Is The Pythagorean Theorem Stolen?
Putting mathematics into context for the Greeks, Egyptians, and Babylonians.
The Pythagorean Theorem, a fundamental concept in geometry, is often misrepresented online by clickbait posts. Some try to assert that the Babylonians and Egyptians not only recognized the theorem but also applied it over a millennium prior to Pythagoras, leading people to think that the Greeks / Pythagoras “stole" the theorem and other mathematics. Topics such as this ignite both Eurocentric and Afrocentric narratives, both of which distort history in harmful ways.
While these ancient civilizations had knowledge of numerical relationships corresponding to the theorem, it is misleading to claim they developed or proved it as Greek mathematicians later did. The key difference lies in their methods: the distinction between empirical knowledge and deductive proof. Understanding this difference is crucial for accurately placing Greek mathematics within its historical context.
The Babylonians, with mathematical records dating back to 1800 BCE, had a sophisticated understanding of right-angled triangles. Artifacts like the Plimpton 322 tablet list Pythagorean triples—whole number sets (e.g., 3, 4, 5) satisfying a^2 + b^2 = c^2. This suggests that the Babylonians understood and used these relationships, likely for architectural and surveying purposes. Another tablet, YBC 7289, approximates the square root of 2, showing their capability to calculate a square's diagonal accurately. These findings show that the Babylonians possessed advanced mathematical techniques but did not prove they had a generalized statement or proof of the Pythagorean Theorem.
This means that although they could accurately compute values for specific instances, they did not express or prove a universal mathematical rule applicable to all right triangles. Rather than framing the theorem as a general principle, they concentrated on individual cases. There is no indication that they recognized the theorem as a basic geometric truth or made any effort to demonstrate why it universally applies. Instead, their method was primarily computational, focusing on results rather than justification.
Like Babylonian mathematics, Egyptian mathematics prioritized practical applications over theoretical concepts. The Rhind Mathematical Papyrus (around 1650 BCE) mentions 3-4-5 triangles, which Egyptian surveyors used to establish right angles in land surveying and construction. However, this was more of a practical guideline than a mathematical theorem. Unlike later Greek mathematicians, the Egyptians did not investigate whether this principle applied to all right triangles, nor did they seek to explain why it was effective. Using Pythagorean triples does not demonstrate a formal understanding of the Pythagorean Theorem; it merely indicates that these civilizations recognized that specific number sets resulted in right angles.
The pivotal shift in mathematics occurred with the Greeks, notably through Pythagoras and Euclid's contributions. Differing from earlier mathematical approaches, Greek mathematics prioritized deduction rather than simple computation. Its goal was to reveal universal truths through logical reasoning, moving away from dependence on empirical data. The Pythagorean school is recognized for formulating the first formal proof of the Pythagorean Theorem. Unlike the Babylonians, who concentrated on numerical patterns, the Pythagoreans sought geometric explanations for mathematical truths. Their approach evolved from analyzing specific cases to establishing general principles, marking a significant leap in mathematical development.
Euclid’s Elements offers the most definitive proof of the Pythagorean Theorem around 300 BCE. In Book I, Proposition 47, he presents a thorough geometric proof that verifies the theorem for every right triangle. Unlike earlier civilizations that may have recognized specific instances, Euclid’s demonstration is grounded in universal logical principles. In addition, his method is axiomatic, starting with basic assumptions (axioms) and carefully constructing a logical framework from there. This systematic and logical reasoning was a hallmark of Greek mathematics, setting it apart from all prior mathematical traditions.
In right-angled triangles the square from the side subtending the right angle is equal to the squares from the sides containing the right angle.
ἐν τοῖς ὀρθογωνίοις τριγώνοις τὸ ἀπὸ τῆς τὴν ὀρθὴν γωνίαν ὑποτεινούσης πλευρᾶς τετράγωνον ἴσον ἐστὶ τοῖς ἀπὸ τῶν τὴν ὀρθὴν γωνίαν περιεχουσῶν πλευρῶν τετραγώνοις.
The tendency to overstate Babylonian and Egyptian knowledge of the Pythagorean Theorem stems from a misunderstanding of mathematical advancement. Although these cultures exhibited notable mathematical skills, knowing how to use something is not the same as understanding why it works. The Babylonians and Egyptians recognized the properties of right triangles—they could measure lengths and develop specific instances—but they did not create a comprehensive, deductive theorem or provide a logical proof. In contrast, the Greeks established formal proof, ensuring that mathematical assertions held universal validity rather than being confined to particular instances.
The difference between empirical and deductive mathematics is not just a minor detail—it marks a fundamental shift in our intellectual history. Had the Greeks not made the transition from simple observation to rigorous proof, mathematics would likely have remained just a set of practical techniques instead of evolving into a structured and theoretical field. This is why the theorem is credited to Pythagoras rather than the Babylonians or Egyptians: his school might not have been the first to discover it, but they were the first to recognize it as a universally provable truth, paving the way for Euclid’s later formalization.
While the Babylonians and Egyptians made notable contributions to mathematics, their approaches differed significantly from those of Greek mathematicians. Greek mathematics stands out because it introduced a system of rigorous logical proof, transitioning from empirical calculation to axiomatic reasoning. This change fundamentally transformed mathematics, explaining why the Pythagorean Theorem, although utilized earlier, is considered a hallmark of Greek, rather than Babylonian or Egyptian, mathematical achievement.
With all this said, it is important to stress that Greek mathematics did not emerge in isolation. I am not by any means arguing for a “Greek miracle” narrative, which is often used to justify white supremacy and Eurocentrism. I aim to properly contextualize mathematical development to better understand the Greek contribution correctly. The systematization of Greek mathematics should be viewed in the larger historical framework of cultural exchange, especially with the Near East. The Greeks did not create mathematics from nothing; rather, they received a rich legacy of mathematical traditions from Babylonian and Egyptian sources, which they then refined through a new emphasis on deductive reasoning and formal proof.
This advancement was not the result of a singular moment of intellectual brilliance. It is not a case of a “Greek miracle” but the outcome of historical conditions. The Greek polis culture emphasized public discourse, education, and civic participation, creating an environment where mathematics thrived within philosophical inquiry. Unlike centralized bureaucratic systems in Egypt and Mesopotamia that used mathematics primarily for administration, Greek mathematics developed through public debate and philosophical schools. Plato and Aristotle established academies promoting mathematics as both practical and theoretical, connected to metaphysics and logic. This open climate encouraged cultural exchanges, notably with Persian and Egyptian networks, integrating older techniques into a deductive framework.
With the rise of Hellenistic kingdoms, especially the Ptolemies in Egypt, mathematics transitioned into an institutional discipline. The Ptolemies fostered mathematical research through centers like the Library and Museum of Alexandria. Mathematicians such as Euclid, Eratosthenes, and Archimedes thrived on royal patronage, which provided essential support for scholarly work. This system enabled unprecedented intellectual specialization and collaboration in Greek history. Unlike informal, polis-based schools reliant on private wealth, royal backing created formal institutions that structured the study of mathematics, differentiating it from decentralized traditions. By merging Near Eastern and Egyptian mathematical knowledge into a theoretical framework, Hellenistic rulers solidified Greek mathematics as an elite tradition that influenced scientific thought for centuries after.
Euclid’s Elements was a monumental contribution. The Elements systematized mathematical principles through axiomatic proofs, serving as the foundational text for mathematics for almost two millennia. Euclid’s influence in mathematics continued until the 19th century, when non-Euclidean geometries by Gauss, Lobachevsky, and Riemann challenged his postulates and expanded mathematical thought.