuning. The method of constructing scales by stacking perfect fifths (3 : 2) became the basis of music education in medieval Europe (the quadrivium: arithmetic, geometry, music, and astronomy).

Neo-Pythagoreanism: From the first century BCE into the Common Era, Pythagoreanism was revived in fusion with Neoplatonism. Nicomachus' Introduction to Arithmetic and Boethius' De Institutione Musica became the standard texts for Pythagorean mathematical education in medieval Europe.

Connections to the Present

First, is mathematics discovered or invented? Are mathematical structures products of human thought, or are we discovering structures inherent in the cosmos? "Mathematical realism" in the philosophy of mathematics — the view that mathematical objects exist objectively — is a direct descendant of Pythagoras. The puzzle that physicist Eugene Wigner called "the unreasonable effectiveness of mathematics" (1960) remains unsolved.

Second, the digital world and the reign of number. Modern civilisation literally "runs on numbers." Music is converted into digital signals, images are numerical arrays of pixels, and AI "thinks" through mathematical optimisation. "All things are number" is drawing closer to a technological fact rather than a metaphor.

Third, the relationship between science and religion. For Pythagoras, rational mathematics and mystical religion coexisted without contradiction. Modernity separated the two, yet when scientists speak of the "beauty" and "harmony" of mathematical laws, the Pythagorean sense of awe has not entirely vanished.

Questions for the Reader

• Is the universe mathematically describable because the universe itself is mathematical, or because human cognition is structured that way?
• Can the experience of finding music "beautiful" be fully explained by numerical ratios? Is there a "beauty" that number cannot capture?
• For Pythagoras, mathematics and religion were inseparable. In the modern world, can scientific inquiry be purely "secular," or does it require some form of "faith"?

References

• (Primary sources): Aristotle, Metaphysics I (985b23–986a3, 986a15–21); Physics III (203a10–15); De Caelo II (290b12–291a6); Nicomachean Ethics IX.4 (1166a31–32). — Key passages cited in this article
• (Primary sources): Diogenes Laërtius, Lives of the Eminent Philosophers VIII.10. — Transmission of Pythagorean maxims and communal norms
• (Primary sources): Diels, H. & Kranz, W. Die Fragmente der Vorsokratiker, 6th ed., 1951. (DK 21B7, 22B40, 44A16, etc.) — Standard edition of Pre-Socratic fragments
• (Study): Burkert, Walter. Lore and Science in Ancient Pythagoreanism. Cambridge, MA: Harvard UP, 1972. — Landmark study of Pythagoreanism
• (Study): Huffman, Carl A. Philolaus of Croton: Pythagorean and Presocratic. Cambridge UP, 1993. — Critical edition and commentary of Philolaus' fragments
• (Study): Kahn, Charles H. Pythagoras and the Pythagoreans: A Brief History. Indianapolis: Hackett, 2001. — Concise and reliable overview
• (Overview): Stanford Encyclopedia of Philosophy, "Pythagoras" (Carl Huffman). https://plato.stanford.edu/entries/pythagoras/