History of mathematics

• 1. Introduction

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  From 3000 BC, the Mesopotamian states of Sumer, Akkad and Assyria began using arithmetic, algebra and geometry to handle taxation, commerce and the recording of time. The history of mathematics traces how those scattered beginnings became a single global language. Yet for most of that span, written examples of new mathematical developments have surfaced in only a few locales. Clay tablets in Iraq, papyrus in Egypt, carved bone near the Nile, knotted yarn left by Neanderthals. Who first proved a theorem instead of merely observing a pattern? Why did one civilization give zero a symbol while another could only infer it? How did a numeral system from India come to displace every older system on Earth? And why did long bursts of discovery so often give way to centuries of stagnation? The answers run through Babylon and Alexandria, through Song-dynasty China and the Kerala coast, and into a list of 23 problems read aloud in the year 1900.

• 2. Before The First Number

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  Modern studies of animal cognition show that the concepts of number, pattern, magnitude and form are not unique to humans. Some languages still preserve a distinction between one, two, and many, with no words for numbers larger than two, which supports the idea that the number concept evolved gradually. The use of yarn by Neanderthals roughly 40,000 years ago at Abri du Maras in the south of France suggests they grasped basic mathematical concepts. The Ishango bone, found near the headwaters of the Nile in northeastern Congo, may be more than 20,000 years old. It carries a series of marks carved in three columns running the length of the bone. Some read those marks as the earliest known demonstration of sequences of prime numbers, others as a six-month lunar calendar. Peter Rudman argues that prime numbers probably were not understood until about 500 BC, since the concept of division he dates only to after 10,000 BC. He notes that no one has explained why a simple tally would happen to show multiples of two, primes between 10 and 20, and near-multiples of 10. Claims have been made that megalithic monuments in England and Scotland from the 3rd millennium BC encode circles, ellipses and Pythagorean triples, but every such early claim is disputed. The currently oldest undisputed mathematical documents come from Babylonian and dynastic Egyptian sources.

• 3. Clay, Sixty, And The Missing Point

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  More than 400 clay tablets unearthed since the 1850s are the source of nearly everything known about Babylonian mathematics. Inscribed while the clay was moist and then baked hard, some of them appear to be graded homework. The earliest written mathematics dates back to the ancient Sumerians, who from 3000 BC built a complex metrology for grain allotments, workers, weights of silver and liquids. From around 2500 BC the Sumerians wrote multiplication tables and worked through geometrical exercises and division problems. The Babylonian system was sexagesimal, built on base 60, the reason we still count 60 seconds in a minute, 60 minutes in an hour, and 360 degrees in a circle. One explanation is that 60 divides evenly by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30, making it easy to calculate by hand. Unlike the Egyptians, Greeks and Romans, the Babylonians used a place-value system, where a digit's column set its size, much as in decimals. That power let them treat fractions exactly like whole numbers, and the tablet YBC 7289 gives an approximation accurate to five decimal places. The Babylonians lacked any equivalent of the decimal point, so a symbol's place value often had to be guessed from context. By the Seleucid period they had a zero sign for empty interior positions, but never used it in terminal positions, so they came close to a true place-value system without reaching it. Tablets from the Old Babylonian period also carry the earliest known statement of the Pythagorean theorem, yet show no awareness of the difference between exact and approximate solutions, and no explicit demand for proof.

• 4. Papyrus And Word Problems

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  The Rhind papyrus, sometimes called the Ahmes papyrus after its author, is the most extensive Egyptian mathematical text, dated to about 1650 BC but likely copied from a document of around 2000 to 1800 BC. It served as an instruction manual in arithmetic and geometry, with area formulas, methods for multiplication and division, and work with unit fractions. Within it lie traces of composite and prime numbers, of arithmetic, geometric and harmonic means, and an early sense of the Sieve of Eratosthenes and of perfect numbers, illustrated by 6. The Moscow papyrus, dated to about 1890 BC, gathers what we would now call word problems, apparently meant as entertainment. One of them matters especially because it gives a method for finding the volume of a frustum, a truncated pyramid. The Berlin Papyrus 6619, from about 1800 BC, shows that ancient Egyptians could solve a second-order algebraic equation. Archaeological evidence suggests the Egyptian counting system had origins in Sub-Saharan Africa, and the megalithic structures at Nabta Playa in Upper Egypt aligned calendar arrangements with the heliacal rising of Sirius to time the annual Nile flood.

• 5. The Birth Of Proof

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  Thales of Miletus, born about 624 BC, used geometry to calculate the height of pyramids and the distance of ships from shore. All surviving records of pre-Greek mathematics rely on inductive reasoning, rules of thumb drawn from repeated observation. The Greeks broke with that. They used logic to derive conclusions from definitions and axioms, then used rigor to prove them. Thales is credited with the first deductive reasoning applied to geometry through four corollaries to his theorem, which has earned him the title of first true mathematician. Pythagoras of Samos, born about 582 BC, founded a school whose doctrine held that mathematics ruled the universe and whose motto was All is number. The Pythagoreans coined the term mathematics, from the Greek mathema meaning subject of instruction, and are credited with the first proof of the Pythagorean theorem and a proof that irrational numbers exist. Plato, who lived from 428 or 427 BC to 348 or 347 BC, mattered less for technical results than for guidance. His Academy in Athens became the mathematical center of the world in the 4th century BC, producing figures such as Eudoxus of Cnidus, born about 390 BC, whose method of exhaustion was a precursor of modern integration. Aristotle, who lived from 384 to about 322 BC, made no specific technical discovery but laid the foundations of logic. The line Plato clarified as breadthless length still sits near the start of the subject.

• 6. Alexandria's Golden Age

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  Euclid, working around 300 BC at the Musaeum of Alexandria, wrote the Elements, widely considered the most successful and influential textbook of all time. It introduced rigor through the axiomatic method and set the still-used format of definition, axiom, theorem and proof. Most of its contents were already known, but Euclid arranged them into one coherent logical framework, and it remained known to educated people in the West into the middle of the 20th century. Among its proofs are that the square root of two is irrational and that there are infinitely many prime numbers. Archimedes of Syracuse, who lived from about 287 to 212 BC and is widely considered the greatest mathematician of antiquity, used the method of exhaustion to find the area under a parabola through the summation of an infinite series. He obtained the most accurate value of pi then known, studied the spiral that bears his name, and prized above all his proof that a sphere's surface area and volume are two-thirds those of the circumscribing cylinder. Apollonius of Perga, who lived from about 262 to 190 BC, showed that all three conic sections come from varying the angle of a plane cutting a double cone, and he coined the terms parabola, ellipse and hyperbola. His treatment of curves anticipated, in some ways, the analytical geometry Descartes would develop some 1800 years later. Eratosthenes of Cyrene, who lived from about 276 to 194 BC, devised his sieve for finding primes around the same time. After this 3rd-century peak, pure mathematics entered a relative decline, while applied work flourished. Hipparchus of Nicaea compiled the first known trigonometric table and gave us the systematic 360-degree circle. The Almagest of Ptolemy, who lived from about AD 90 to 168, carried trigonometric tables that astronomers would use for the next thousand years.

• 7. Twilight In Athens

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  Diophantus made his advances in algebra during the period between 250 and 350 AD, sometimes called the Silver Age of Greek mathematics. His Arithmetica gathered 150 algebraic problems on exact solutions to determinate and indeterminate equations, and it was the first instance of algebraic symbolism and syncopation. Pierre de Fermat arrived at his famous Last Theorem after trying to generalize a problem he read in the Arithmetica, that of dividing a square into two squares. Pappus of Alexandria, in the 4th century AD, is considered the last major innovator in Greek mathematics, known for his hexagon theorem and centroid theorem; after him the work was mostly commentary. Hypatia of Alexandria, who lived from AD 350 to 415, is the first recorded woman mathematician and wrote many works on applied mathematics. Because of a political dispute, the Christian community in Alexandria had her stripped publicly and executed, a death sometimes taken as the end of Alexandrian Greek mathematics. The closure of the neo-Platonic Academy of Athens by the emperor Justinian in 529 AD traditionally marks the end of the Greek era, though the tradition continued unbroken in the Byzantine empire with figures such as Anthemius of Tralles and Isidore of Miletus, the architects of the Hagia Sophia.

• 8. The Numbers That Conquered The World

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  Around 500 AD, Aryabhata wrote the Aryabhatiya, a slim volume in verse, and it is there that the decimal place-value system first appears; centuries later the mathematician Abu Rayhan Biruni called the work a mix of common pebbles and costly crystals. In the 7th century, Brahmagupta, in his Brahma-sphuta-siddhanta, for the first time lucidly explained the use of zero as both placeholder and decimal digit. It was from a translation of that Indian text, around 770, that Islamic mathematicians met the Hindu-Arabic numeral system, which they adapted as Arabic numerals. In the 9th century, the Persian mathematician Muhammad ibn Musa al-Khwarizmi wrote a book on these numerals and another on solving equations; the word algorithm comes from the Latinization of his name, and algebra from the title of his work on calculation by completion and balancing. The Maya civilization of Mexico and Central America, geographically isolated and entirely independent, used a base-twenty vigesimal system and, unlike many contemporary cultures, gave zero a standard symbol. Far to the east, Chinese mathematics developed independently too, with a decimal positional system of rod numerals in use several centuries before the common era. The Nine Chapters on the Mathematical Art, its full title appearing by AD 179, collected 246 word problems and created a proof for the Pythagorean theorem and a formula for Gaussian elimination. In the 5th century AD, Zu Chongzhi computed pi to seven decimal places, between 3.1415926 and 3.1415927, a value unmatched for almost a thousand years. Madhava of Sangamagrama, founder of the Kerala School, used the first 21 terms of a transformed series to compute pi as 3.14159265359.

  Leonardo of Pisa, now known as Fibonacci, learned the Hindu-Arabic numerals on a trip to what is now Bejaia, Algeria, while Europe still used Roman numerals, and his Liber Abaci of 1202 began a long campaign to popularize them. During the Renaissance, Luca Pacioli's Summa de Arithmetica, printed in Venice in 1494, introduced symbols for plus and minus for the first time in a printed book and was the first book printed in Italy to contain algebra. In Italy, Scipione del Ferro and Niccolo Fontana Tartaglia found solutions for cubic equations, which Gerolamo Cardano published in his 1545 Ars Magna alongside his student Lodovico Ferrari's solution for quartics. The 17th century brought an unprecedented surge: Johannes Kepler formulated mathematical laws of planetary motion, aided by the logarithms of John Napier and Jost Burgi, and Rene Descartes, who lived from 1596 to 1650, developed the analytic geometry that let orbits be plotted in Cartesian coordinates. Isaac Newton discovered the laws of physics explaining Kepler's laws and brought together calculus, while Gottfried Wilhelm Leibniz independently developed calculus and much of its still-used notation. In the 19th century, mathematics grew increasingly abstract, with Nikolai Lobachevsky and Janos Bolyai independently defining hyperbolic geometry and Bernhard Riemann later developing the elliptic and Riemannian geometry that underpinned general relativity. In a 1900 speech to the International Congress of Mathematicians, David Hilbert set out 23 unsolved problems, of which 10 have since been solved and 2 remain open. Then in 1931, Kurt Godel found that the natural numbers with both addition and multiplication form an incomplete system, proving that within any system containing Peano arithmetic, truth necessarily outruns proof. In 2000, the Clay Mathematics Institute announced seven Millennium Prize Problems, and in 2003 Grigori Perelman solved the Poincare conjecture, then declined the award.