Mathematics did not begin in Greece. Counting, measuring, and geometric reasoning appear in every early civilization, driven by the practical demands of agriculture, trade, and monument-building. What the Greeks added — and what makes “mathematics” recognizable as a discipline rather than a collection of techniques — was the idea of proof.
Before the Greeks
Mesopotamia
Babylonian mathematics, from around 2000 BCE, was remarkably sophisticated. Babylonian scribes used a base-60 (sexagesimal) number system, which we still use for minutes, seconds, and degrees. They had multiplication tables, reciprocal tables, and procedures for solving linear and quadratic equations. The clay tablet Plimpton 322 (ca. 1800 BCE) lists fifteen Pythagorean triples — integer solutions to — a thousand years before Pythagoras.
Babylonian mathematics was procedural: here is a recipe that works. It was not axiomatic: there were no proofs, no statements of generality.
Egypt
Egyptian mathematics is known primarily from two surviving papyri: the Rhind Papyrus (ca. 1650 BCE) and the Moscow Papyrus (ca. 1850 BCE). Egyptian mathematicians computed areas, volumes, and worked with unit fractions. They knew a formula for the volume of a frustum (a truncated pyramid) that is essentially correct. Their methods, like Babylon’s, were practical and procedural.
India and China
Indian mathematics of the Vedic period (ca. 1500–500 BCE) included geometric constructions for altar-building and arithmetic in the Sulbasutras. Chinese mathematics of the Han dynasty (206 BCE – 220 CE) produced the Nine Chapters on the Mathematical Art, a compendium that treated linear systems, Pythagorean theorems, and approximations to .
The Greek invention of proof
Around 600 BCE, something changed. Greek mathematicians — starting with Thales and Pythagoras — began to ask not just what is true but why it is true, and to answer with deductive arguments. This move from observation to demonstration is the most important event in the history of mathematics.
The Pythagorean school
Pythagoras and his followers proved the theorem that bears his name, discovered the connection between musical consonance and numerical ratios, and — uncomfortably — the existence of irrational numbers. Their commitment to proof turned mathematics into a deductive science.
Plato and the Academy
Plato, though not a research mathematician, elevated mathematics to philosophical centrality. The motto above the Academy’s door was reportedly “Let no one ignorant of geometry enter here.” The five Platonic solids — the only convex regular polyhedra — appear in his dialogue Timaeus. Plato’s view that mathematical objects are real, eternal, and more perfect than physical reality has shaped philosophy of mathematics ever since.
Euclid
Around 300 BCE in Alexandria, Euclid compiled the Elements — the most influential textbook in mathematical history. In thirteen books he organized the known mathematics of his time into a single deductive system. Starting from explicit definitions, postulates, and common notions, he derived proposition after proposition in strict logical order.
Archimedes
Archimedes of Syracuse (ca. 287–212 BCE) was probably the most inventive mathematician of antiquity. He computed the area and volume of spheres and cylinders, approximated with remarkable accuracy, anticipated integral calculus with his “method of exhaustion,” and invented machines that held off the Roman siege of Syracuse. He was killed by a Roman soldier during the city’s fall.
Apollonius and Diophantus
Apollonius of Perga wrote the definitive ancient treatise on conic sections. Diophantus of Alexandria (ca. 250 CE) gave his name to Diophantine equations — equations seeking integer solutions — through his book Arithmetica, which Pierre de Fermat was reading when he wrote the famous margin note about his Last Theorem.
The end of antiquity
Greek mathematics declined after the Roman conquest and effectively ended with the closure of the Academy in Athens (529 CE) and the destruction of the Library of Alexandria (various episodes from 48 BCE through 642 CE). The mathematical tradition was preserved and extended in the Islamic world, beginning with the translation movement in 9th-century Baghdad — a period that brought figures like al-Khwarizmi and the birth of algebra.
What antiquity bequeathed to mathematics is not primarily a collection of results. It is the method: definitions, axioms, proofs. That method, refined and extended, has defined the discipline ever since.
Frequently asked
Who invented mathematics?
No one person — and no one civilization. Counting systems appear independently in Mesopotamia, Egypt, China, India, and the Americas. What the Greeks added, around 600 BCE, was the idea of proof: the insistence that mathematical claims must be justified by logical argument.
What did the Babylonians know?
More than often recognized. The tablet Plimpton 322 (ca. 1800 BCE) lists Pythagorean triples. Babylonian mathematicians solved quadratic equations by geometric methods, used a base-60 number system (the source of our 60-minute hour and 360-degree circle), and had sophisticated astronomical tables.