A history of mathematics.

OpeningWhat this is.

A condensed history of the longest-running argument in human thought.

Mathematics is older than writing. The first arithmetic was probably tally marks on bones; the first geometry was the surveying of post-flood Nile fields. By 2000 BCE the Babylonians could solve quadratic equations and compute square roots accurately to four decimal places. By 300 BCE Euclid had organised geometry into the form it would keep for two thousand years.

The thread that runs through is proof — the discipline of deriving consequences from premises by logically necessary steps. Mathematics is the only field of human knowledge in which a result, once proved, stays proved. Pythagoras's theorem was true in 500 BCE, is true now, will be true in 4000 CE.

This deck moves chronologically through the major figures and turning points. It tries to convey the texture of the discipline at each period — what mathematicians cared about, what they could prove, what they could not.

Chapter IThe Ishango bone.

Discovered in 1960 by Belgian geologist Jean de Heinzelin de Braucourt at Ishango on the upper Nile, in what is now the Democratic Republic of the Congo. A baboon fibula, about 10 cm long, with three columns of notches arranged in groups. Approximately 20,000 years old.

The interpretation is contested. The middle column shows the prime numbers between 10 and 20 (11, 13, 17, 19); the right column doubles a sequence (3, 6, 4, 8, 10, 5, 5, 7); the left column shows ten plus or minus 1. De Heinzelin argued for a tally of lunar phases. Others propose an arithmetical reckoner.

The conservative reading: the bone shows that someone, twenty thousand years ago, took the trouble to record numerical patterns. Whether these patterns count as mathematics in any modern sense is a definitional question. They are the earliest physical evidence of mathematical thought.

Chapter IIBabylon and Egypt.

The first cultures with written mathematics. Babylonian mathematics (c. 1900–500 BCE) was written in cuneiform on clay tablets and used a sexagesimal (base-60) place-value system that survives in our minutes, seconds, and degrees.

The Babylonians could solve quadratic equations and certain cubics. The tablet Plimpton 322 (c. 1800 BCE) lists fifteen Pythagorean triples — a, b, c with a² + b² = c² — a thousand years before Pythagoras. The tablet YBC 7289 records √2 to six decimal places (1.414213…), accurate to the precision available.

Egyptian mathematics was practical and arithmetical. The Rhind Papyrus (c. 1650 BCE, copied from c. 2000 BCE) contains 84 problems involving fractions, areas, and volumes. The Egyptians used unit fractions exclusively (1/n) — every fraction had to be written as a sum of distinct unit fractions, an inefficient convention that nonetheless produced sophisticated computational methods.

Chapter IIIPythagoras and the Greeks.

The decisive innovation. Greek mathematics, beginning in the sixth century BCE with Thales and Pythagoras, made proof the central activity. Earlier traditions had answers; the Greeks demanded justifications.

Pythagoras (c. 570–495 BCE) and the Pythagorean school discovered (or transmitted from earlier sources) the theorem that bears his name and the irrationality of √2. The latter was a genuine philosophical scandal — the Pythagorean worldview held that all magnitudes were ratios of integers, and the diagonal of a unit square turned out not to be.

Eudoxus of Cnidus (c. 408–355 BCE) developed the method of exhaustion — a precursor of integration — and a theory of proportions adequate to handle irrational ratios. Plato's Academy made mathematics central to philosophical education; the apocryphal motto over its door, μηδεὶς ἀγεωμέτρητος εἰσίτω, "Let no one ignorant of geometry enter."

Chapter IVEuclid.

The most influential textbook ever written. Euclid of Alexandria's Elements (c. 300 BCE) collected the Greek geometric tradition into thirteen books, organised around five postulates from which the rest is derived.

The achievement is more pedagogical than original — most of the contents predate Euclid — but the organisation is decisive. Elements demonstrated that a complex body of knowledge could be derived from a small number of explicit assumptions by chains of demonstrative reasoning. The model has dominated mathematical exposition ever since.

The fifth postulate — given a line and a point not on it, exactly one parallel exists through the point — was felt by mathematicians for two millennia to be less self-evident than the others, and many tried to derive it. The eventual conclusion (1820s onward) that it could be denied without contradiction founded non-Euclidean geometry.

Elements went through more than 1,000 editions. Until the 20th century it was the standard geometry textbook in many countries.

Chapter VArchimedes.

The greatest of the ancient mathematicians and arguably the greatest scientist before Newton. Archimedes of Syracuse (c. 287–212 BCE) advanced essentially every branch of the mathematics of his day.

He found accurate bounds on π by inscribing and circumscribing polygons in a circle (3 + 10/71 < π < 3 + 1/7). He computed the volumes and surface areas of spheres, cones, and paraboloids, including the result that the volume of a sphere is two-thirds that of its circumscribing cylinder — a theorem he asked to be engraved on his tomb.

He invented the science of statics — the law of the lever, the principle of buoyancy ("Eureka!"). He proposed a heliocentric system in his lost work The Sand Reckoner and defended it against Aristarchus.

The Archimedes Palimpsest — a 10th-century parchment containing copies of his works that was overwritten with prayers in the 13th century and rediscovered in 1906 — contains The Method, in which Archimedes uses infinitesimal arguments two thousand years before calculus. He was killed by a Roman soldier during the sack of Syracuse, allegedly while drawing geometric figures in the sand.

Chapter VIDiophantus.

The father of algebra in the European tradition. Diophantus of Alexandria (c. 200–284 CE) wrote Arithmetica, thirteen books on the solution of polynomial equations in integers and rationals.

The book introduced symbolic notation for unknowns and powers, two thousand years before such notation became standard. Diophantus solved hundreds of problems involving what we now call Diophantine equations — polynomial equations sought in integer or rational solutions. His methods were ad hoc; his examples were systematic.

His most lasting influence came indirectly. Pierre de Fermat, reading Bachet's 1621 Latin translation of Arithmetica, scribbled in the margin next to Diophantus's discussion of Pythagorean triples his famous note that no analogous triples exist for higher powers. The proof, he claimed, was too long for the margin. Fermat's Last Theorem stayed open for 358 years.

Chapter VIIIndian mathematics.

The discipline that gave us our number system. The mathematicians of classical India worked at a level fully comparable to their Greek and Islamic contemporaries, and at several points ahead.

Aryabhata (476–550) computed the value of π accurate to four decimal places, gave the formula for the sum of squares and cubes, and proposed (in passing) that the Earth rotates on its axis daily. Brahmagupta (598–668) wrote the first systematic treatment of zero as a number and gave rules for arithmetic with positive and negative quantities. He also gave a general solution to the Pell equation x² − Ny² = 1.

Madhava of Sangamagrama (c. 1340–1425) and the Kerala school discovered infinite series for π, sine, and cosine — including what would later be called the Madhava-Leibniz series for π — three centuries before Newton and Leibniz. The discovery did not propagate to Europe; the European calculus was independent.

Chapter VIIIThe decimal place-value system.

The single most consequential mathematical invention. The decimal place-value system with zero as a placeholder developed in India between the 4th and 7th centuries. Brahmagupta's 628 CE Brāhmasphuṭasiddhānta contains the earliest known explicit treatment.

The system spread to the Islamic world by the 9th century. Al-Khwarizmi's On the Calculation with Hindu Numerals (c. 825) was the conduit. From Arabic translations the system reached Europe through Fibonacci's Liber Abaci (1202) and slowly displaced Roman numerals over the following four centuries.

The reason the system matters: arithmetic with Roman numerals is essentially impossible at scale, while the place-value system reduces multiplication and division to mechanical procedures any literate person can execute. The development of European commerce, science, and engineering after 1500 depended on having a workable arithmetic. The Hindu-Arabic numerals provided it.

Chapter IXThe Islamic golden age.

From the 9th to the 14th century, the centre of mathematical activity moved to the Islamic world. The translation movement at the House of Wisdom in Baghdad preserved the Greek mathematical inheritance and the work that followed extended it substantially.

Muḥammad ibn Mūsā al-Khwārizmī (c. 780–850) wrote The Compendious Book on Calculation by Completion and Balancing, founding algebra as a separate discipline. The Latin algebra derives from the Arabic al-jabr; the word algorithm derives from his name.

Abū Bakr al-Karajī (c. 953–1029) treated polynomials systematically and proved binomial expansions by induction. Omar Khayyam (1048–1131) gave a geometric solution to cubic equations using intersecting conic sections — a substantial advance not exceeded until the 16th-century Italian algebraists. He also produced critical commentary on Euclid's parallel postulate that anticipates non-Euclidean geometry by seven centuries.

The transmission of this tradition into medieval Europe was decisive for the Renaissance.

Chapter XFibonacci.

The man who carried Hindu-Arabic mathematics into Europe. Leonardo of Pisa (c. 1170–1250), known as Fibonacci, travelled with his merchant father in North Africa, learned Arabic mathematics, and on his return wrote Liber Abaci (1202).

The book introduced Hindu-Arabic numerals to European commerce, taught the algorithms of arithmetic with them, and presented a wide range of practical problems — currency conversion, weights and measures, profit calculation. It also, almost incidentally, introduced the rabbit-population problem from which the Fibonacci sequence takes its name.

The sequence — 0, 1, 1, 2, 3, 5, 8, 13, 21, … — has been studied ever since for its remarkable properties. The ratios of consecutive Fibonacci numbers converge to the golden ratio φ. The sequence appears in plant phyllotaxis, in the analysis of the Euclidean algorithm, and in countless places nobody expected.

Fibonacci was the European mathematician of his century. Italian mathematics did not advance significantly again until the 16th century.

Chapter XIThe cubic equation.

The first major mathematical result not anticipated by the Greeks. The general solution of the cubic equation x³ + px = q was found, around 1500, by Scipione del Ferro, professor at Bologna. He kept his solution secret to use against challengers in mathematical duels — the standard professional contest of the period.

Niccolò Fontana (Tartaglia) rediscovered the solution independently in 1535 and used it to win a public duel. Gerolamo Cardano obtained Tartaglia's method under oath of secrecy, broke the oath, and published the solution along with extensions of his own (and his student Lodovico Ferrari's solution to the quartic) in Ars Magna (1545). The book is the first clear advance in algebra after the Islamic mathematicians.

Cardano's solution required taking square roots of negative numbers as intermediate steps. Even when all three roots of the cubic were real, the algebraic manipulation passed through what would later be called complex numbers. Rafael Bombelli's Algebra (1572) accepted these quantities as legitimate; the foundations of complex analysis were laid.

Chapter XIIDescartes.

René Descartes's 1637 La Géométrie — published as an appendix to the philosophical Discourse on the Method — founded analytic geometry. The single insight: identify points in the plane with pairs of numbers, and curves with equations relating those numbers.

The fusion was decisive. Geometric problems became algebraic problems and conversely. Descartes's notation — x, y, z for unknowns; superscripts for exponents; the convention that a, b, c denote constants — is essentially the notation used today.

The implication: every conic section is described by a quadratic equation in two variables, every cubic curve by a cubic equation, and the algebraic complexity of a curve maps cleanly to the geometric complexity. The unification trained the next generation of mathematicians to think both algebraically and geometrically about the same objects.

Pierre de Fermat independently developed similar ideas around 1636 but did not publish them. The two were rivals throughout the period; Descartes took most of the credit, partly because he published.

Chapter XIIINewton vs Leibniz.

Calculus was invented twice, independently, in the 1660s and 1670s. Isaac Newton developed his "method of fluxions" between 1665 and 1667, while sheltering at his mother's farm during the plague closure of Cambridge. He did not publish for over twenty years.

Gottfried Wilhelm Leibniz, working in Paris in the 1670s, developed his own version with superior notation — the integral sign ∫ and the differential dx are his — and published in 1684 ("Nova Methodus pro Maximis et Minimis").

The priority dispute that followed was vicious. The Royal Society, with Newton as president, ruled in his favour in 1712. Modern scholarship treats both as independent discoverers; Newton with priority of discovery, Leibniz with the better notation that became standard.

The discovery itself transformed mathematics. Calculus made tractable problems — the path of a planet, the shape of a hanging chain, the volume of a solid of revolution — that had defeated the Greeks. Within a century it had grown into the dominant branch of mathematics. Principia Mathematica (1687) demonstrated what calculus could do for physics; the rest followed.

Chapter XIVEuler.

The most prolific mathematician of all time. Leonhard Euler (1707–1783), Swiss-born, working at Saint Petersburg and Berlin, contributed essentially to every branch of the mathematics of his day and founded several.

The list is barely possible to compress. Calculus of variations. Graph theory (Königsberg bridges, 1736). Analytic number theory (the zeta function, the prime-counting function). Topology (the V − E + F = 2 formula). Mechanics (the Euler equations of fluid flow). Modern notation: e, i, π, sin, cos, f(x), Σ — much of it is his standardisation. The identity e^iπ + 1 = 0 connects five fundamental constants in a single formula.

Euler went blind in his right eye at 31 and lost his left eye at 59. His productivity, by the testimony of contemporaries, increased after the latter event. His collected works Opera Omnia ran to 80 volumes in the original modern edition; the project remains incomplete.

Pierre-Simon Laplace told his students: "Read Euler, read Euler. He is the master of us all."

Chapter XVGauss.

Carl Friedrich Gauss (1777–1855) — the Princeps Mathematicorum — was, with Archimedes and Newton, generally counted among the three greatest mathematicians in history.

The childhood story (perhaps apocryphal): asked at school to sum the integers 1 to 100, Gauss saw immediately that the sum was 50 × 101 = 5050. By 19 he had constructed the regular 17-gon by compass and straightedge, the first new constructible regular polygon since the Greeks. His doctoral dissertation (1799) gave the first essentially correct proof of the fundamental theorem of algebra.

Disquisitiones Arithmeticae (1801) founded modern number theory. He invented the method of least squares (publishing in 1809, anticipated by Legendre in 1805 — priority dispute). He developed the theory of curvature for surfaces, anticipating the language of differential geometry that Riemann would systematise.

Gauss anticipated non-Euclidean geometry — he investigated geometries in which the parallel postulate fails — but did not publish, fearing the controversy. Bolyai and Lobachevsky published independently in 1832 and 1829.

Chapter XVICauchy.

The mathematician who put calculus on rigorous foundations. Augustin-Louis Cauchy (1789–1857), professor at the École Polytechnique in Paris, replaced the loose 18th-century calculus with ε-δ definitions and properly stated theorems.

His Cours d'Analyse (1821) and the lectures that followed gave precise definitions of limit, continuity, derivative, and integral. The concept of a Cauchy sequence — one whose terms become arbitrarily close — is named for him. He proved the Cauchy integral theorem and the Cauchy integral formula, founding complex analysis.

Cauchy was politically and personally difficult — a stiff Catholic royalist who alienated colleagues and rejected work he disliked. His refereeing rejected (or lost) the early manuscripts of Galois and Abel. Both died young (Galois in a duel at 20; Abel of tuberculosis at 26) without seeing their work published.

Cauchy nonetheless produced more papers than any mathematician except Euler — about 800 — and the rigour he imposed was decisive. After Cauchy, sloppy 18th-century arguments were no longer acceptable.

Chapter XVIIRiemann.

One of the great mathematicians of the 19th century, dead at 39 of tuberculosis. Bernhard Riemann (1826–1866), Gauss's last great student at Göttingen, transformed several fields in a short career.

His 1851 dissertation founded modern complex analysis through the concept of Riemann surfaces — multi-sheeted complex domains on which multi-valued functions become single-valued. His 1854 Habilitation lecture "On the Hypotheses which lie at the Foundations of Geometry" generalised Gauss's surface theory to arbitrary dimensions, founding the differential geometry that Einstein would use sixty years later for general relativity.

His 1859 paper "On the Number of Primes Less than a Given Magnitude" introduced the Riemann zeta function and stated the Riemann hypothesis — that all non-trivial zeros of ζ(s) lie on the critical line Re(s) = 1/2. The hypothesis is the most famous unsolved problem in mathematics, with deep implications for the distribution of primes. It is one of the seven Clay Millennium Prize Problems.

Chapter XVIIIHilbert and the 1900 problems.

At the Second International Congress of Mathematicians in Paris in August 1900, David Hilbert delivered an address listing 23 unsolved problems whose resolution he believed would shape mathematics in the coming century. He was largely correct.

The list ranged across analysis, geometry, number theory, foundations, and mathematical physics. The first was the continuum hypothesis. The second asked for a proof of the consistency of the axioms of arithmetic. The eighth was the Riemann hypothesis (and the Goldbach and twin-prime conjectures). The tenth asked for an algorithm to determine whether a Diophantine equation has integer solutions.

By 2026, most of the problems have been resolved; some negatively (Gödel's incompleteness theorems killed the second; Matiyasevich's 1970 work the tenth). The Riemann hypothesis remains open. The list defined a research programme for the 20th century with extraordinary success — Hilbert understood, in 1900, where the deep problems lay.

Chapter XIXCantor and the transfinite.

The mathematician who proved that infinity comes in different sizes. Georg Cantor (1845–1918), professor at Halle, founded set theory in the 1870s and 1880s.

His 1874 paper proved that the algebraic numbers are countable while the real numbers are not — there are strictly more reals than rationals, even though both sets are infinite. The 1891 diagonal argument gave a transparent proof of the same uncountability and became one of the standard techniques of mathematics.

Cantor introduced the hierarchy of cardinal numbers: ℵ₀ for countable sets, ℵ₁ for the next size, and so on. The continuum hypothesis — that the cardinality of the real line is ℵ₁ — was Cantor's central conjecture and remained open until proved independent of ZFC by Cohen in 1963.

Cantor's work was attacked viciously in his lifetime — Kronecker called transfinite numbers "nonsense" and blocked Cantor's career — and Cantor suffered repeated mental breakdowns. The opposition collapsed within twenty years of his death. Hilbert: "No one shall expel us from the paradise that Cantor has created."

Chapter XXNoether.

The greatest woman mathematician of her time and one of the great mathematicians of the 20th century. Emmy Noether (1882–1935), denied a paid position at Göttingen for years on the grounds that she was a woman, founded modern abstract algebra and proved one of the great theorems of mathematical physics.

Noether's theorem (1918) connects continuous symmetries to conserved quantities — translation symmetry implies conservation of momentum, time-translation symmetry implies conservation of energy, rotational symmetry implies conservation of angular momentum. The theorem is foundational to all of modern theoretical physics.

Her abstract algebra reorganised the field around rings, ideals, and modules. The structural approach she introduced, in Idealtheorie in Ringbereichen (1921) and through her teaching, became the dominant framework. The Bourbaki project would systematise her programme.

Forced from Göttingen by the 1933 Nazi laws, she emigrated to Bryn Mawr and died of complications from cancer surgery at 53. Einstein wrote to Pauli that she was the most significant creative mathematical genius produced since the higher education of women began.

Chapter XXIHardy and Ramanujan.

The most extraordinary mathematical correspondence in modern history. In 1913, G. H. Hardy, then 36, professor at Cambridge, received a letter from a clerk in the Madras Port Trust offering theorems on infinite series and number theory. The clerk was Srinivasa Ramanujan, 25, largely self-taught.

Hardy initially suspected fraud — some of the formulas he could prove, others he could not, and the rest seemed too unlikely to be genuine. After consultation with Littlewood, Hardy concluded that the formulas had to be the work of a mathematician of "the highest class." He brought Ramanujan to Cambridge.

The collaboration that followed produced major work in partition theory, modular forms, and the theory of mock theta functions. Ramanujan's notebooks — recording thousands of identities, many provable by methods unknown to him — continue to be mined by researchers a century later. The 2012 Ono-Bringmann theory of mock modular forms grew out of Ramanujan's last letters.

Ramanujan died in 1920, aged 32, of tuberculosis or amoebic dysentery. Hardy listed his discovery of Ramanujan as the single greatest "romantic incident" of his life.

Chapter XXIIBourbaki.

The most consequential mathematical pseudonym in history. Nicolas Bourbaki was the collective pen-name of a group of mostly French mathematicians who, beginning in 1935, set out to write a complete, rigorous, modern treatise on the foundations of pure mathematics.

The founding members included Henri Cartan, Claude Chevalley, Jean Delsarte, Jean Dieudonné, and André Weil; later members included Laurent Schwartz, Jean-Pierre Serre, Alexander Grothendieck, and Pierre Cartier. The group met regularly, fought constantly, and over forty years produced the Éléments de mathématique — ten core volumes covering set theory, algebra, topology, integration, Lie groups.

The Bourbaki style — abstract, structural, prose-heavy, definition-rich — dominated mathematical writing from 1950 to 1980. Working mathematicians of every generation have complained about it; the same mathematicians have used the books constantly.

The project's influence on mathematical pedagogy — particularly the "New Math" movement of the 1960s — has been more controversial. The view that abstract structures are the proper starting point for elementary mathematics education was tested and largely retracted.

Chapter XXIIIThe first computer proof.

In 1976 Kenneth Appel and Wolfgang Haken announced a proof of the four-color theorem using approximately 1,200 hours of IBM 360 computation. The proof reduced the problem to checking 1,936 specific configurations, which the computer verified.

The result was historically and philosophically novel. For the first time, a major mathematical theorem had been proved by an argument that no human could check by hand. Some mathematicians questioned whether such a proof was a proof in the traditional sense.

The objection has aged poorly. Computer-assisted proofs are now standard. The Kepler conjecture (Hales 1998, formalised 2017) on the densest sphere packing in three dimensions; the Robbins conjecture (McCune 1996) in Boolean algebra; the Feit-Thompson theorem on solvability of finite groups of odd order (Gonthier 2012, in Coq); the four-color theorem itself (Gonthier 2005, in Coq). The trend is one direction only.

By 2026, the leading edge of formalisation efforts — Lean, Coq, Isabelle — has the active engagement of working research mathematicians including Tao, Scholze, and Buzzard. The mathematics of 2050 will be largely machine-checked.

Chapter XXIVWiles and Fermat.

The most famous problem in mathematics, open for 358 years, closed in the 1990s. Andrew Wiles, Princeton, worked in essential isolation for seven years on a proof of the Taniyama-Shimura conjecture in the case of semistable elliptic curves — sufficient, by Ribet's 1986 theorem, to imply Fermat's Last Theorem.

Wiles announced the proof in three lectures at Cambridge in June 1993. A gap was identified by Nick Katz during refereeing. Wiles, working with his former student Richard Taylor, repaired the gap over the following year. The corrected proof was published in 1995 in two papers in the Annals of Mathematics.

The proof connects two apparently unrelated areas — modular forms and elliptic curves — at a depth that took most of 20th-century algebraic number theory to establish. Fermat's claimed proof, written in 1637, almost certainly contained an error; the techniques required for a correct proof did not exist until the late 1980s.

Wiles received numerous honours, including the Abel Prize in 2016. He was 41 at the time of the proof — older than the cutoff for the Fields Medal — and so missed the latter, an oversight that produced the Silver Plaque he was awarded as compensation.

Chapter XXVPerelman and Poincaré.

The other great proof of the era. Grigori Perelman, working alone at the Steklov Institute in Saint Petersburg, posted three preprints to the arXiv between November 2002 and July 2003 that resolved the Poincaré conjecture and the more general geometrisation conjecture due to Thurston.

The technique was Ricci flow with surgery, building on Richard Hamilton's twenty-year programme. Three independent verification teams confirmed the proof through 2006.

Perelman was awarded the Fields Medal in 2006 and the Clay Millennium Prize in 2010. He declined both. He gave no detailed explanation, but did say in interviews that he was uncomfortable with the celebrity culture of contemporary mathematics. He has not published mathematics since.

The Poincaré conjecture is the only Millennium Prize Problem solved in the quarter-century since the prizes were announced. It is also the only one of Perelman's results he ever discussed publicly.

Chapter XXVITao, Green, and the new style.

Modern mathematics has a public face: Terence Tao. UCLA professor since 1996, Fields Medalist 2006, Tao has worked across analysis, partial differential equations, combinatorics, number theory, and (since 2020) the formalisation of mathematics in Lean.

His 2004 paper with Ben Green proved that the primes contain arbitrarily long arithmetic progressions — combining ergodic theory with number theory in ways that opened a substantial new research area. The Green-Tao theorem is one of the major results of 21st-century mathematics.

Tao runs an actively read mathematical blog (since 2007) and has done as much as anyone to make modern research mathematics legible to outsiders. His 2024 work with collaborators on the Polynomial Freiman-Ruzsa conjecture, formalised in Lean within months of completion, is a model of where the discipline is heading: rigorous, collaborative, machine-verified, public.

Tao's career is the contemporary analogue of Hilbert's at Göttingen: a centre of gravity for the field, generative across multiple sub-disciplines, organising research questions across communities.

Chapter XXVIIRecent breakthroughs.

A rough sample of major mathematical events of the past decade.

Sphere packing. Maryna Viazovska, in 2016, solved the sphere-packing problem in eight dimensions — proving that the E₈ lattice gives the densest possible packing. With Cohn, Kumar, Miller, and Radchenko, she extended the result to 24 dimensions (the Leech lattice). She received the Fields Medal in 2022, only the second woman after Mirzakhani.

The bounded gaps theorem. Yitang Zhang, in 2013, proved that there are infinitely many pairs of primes whose difference is at most 70 million. The Polymath project, coordinated online, drove the bound down to 246 within months. James Maynard (Fields 2022) gave a different approach achieving comparable bounds.

The abc conjecture controversy. Shinichi Mochizuki's 2012 papers claim a proof of the abc conjecture using inter-universal Teichmüller theory — a 500-page apparatus he developed alone. The mathematical community has not accepted the proof. The dispute, particularly with Peter Scholze and Jakob Stix, remains unresolved at the time of writing.

Chapter XXVIIIReading list.

• 1937Men of MathematicsE. T. Bell
• 1953The Development of MathematicsE. T. Bell
• 1972Mathematical Thought from Ancient to Modern TimesMorris Kline
• 1977A Concise History of MathematicsDirk Struik
• 1990Journey through GeniusWilliam Dunham
• 1991The Mathematical ExperienceDavis & Hersh
• 1996The History of Mathematics: A Brief CourseRoger Cooke
• 1997Fermat's EnigmaSimon Singh
• 2007The Princeton Companion to Mathematicsed. Tim Gowers
• 2007Mathematics: A Very Short IntroductionTim Gowers
• 2007Manifold Destiny (Perelman profile)Nasar & Gruber
• 2010The Princeton Companion (2nd ed.)ed. Gowers
• 2014Love and MathEdward Frenkel
• 2017The Calculus of FriendshipSteven Strogatz
• 2022What's Next? (collected essays on the field)ed. Wood & Cook

Chapter XXIXWatch & read.

↑ The man who loved numbers · Ramanujan documentary (1988)

More on YouTube

Watch · Archimedes — the greatest inventor of the ancient world
Watch · Euler — the blind mathematician who became the world's greatest

And read

The single best one-volume guide to the field is the Princeton Companion to Mathematics, edited by Tim Gowers — essays by working mathematicians on every major branch and many of the great figures. Dunham's Journey through Genius is the most accessible historical narrative. Singh's Fermat's Enigma is the standard popular account of the Wiles proof. For the long sweep, Kline's Mathematical Thought from Ancient to Modern Times remains unmatched.