From prehistoric tally marks to proofs verified by computers
Numbers came before writing. People left marks on bone, stone and clay to count and record quantities. Much of how we read this era, though, rests on conjecture.
Notches carved in bone, one of the oldest traces of counting.
A bone found near Ishango in Congo bears several groups of notches. Interpretations range from a simple tally to a calendar or even a list of primes, but one thing is certain: people were recording numbers a very long time ago.
Small pieces of clay for counting grain and livestock.
Across Mesopotamia, people marked quantities with clay tokens shaped like cones, spheres and disks. One theory holds that the practice of sealing these tokens in clay envelopes and pressing their shapes on the outside later developed into numerals and writing.
In Egypt and Mesopotamia, mathematics was a craft for surveying land, collecting taxes, building and tracking the heavens. It was remarkably sophisticated, but it did not prove why its methods worked.
The Nile's floods give birth to geometry.
Egypt used a base-10 system with a different picture sign for 1, 10, 100 and so on. Re-measuring field boundaries washed away by the yearly floods, and building the pyramids, drove the development of practical geometry.
Numbers whose value depends on position.
The Babylonians applied place value to a base-60 system and extended it to fractions, allowing calculations that rival modern decimal notation. Sixty minutes in an hour and 360 degrees in a circle are traces of this system.
The exact volume of a truncated pyramid.
This Egyptian problem book includes a problem on the volume of a pyramid with its top cut off. The formula is exactly the one we use today, but there is no explanation of how it was found.
A table of Pythagorean triples, and √2 to six decimal places.
The clay tablet Plimpton 322 lists whole-number triples satisfying a² + b² = c² in systematic order, and YBC 7289 shows the diagonal of a square with an approximation of √2 accurate to about six decimal places. Both are more than a thousand years older than Pythagoras. The Babylonians also solved quadratic equations by essentially the same steps as today's quadratic formula.
An Egyptian math textbook copied by the scribe Ahmes.
With 84 problems, this papyrus is the key source for Egyptian mathematics. It expressed every fraction as a sum of unit fractions with numerator 1, and computed the area of a circle as a square whose side is 8/9 of the diameter, which works out to π ≈ 3.16.
A decisive shift happened in Greece. Mathematics turned from a craft of calculation into a discipline of proof, and number became a philosophy for explaining the world. Then came the first crisis to shake that philosophy.
The first to demand proof in geometry.
According to tradition, Thales showed by logic that an angle inscribed in a semicircle is a right angle, among other facts, and predicted a solar eclipse. Most of what we know about the early Greek mathematicians, however, comes from records written centuries later.
"All is number."
Pythagoras and his followers believed everything, from musical harmony to the motion of the heavens, could be explained by ratios of whole numbers. Number became more than a tool for calculation; it became the principle of the cosmos. The theorem that bears his name was already known in Babylon, but proving it in general is credited to Greek mathematics.
The diagonal of a square cannot be written as a ratio of whole numbers.
It was shown that the diagonal of a square with side 1, √2, cannot be expressed as a ratio of any two whole numbers. The worldview that "all is a ratio of whole numbers" collapsed, and a legend even arose that Hippasus drowned at sea for revealing the secret.
Achilles can never catch the tortoise?
Zeno raised the problem of infinity with paradoxes showing that infinitely divisible motion leads to contradiction. In the same period, three problems were posed for straightedge and compass alone: squaring the circle, doubling the cube and trisecting an angle. All three were proved impossible only in the 19th century.
After the shock of irrational numbers, Greek mathematics turned from numbers to geometry, working with lengths, areas and volumes. The standard form of mathematics, deriving theorems from axioms, was perfected in this era.
"Let no one ignorant of geometry enter."
Plato's school, said to have borne these words over its entrance, was the center of mathematical research in its day. Plato saw mathematics as training for reaching truths beyond the senses.
A theory of ratio that handles irrationals, and the forerunner of the integral.
Eudoxus built a theory of proportion that held even between incommensurable magnitudes, overcoming the crisis of irrationals geometrically. His method of exhaustion, filling a shape with ever smaller pieces to find its area, became the forerunner of integration 2,000 years later.
Hundreds of theorems from five postulates.
Euclid organized the geometry and number theory of his time into a system proved step by step from a few definitions, axioms and postulates. Its greatness lies less in original discoveries than in its structure, and it served as a textbook for over 2,000 years, said to be second only to the Bible in number of editions. The proof that there are infinitely many primes is in it.
The greatest mathematician of antiquity.
Using a 96-sided polygon, he showed that 3 10/71 < π < 3 1/7, and he found the area of a parabolic segment and the volume of a sphere. He was proudest of discovering that a sphere and the cylinder enclosing it have volumes in the ratio 2:3, and is said to have had it carved on his tomb. His Method, found beneath a prayer book in 1906, shows that he first "discovered" results by adding up infinitely thin slices.
Measuring the Earth with the shadow of a stick.
Eratosthenes, librarian of Alexandria, compared the angle of the noon shadow in two cities on the same day and calculated the Earth's circumference with remarkable accuracy. The "sieve of Eratosthenes," a way of filtering out primes, also carries his name.
Ellipse, parabola and hyperbola from a single cone.
Apollonius systematically showed that cutting a cone at different angles produces three curves, and gave them the names we still use. Some 1,800 years later, Kepler used this very theory when he discovered that planetary orbits are ellipses.
Greek mathematics leaned toward applications such as astronomy and surveying, while China developed its own mathematics of calculation. The scholarly tradition of Alexandria gradually faded.
Solving simultaneous equations by elimination.
Built up by generations of Chinese scholars, this problem collection took its present form around the 1st century. Its method of arranging counting rods in a grid to solve systems of linear equations is the same as today's Gaussian elimination, and it handled negative numbers naturally.
The first full-scale trigonometric table.
In the Almagest, Ptolemy tabulated the length of the chord for each central angle in half-degree steps. Built for calculating the positions of heavenly bodies, the table was the foundation of astronomy for the next 1,400 years.
Writing unknowns as symbols.
This collection of problems seeking whole-number and rational solutions to equations used abbreviated symbols for the unknown and its powers. Some 1,400 years later, Fermat would scribble his famous note in its margin.
The last mathematician of Alexandria.
A mathematician and philosopher, Hypatia taught and wrote commentaries on the works of Diophantus and Apollonius. In 415 she was murdered by a mob amid political and religious strife, and her death came to symbolize the decline of ancient Alexandrian learning.
After the fall of the Western Roman Empire, the center of mathematics moved to India and the Islamic world. Zero and decimal place value were perfected, and algebra got its name.
π ≈ 355/113, a precision unbeaten for nearly a thousand years.
China's Zu Chongzhi placed π between 3.1415926 and 3.1415927 and gave the remarkably accurate fraction 355/113. His precision remained the world record for about 900 years.
The half-chord, origin of the sine.
India's Aryabhata built a table of sines using the half-chord instead of the Greek chord. As the word passed into Arabic and then Latin, it became today's "sine."
Treating zero as a number.
Brahmagupta treated zero not as a placeholder but as a number in its own right, and set out rules for adding, subtracting and multiplying with zero and negative numbers. The decimal place-value system with zero, perfected in India, traveled through the Islamic world to become the global standard.
The origin of the words "algebra" and "algorithm."
Al-Khwarizmi, a scholar at Baghdad's House of Wisdom, explained systematically how to set up and solve equations. "Algebra" comes from al-jabr (restoration) in his book's title, and "algorithm" from his name. Islamic scholars translated and preserved Greek and Indian texts, becoming the bridge between the two.
A poet and mathematician solves equations with conic sections.
Better known as the poet of the Rubaiyat, Khayyam classified cubic equations and systematized a geometric method of solving them by intersecting two conic sections. He also attempted to prove Euclid's parallel postulate.
As Arabic texts were translated into Latin, Hindu-Arabic numerals and algebra entered Europe. In the same period, China and southern India produced remarkable results of their own.
Spreading Hindu-Arabic numerals through Europe.
Trained as a merchant in North Africa, Fibonacci showed how much easier the new numerals, zero included, made calculation compared with Roman numerals. The sequence from his rabbit-breeding problem still bears his name.
"Pascal's triangle" and higher-degree equations.
Chinese mathematics of the 13th and 14th centuries solved higher-degree equations numerically (by what we now call Horner's method) and worked with the triangle of binomial coefficients. Zhu Shijie handled systems of higher-degree equations in as many as four unknowns, and the triangle had already appeared in the earlier work of Yang Hui.
Graphing change, and proving the harmonic series diverges.
France's Nicole Oresme drew changing quantities such as speed as horizontal and vertical lengths, anticipating the idea of the coordinate graph. He also proved that 1 + 1/2 + 1/3 + … grows without bound. But the Black Death and the Hundred Years' War cut European research short once more.
Sine and cosine series 250 years before Newton.
Madhava of Kerala in southern India and his students expanded sine, cosine and arctangent as infinite series and used them to calculate π. Their work long remained unknown in Europe.
Sixteen decimal places.
At the Samarkand observatory, al-Kashi used a polygon with 3×2²⁸ sides to calculate π correctly to 16 decimal places. The record stood for about 170 years.
Modern mathematics began when solutions to cubic and quartic equations were found in Renaissance Italy. Imaginary numbers made their first appearance, and mathematics gained a new language of letter symbols.
General solutions to cubic and quartic equations.
The solution to the cubic, first found by del Ferro and rediscovered by Tartaglia, who kept it secret, was published by Cardano together with his student Ferrari's solution to the quartic. The publication broke a promise and set off a bitter dispute, but the book is counted as the starting point of modern mathematics.
Calculating with square roots of negatives.
Using the cubic formula, square roots of negative numbers sometimes appeared midway even when the actual answer was real. Bombelli worked out rules for calculating with these "impossible numbers" and showed they could lead to correct real answers.
Decimal fractions for everyday calculation.
In a short book, Simon Stevin of Flanders explained decimal notation and how to calculate with it, and argued that currency and measures should be decimal too.
Vowels for unknowns, consonants for knowns.
France's François Viète wrote both unknowns and known quantities as letters, making it possible to write general equations rather than particular ones. It was the moment algebra turned from solving individual problems into a general theory.
Analytic geometry turned curves into equations, and on that foundation calculus was born to handle tangents and areas. Mathematics became the language of physics.
Tables that turn multiplication into addition.
Napier's logarithms greatly reduced the time astronomers spent calculating. Laplace later said they had doubled the life of the astronomer. Log tables and slide rules remained in use for more than 350 years, until the electronic calculator.
Curves as equations, equations as curves.
Descartes, in La Géométrie, an appendix to his Discourse on Method, and Fermat, independently and at almost the same time, used coordinates to turn geometric problems into algebraic ones. Algebra and geometry, separate since the Greeks, became one.
"This margin is too narrow to contain the proof."
In the margin of Diophantus's Arithmetica, Fermat claimed that xⁿ + yⁿ = zⁿ has no solutions in whole numbers when n is 3 or more, and added this remark. The problem would torment mathematicians for 358 years.
If a game is interrupted, how should the stakes be divided?
In letters exchanged while solving the problem of fairly dividing the stakes of an unfinished game, Pascal and Fermat laid down the basic principles of probability. Three years later Huygens wrote the first book on the subject.
Calculus invented while the plague closed the university.
While Cambridge was shut by the plague and he stayed in his home village, Newton developed a method for rates of change (fluxions) and realized that differentiation and integration are inverse operations. He did not publish for many years.
dx and ∫, the calculus notation still in use.
Leibniz developed calculus independently and published first. His superior notation spread quickly among mathematicians on the Continent. The priority dispute with Newton's camp that followed left British mathematics cut off from the Continent for nearly a century.
Mathematics grew explosively on the flood of results from calculus. At the center of the era stood Euler, who wrote more papers than any mathematician in history. The foundations of infinitesimals, however, remained murky.
Jacob Bernoulli: the more you throw, the closer to the probability.
In his posthumously published Ars Conjectandi, Bernoulli proved that as the number of trials grows, the observed frequency approaches the true probability. Twenty years later de Moivre showed that the binomial distribution approaches a bell-shaped curve, the normal distribution.
"The ghosts of departed quantities."
The philosopher Bishop Berkeley sharply criticized calculus for relying on infinitesimals that were somehow zero and not zero at once. Calculus worked, but no one could rigorously explain why.
Can you cross all seven bridges exactly once?
Euler ignored the shape of the land and the distances, keeping only what was connected to what, and proved it impossible. It was the starting point of graph theory and the seed of topology, which looks at structure of connections rather than shape.
Analysis built on functions, and e^(ix) = cos x + i sin x.
Euler reorganized analysis around the function and linked exponential and trigonometric functions through complex numbers. Notations such as e, i, π, f(x) and Σ became established through his work. He kept working even after losing his sight late in life.
Every polynomial equation has a complex solution.
Gauss presented a proof of this theorem in his doctoral dissertation and gave several more proofs over his lifetime. In the same era, revolutionary France created the metric system and founded the École Polytechnique, which became a center of mathematics led by Lagrange, Laplace and Monge.
Work began to rebuild the foundations of calculus rigorously, and Euclidean geometry, considered self-evident for 2,000 years, turned out not to be the only geometry. The theory of equations gave rise to a new concept: the group.
The Prince of Mathematics systematizes number theory.
At 24, Gauss organized number theory, including modular arithmetic, into a single system. At 19 he had proved that a regular 17-sided polygon can be constructed with straightedge and compass. He made major contributions to differential geometry, statistics and astronomy too, but he would not publish anything less than perfect, and often remarked of others' discoveries that he had known them already.
Limits, continuity and convergence made rigorous.
Cauchy tried to define the basic concepts of calculus using limits instead of infinitesimals. The work was completed by Weierstrass's ε-δ definitions and became the form of today's analysis textbooks.
No formula for equations of degree five or higher.
The young Norwegian mathematician Niels Henrik Abel proved that the general fifth-degree equation cannot be solved using only arithmetic and roots. He died in poverty and illness at 26.
A world with many parallel lines is free of contradiction too.
Lobachevsky and Bolyai each showed independently that changing Euclid's parallel postulate still yields a consistent geometry. Gauss had reached similar conclusions but never published them. The belief that axioms are self-evident truths was broken.
Group theory, written down the night before a duel.
The night before he died in a duel at 20, Galois summarized his research in a letter. His theory, which analyzed the symmetries among an equation's solutions as a "group" to decide which equations can be solved by formula, was properly understood only decades after his death.
Numbers that abandon commutative multiplication.
Hamilton is said to have carved the formula that came to him on a walk, i² = j² = k² = ijk = −1, into the stone of a Dublin bridge. A number system where ab can differ from ba broke assumptions about what numbers must be, and today it is used to calculate rotations in 3D graphics and robotics.
Logic as calculation.
George Boole created a system for handling logic, "and," "or," "not," like algebraic equations. Ninety years later Shannon applied it to electrical circuits, and it became the foundation of every digital computer.
Mathematics turned from the study of numbers and shapes into the study of structures. Space, infinity and symmetry were redefined in a new language.
The geometry of curved space.
In his habilitation lecture, Riemann presented a geometry of general spaces whose curvature can vary from point to point. Sixty years later, Einstein's general theory of relativity was written in exactly this language.
A hidden order in the distribution of primes.
In his only paper on number theory, Riemann connected the distribution of primes to the zeros of the zeta function and conjectured that those zeros all lie on a single line. It remains the most important unsolved problem in mathematics.
Defining the real numbers, and classifying geometries.
Dedekind defined the real numbers rigorously with "cuts" that divide the rationals in two, and Weierstrass revealed the limits of intuition with a continuous function that is differentiable nowhere. The same year Klein announced the Erlangen program, classifying geometries by the properties that stay unchanged under given transformations.
Some infinities are bigger than others.
Cantor proved that although the natural numbers and the real numbers are both infinite, there are "more" reals. In 1891 he showed it again, more simply, with his diagonal argument. Amid fierce opposition from some mathematicians, the set theory he created became the language of all 20th-century mathematics.
Properties that survive stretching and bending.
Poincaré laid the foundations of topology, which studies the properties that remain when a shape is continuously deformed without tearing. A question he raised along the way survived for more than a century as the Poincaré conjecture.
Attempts to build all of mathematics completely on set theory and logic continued, but paradoxes and the incompleteness theorems drew limits around that dream. Along the way probability was axiomatized and the theory of computability gave birth to computer science.
A map for 20th-century mathematics.
At the International Congress of Mathematicians in Paris, Hilbert presented 23 problems for the new century. Including the continuum hypothesis and the Riemann hypothesis, the list guided mathematical research for a hundred years.
The set of all sets that do not contain themselves.
Russell discovered that such a set leads to contradiction and told Frege in a letter just as Frege was finishing his book. Set theory, thought to be the foundation of mathematics, had cracked. Responses such as axiomatic set theory and Principia Mathematica followed.
A new integral for many more functions.
Lebesgue redefined the integral by dividing up the range of a function's values instead of its domain. His concept of measure became the foundation of modern analysis and probability.
Formulas sent to Cambridge by a self-taught genius.
Seeing the astonishing formulas in a letter from Ramanujan, a clerk in India, Hardy invited him to Cambridge. Without formal training, Ramanujan left thousands of formulas found by intuition before dying at 32.
Studying structure itself, not particular numbers.
Emmy Noether established the theory of rings and ideals, giving modern abstract algebra its shape. Earlier, Noether's theorem, linking symmetries to conservation laws, had left a deep mark on physics. For years she was denied a regular professorship because she was a woman.
There are true statements that cannot be proved.
Gödel showed that any consistent formal system that includes arithmetic contains true statements it cannot prove, and cannot prove its own consistency. He drew a limit in principle around Hilbert's plan to found all of mathematics on a complete formal system.
Building probability on measure theory.
Kolmogorov axiomatized probability in the language of measure, making it a rigorous mathematical field. Around the same time Fisher, Neyman, Pearson and others established the methods of modern statistics.
What does it mean to be computable?
Turing and Church each answered Hilbert's decision problem by showing that some problems cannot be solved by any mechanical procedure. Turing's imaginary machine became the starting point of computer science.
Relations between objects, more than the objects themselves.
Eilenberg and Mac Lane introduced category theory, which ties together the structures of different fields of mathematics in a single language of objects and morphisms. At the same time the Bourbaki group was rewriting all of mathematics around structures.
Measuring information in bits.
Shannon defined the quantity of information as entropy and worked out mathematically the limit of how much can be sent without error over a noisy channel. It is the theoretical foundation of digital communication and data compression.
Mathematics became extremely specialized, and centuries-old problems were solved one by one. As computers began to be used in proofs, the question "what is a proof?" came back.
Neither provable nor disprovable.
The continuum hypothesis, raised by Cantor and first on Hilbert's list, turned out to be impossible to prove or disprove from the standard axioms of set theory. Gödel showed one half, and Paul Cohen the other.
A grand unified theory of mathematics.
In a letter to André Weil, Robert Langlands proposed a vast web of conjectures predicting deep correspondences between number theory, representation theory and analysis. Many of mathematics' great results since have come to be understood as parts of this program.
How long is a coastline?
Benoit Mandelbrot gave the name "fractal" to shapes that repeat similar patterns at every scale. The Mandelbrot set, drawn with computer graphics, became one of the most beautiful images mathematics has ever shown the public.
The first major proof with a computer's help.
Appel and Haken proved that any map can be colored with four colors so that no neighboring countries share a color, using a computer to check a vast number of cases. A debate followed over whether a proof no human could check by hand should count as a proof.
The difficulty of factoring keeps secrets safe.
Rivest, Shamir and Adleman published a public-key cryptosystem based on the difficulty of factoring large numbers. Number theory, long called the most useless branch of mathematics, became the basis of internet security.
Solved after 358 years.
After seven years of working alone, Andrew Wiles announced a proof in 1993, but a gap was soon found. He and his former student Richard Taylor worked on it for more than a year to close the gap, and the proof was published in 1995. It relied on part of the Langlands program linking elliptic curves and modular forms.
A list of problems for the new century was announced, and major conjectures fell one after another. Formalization, in which computers check proofs, spread, and AI began solving math problems.
A million dollars per problem.
The Clay Mathematics Institute offered a million dollars each for seven hard problems, including the Riemann hypothesis and P versus NP. The announcement deliberately echoed Hilbert's 23 problems a century earlier.
Papers posted online, and a Fields Medal declined.
Using Richard Hamilton's Ricci flow, Grigori Perelman proved the century-old Poincaré conjecture in papers he posted to a preprint server rather than a journal. After his proof was verified, he turned down both the Fields Medal and the Millennium Prize. It is still the only Millennium Problem to be solved.
A proof assistant checks the entire proof.
Using the proof assistant Coq, Georges Gonthier rewrote the whole proof of the four color theorem in a form a computer could check step by step. It answered thirty years of doubt about computer proofs.
A surprise proof from an unknown mathematician.
Yitang Zhang, working as a university lecturer, stunned mathematicians by proving that there are infinitely many pairs of primes less than 70 million apart. An online collaboration (Polymath) led by Terence Tao and others then brought the gap down to 246.
The geometry and dynamics of surfaces.
For her work on the geometry and dynamics of Riemann surfaces, Iranian-born Maryam Mirzakhani became the first woman ever to win the Fields Medal. She died three years later, at 40.
The densest way to stack balls.
Maryna Viazovska gave a strikingly short proof of the densest way to pack spheres in eight-dimensional space, and soon extended it with colleagues to 24 dimensions. She received the Fields Medal in 2022.
Verifying frontier mathematics in Lean.
Peter Scholze publicly asked for a computer check of a key theorem whose proof even he was not fully sure of, and the Lean community completed it in a year and a half. It showed that formalization could serve current research, not just confirm old theorems.
AI reaches silver-medal level at the Math Olympiad.
Google DeepMind's AlphaProof and AlphaGeometry 2 solved International Mathematical Olympiad problems by formalizing them in Lean, scoring at silver-medal level. The following year, several AI models reached gold-medal level with solutions written in natural language.