Mathematics

• 1. Introduction

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  Mathematics is a field of knowledge built on abstract things you cannot hold in your hand: numbers, geometric shapes, sets, functions, and probabilities. It does not measure them with a ruler or weigh them on a scale. Instead it reasons about them, and it proves what is true through theorems, formulas, and equations. A mathematician who claims two lengths are equal cannot simply lay them side by side. The claim has to be proven, deduced step by step from results already established.

  The word itself carries a buried surprise. For centuries, in Latin and English up until around 1700, the term mathematics more often meant astrology than the discipline we know today. Saint Augustine once warned Christians to beware of the mathematici, by which he meant astrologers. That warning is sometimes mistranslated as a condemnation of mathematicians. The meaning shifted slowly, roughly between 1500 and 1800, into the sense we use now.

  This is a subject old enough that its first written records appear in Ancient Egypt and Mesopotamia, yet vast enough that the contemporary Mathematics Subject Classification lists more than sixty first-level areas. How did counting marks on a bone grow into more than sixty fields? Why does a problem about prime numbers stay unsolved for centuries while another, stated in the same breath, gets cracked? And why does mathematics, even at its most abstract and useless-seeming, keep turning out to describe the physical world with uncanny accuracy? Those questions run through everything that follows.

• 2. The Birth Of Proof

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  Around 300 BC, a man named Euclid organized the mathematical knowledge of his world into a book called Elements, and in doing so set a standard that still governs the subject. His method was simple to state and demanding to follow. Every assertion must be proved. It is not enough to measure two lengths and see they match. Their equality has to be established by reasoning from results already accepted, from postulates that are self-evident, and from axioms that belong to the definition of the subject.

  Elements covers both geometry and number theory, and it is widely considered the most successful and influential textbook ever written. From its structure grew the axiomatic method that mathematics still uses today: definition, axiom, theorem, and proof. The Pythagoreans, working before Euclid, had already begun treating mathematics as a subject in its own right, and by the time of Aristotle, who lived from 384 to 322 BC, that meaning was fully established.

  Greece was not the only place doing serious mathematics, but it was where rigor took root. Archimedes of Syracuse, born around 287 BC, found methods for the surface area and volume of solids of revolution. He calculated the area under the arc of a parabola by summing an infinite series, in a way that looks remarkably like modern calculus, almost two thousand years early. Apollonius of Perga gave the world conic sections in the 3rd century BC, and Hipparchus of Nicaea gave it trigonometry in the 2nd.

  The rigor the Greeks prized had a cost. Their insistence on airtight proof tended to discourage exploration of unsettling new ideas, such as irrational numbers and the concept of infinity. That tension between certainty and discovery would haunt the subject for the next two thousand years, and it eventually broke open into a crisis that reshaped what mathematics even is.

• 3. From Arithmetic To Abstraction

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  Before the Renaissance, mathematics had only two main rooms: arithmetic, the study and manipulation of numbers, and geometry, the study of shapes. The boundaries were so loose that numerology and astrology were not yet clearly separated from the genuine article. The Hindu-Arabic numeral system that the whole world now uses took shape over the first millennium AD in India, then reached Europe through Islamic mathematics.

  Diophantus in the 3rd century and al-Khwarizmi in the 9th were the two great forerunners of algebra. Al-Khwarizmi introduced systematic ways of transforming equations, such as moving a term from one side to the other. The word algebra itself comes from the Arabic al-jabr, meaning the reunion of broken parts, a phrase he used to name one of his methods in the title of his main treatise. During the Golden Age of Islam, especially in the 9th and 10th centuries, Persian mathematicians like al-Khwarizmi, Omar Khayyam, and Sharaf al-Din al-Tusi pushed these ideas forward.

  Francois Viete, who lived from 1540 to 1603, turned algebra into a discipline of its own by introducing variables to stand for unknown or unspecified numbers. With variables, a mathematician could describe operations on numbers using formulas rather than words. John Napier added logarithms in 1614, which dramatically simplified the heavy calculations needed for astronomy and marine navigation.

  Rene Descartes, who lived from 1596 to 1650, introduced what we now call Cartesian coordinates, and the move changed everything. Instead of defining numbers as the lengths of line segments, points could be named by their coordinates. Suddenly algebra could solve geometry, and geometry split into two branches: synthetic geometry using purely geometric methods, and analytic geometry using coordinates. The 19th century then watched mathematicians use variables for things that were not numbers at all: matrices, modular integers, geometric transformations. That leap produced abstract algebra, shaped by the work of Emmy Noether and popularized by Van der Waerden's book Moderne Algebra.

• 4. The Crisis Over Infinity

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  For most of its history, mathematics refused to take infinity seriously as a finished thing. Mathematicians treated it as the result of endless enumeration, a process that never stops, rather than a collection you could hold in your mind all at once. Then Georg Cantor studied infinite sets directly, and worse, he showed there are different sizes of infinity, a result he established through what is called his diagonal argument.

  Cantor's work offended many of his contemporaries. The idea of actually infinite sets was bad enough; the idea that some infinities are bigger than others provoked open controversy. At the same time, mathematicians in several areas were discovering that the old intuitive definitions of basic objects were not solid enough to guarantee rigor. Non-Euclidean geometries, which abandon the parallel postulate, and Russell's paradox both exposed cracks in the foundation. This became known as the foundational crisis of mathematics.

  The resolution came through formalized set theory and the systematic return of the axiomatic method. Each mathematical object would now be defined by the set of all similar objects and the properties they must satisfy. In Peano arithmetic, for instance, the natural numbers are pinned down by rules: zero is a number, each number has a unique successor, each number except zero has a unique predecessor. David Hilbert, around 1910, founded the philosophy of formalism that gave this approach its name.

  Not everyone accepted the truce. A group of mathematicians led by Brouwer promoted intuitionistic logic, which deliberately rejects the law of excluded middle. And then Kurt Godel published his incompleteness theorems, which proved something humbling. In any consistent formal system powerful enough to describe arithmetic, there will be statements that are true but cannot be proved inside the system itself.

• 5. The Unreasonable Effectiveness

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  The physicist Eugene Wigner gave a name to one of the strangest features of mathematics: its unreasonable effectiveness. Theories invented for no practical purpose, even the purest ones, keep turning out to describe the physical world, sometimes phenomena that were completely unknown when the theory was born. The pattern repeats so often that it starts to feel less like coincidence and more like a mystery nobody has fully explained.

  The prime factorization of natural numbers is a clean example. It was studied for more than two thousand years as pure curiosity before becoming the backbone of secure internet communication through the RSA cryptosystem. The problem of integer factorization goes all the way back to Euclid around 300 BC, and it had no practical application whatsoever until cryptography needed it. Ellipses tell a similar story. Ancient Greek mathematicians studied them as conic sections, and roughly two thousand years passed before Johannes Kepler realized the planets travel along them.

  In the 19th century, the internal logic of geometry produced non-Euclidean geometries, spaces of more than three dimensions, and manifolds, all of which looked utterly disconnected from reality. Then Albert Einstein built the theory of relativity on exactly those concepts. The spacetime of special relativity is a non-Euclidean space of four dimensions, and the spacetime of general relativity is a curved four-dimensional manifold.

  Sometimes the mathematics runs ahead and physics chases it. The equations describing certain theories carried unexplained solutions, which led physicists to conjecture that an unknown particle must exist. That is how the positron and the baryon were predicted, and in both cases the particles were found a few years later by deliberate experiments. The accuracy of mathematics, when it models the world, depends only on whether the model fits. When Newton's law of gravitation failed to explain the slow shift in Mercury's orbit, the fault lay not in the math but in the model, and Einstein's general relativity replaced it with a better one.

• 6. What Mathematicians Actually Do

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  There is no agreed definition of mathematics, and many professional mathematicians simply do not care. Some consider it undefinable. There is not even consensus on whether mathematics is an art or a science. One common shrug of an answer is that mathematics is what mathematicians do. Aristotle called it the science of quantity, and that definition held until the 18th century, until mathematicians began studying things like infinite sets that have no clear relation to physical quantity at all.

  Whatever the definition, the daily work runs on a precise vocabulary. A statement taken as true without proof is an axiom or postulate. A statement not yet proven or disproven is a conjecture. Once deductive reasoning establishes it, it becomes a theorem. A smaller theorem used mainly to prove a bigger one is a lemma, and a result that follows directly from a larger finding is a corollary. Some terms are pure inventions, like polynomial and homeomorphism. Others are ordinary words bent to precise meaning, which is why a sentence like every free module is flat can be a true mathematical claim and sound like nonsense to an outsider.

  Rigor is the spine of the whole enterprise. Definitions must be utterly unambiguous, and proofs must reduce to a chain of inference rules with no appeal to intuition or empirical evidence. That standard can demand staggering length. The Feit-Thompson theorem runs to 255 pages, and computer-assisted proofs have stretched them further still. When Gauss was asked how he arrived at his theorems, he answered durch planmassiges Tattonieren, through systematic experimentation, hinting that even pure reason starts with guessing and testing.

  The work is not all logic. Creativity lives inside it, often in the flash of insight that solves a problem others failed at. Roger Apery, in the theorem that bears his name, supplied only the ideas for a proof; three other mathematicians worked out the formal version several months later. Some mathematicians treat their craft as a game of puzzles, the spirit captured in recreational mathematics. And many find genuine beauty in it, an elegance tied to simplicity, symmetry, and unexpectedness. Paul Erdos spoke half-jokingly of The Book, a divine collection of the most beautiful proofs, including Euclid's argument that the primes never run out.

• 7. The Long Unsolved

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  In 1637, Pierre de Fermat stated a conjecture now called Fermat's Last Theorem, and it resisted every assault for more than three centuries. Andrew Wiles finally proved it in 1994, and the proof was anything but simple. He reached across the whole of mathematics for his tools, drawing on scheme theory from algebraic geometry, category theory, and homological algebra. An easily stated problem had demanded the heaviest machinery the subject could offer.

  Goldbach's conjecture shows the same gap between simplicity and difficulty. Stated in 1742 by Christian Goldbach, it claims that every even integer greater than 2 is the sum of two prime numbers. Anyone can understand it, and despite enormous effort it remains unproven. Discrete mathematics carries its own celebrated puzzles. The four color theorem and the question of optimal sphere packing were both major problems solved in the second half of the 20th century, while the P versus NP problem stays open and would, if solved, ripple through a vast number of hard computational problems.

  David Hilbert turned the unsolved into a program. In 1900 he compiled a list of 23 open problems, and that list became famous among mathematicians, with at least thirteen of them solved depending on how some are interpreted. A century later, in 2000, a new list of seven Millennium Prize Problems appeared, each carrying a one million dollar reward. Only one of the seven, the Riemann hypothesis, overlaps with Hilbert's original list.

  Glory in mathematics has its own ceremonies. The Fields Medal, established by the Canadian John Charles Fields in 1936 and awarded every four years to up to four people, is treated as the discipline's Nobel Prize. The Abel Prize arrived in 2002 and was first given in 2003, and the Chern Medal followed in 2009. Of the seven Millennium Prize Problems, exactly one has been solved so far: the Poincare conjecture, cracked by the Russian mathematician Grigori Perelman.