Mathematics

Mathematics is a field of knowledge that studies abstract ideas such as numbers, geometric shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and show their properties, which are often written as theorems, formulas, and equations. Mathematics is widely used to model and solve problems in science, engineering, technology, economics, and everyday life.

There are many areas of mathematics. Number theory studies integers and their properties. Algebra studies operations and the structures they form. Geometry studies shapes and the spaces that contain them. Analysis studies approximation and convergence in a quantitative way. Set theory is presently used as a foundation for all mathematics.

Mathematics works with abstract objects that are either abstractions from nature or purely abstract entities given certain properties, called axioms. Using pure reason, mathematicians prove these properties through proofs, which are a series of steps using deductive rules on results that are already established. These results, called theorems, build on earlier theorems, axioms, and, when objects come from nature, some basic properties that are considered true starting points.

Mathematics is essential in science and engineering, and it is used in every field of knowledge and in everyday life whenever data are handled, shapes and patterns are studied, or logical reasoning is needed. Although mathematics is widely used to model real-world events, its results are proved by deduction rather than by experiment. The relationship between mathematical truth, logic, and reality is a subject of philosophical debate. Some areas, such as game theory, are developed closely with their uses and are often grouped under applied mathematics. Other areas grow without any particular use in mind but often find practical applications later.

Mathematical written records first appeared in Ancient Egypt and Mesopotamia, but the idea of proof and its strict standards, called mathematical rigor, began in ancient Greek mathematics, shown in Euclid's Elements. Mathematics was mainly divided into geometry and arithmetic until the 16th and 17th centuries, when algebra and infinitesimal calculus developed into new fields. Since then, new mathematical ideas and scientific discoveries have pushed each other forward. At the end of the 19th century, the foundational crisis of mathematics led to the systematic use of the axiomatic method, which brought a large increase in the number of mathematical areas and their uses. The modern Mathematics Subject Classification lists more than 60 first-level areas of mathematics.

The words "Math" and "Maths" lead to this article. For other meanings, see Mathematics (disambiguation) and Math (disambiguation).

Areas of mathematics

Before the Renaissance, mathematics had two main areas: arithmetic, which studies numbers, and geometry, which studies shapes. Some practices such as numerology and astrology were not then clearly separated from mathematics. Starting with the Renaissance, modern algebra and calculus became important too. Calculus studies how quantities change. These four areas stayed the main division until the end of the 19th century. Topics such as celestial mechanics and solid mechanics are now considered part of physics. The study of combinatorics has a long history, but it did not become its own branch until the 17th century.

At the end of the 19th century, mathematicians faced the foundational crisis in mathematics and began using the axiomatic method, which led to many new areas. The 2020 Mathematics Subject Classification lists 63 main areas. Some of them match the older divisions, such as number theory and geometry. Others, like mathematical logic and foundations, appeared in the 20th century or were not previously counted as mathematics.

History

Main article: History of mathematics

The word mathematics comes from the Ancient Greek word máthēma, meaning 'something learned, knowledge'. It entered English during the Late Middle English period. The Pythagoreans, a group within Pythagoreanism, were likely the first to use the word only for the study of arithmetic and geometry, and by the time of Aristotle, this meaning was fully established. Until about 1700, the English word more often meant astrology or sometimes astronomy. The meaning changed gradually from about 1500 to 1800, which has led to some mistranslations. The English form looks like a plural because it goes back to a Latin neuter plural used by Cicero, and English may have formed the word after the pattern of metaphysics. Prehistoric peoples may have known how to count physical objects, and possibly abstract quantities like days and years, as prehistoric peoples did.

Mathematics became more complex around 3000 BC, when the Babylonians and Egyptians used it for taxes, building, and astronomy. The Pythagorean theorem seems to be among the oldest and most widespread ideas, after basic arithmetic and geometry. The Babylonians used elementary arithmetic, including addition, subtraction, multiplication, and division, and a sexagesimal system that is still used today for measuring angles and time. By the 5th century BC, Greek mathematics had become a distinct subject, and around 300 BC, Euclid organized it into the axiomatic method used today, whose book Elements is widely considered the most successful textbook of all time. Archimedes of Syracuse developed early methods similar to modern calculus, and other achievements included conic sections by Apollonius of Perga, trigonometry by Hipparchus of Nicaea, and early algebra by Diophantus. The Hindu–Arabic numeral system evolved in India, passed to the Western world through Islamic mathematics, and Indian mathematicians also developed early ideas about cosine and infinite series. During the Golden Age of Islam, mathematicians built on Greek work and developed algebra, spherical trigonometry, and the decimal point, with notable figures such as al-Khwarizmi, Omar Khayyam, and Sharaf al-Dīn al-Ṭūsī. In the early modern period, mathematics grew quickly in Western Europe, with symbolic notation by François Viète, logarithms by John Napier to help with marine navigation, Cartesian coordinates by René Descartes, and calculus by Isaac Newton and Gottfried Leibniz. Leonhard Euler later organized these ideas and proved many theorems.

In the 19th century, Carl Gauss made many contributions to fields such as differential geometry, matrix theory, and statistics. In the early 20th century, Kurt Gödel changed mathematics with his incompleteness theorems. Mathematics has since grown greatly, and it has helped science while science has helped mathematics. New discoveries are still being made today. In 2006, Mikhail B. Sevryuk wrote in the Bulletin of the American Mathematical Society that more than 1.9 million papers and books had been recorded in the Mathematical Reviews database since 1940, with more than 75,000 added each year.

Symbolic notation and terminology

Main articles: Mathematical notation, Language of mathematics, and Glossary of mathematics

Mathematical notation is widely used in science and engineering to show complex concepts and properties in a short, clear, and accurate way. It uses symbols for operations, unknown numbers, relations, and other mathematical objects, which are put together into expressions and formulas. Numbers and other objects are shown by variables, usually Latin or Greek letters, often with subscripts. Operations and relations use special signs or glyphs, such as + (plus), × (multiplication), ∫ (integral), and = (equal).

Until the 19th century, there was no clear difference between applied mathematics and pure mathematics as we understand them today. Later, Isaac Newton and Gottfried Wilhelm Leibniz developed infinitesimal calculus. In the 19th century, mathematicians began to focus more on problems within mathematics itself, which split the subject into pure and applied mathematics. Today, the lines between the two are often blurred, and many discoveries in pure mathematics have turned out to be useful outside of mathematics.

Philosophy

Main article: Philosophy of mathematics

Main article: Definitions of mathematics

See also: Logic

People have debated the link between mathematics and the real world since at least the time of Pythagoras. The ancient philosopher Plato argued that ideas which reflect the real world exist outside space and time. The view that mathematical objects exist on their own is often called Platonism. Many modern mathematicians can be seen as Platonists, because they think and talk about their objects of study as real things. Mathematicians do not agree on a single definition of mathematics or its epistemological status, meaning its place in knowledge. Some say that "mathematics is what mathematicians do." Aristotle called mathematics "the science of quantity," but as new areas appeared, defining it became harder. Saunders Mac Lane described how the many areas of mathematics are connected in his book Mathematics, Form and Function. Another way to describe mathematics is by its methods: an area counts as mathematics when people can prove theorems.

Mathematical reasoning requires rigor. Definitions must be clear, and proofs must follow inference rules without relying on evidence or intuition. Rigorous proofs can be very long, such as the 255-page Feit–Thompson theorem, even though mathematics aims for concision. Computer-assisted proofs have made some proofs longer still. Some thinkers describe a quasi-empiricist view, in which proofs are not perfect but have a probability attached. Rigor has been valued since ancient Greece. In the late 19th century, some basic definitions were found to cause problems, such as the Weierstrass function and Russell's paradox. This was fixed by adding axioms, a return to the method of the ancient Greeks. Under this approach, a proof is either correct or wrong, so a "rigorous proof" is simply a pleonasm. Mathematicians check each other's proofs, and after many years a proof can be considered reliable. The idea of rigor may still help beginners learn what a mathematical proof is.

Training and practice

See also: Computer-based mathematics education

Mathematics can cross cultural boundaries and time periods. As a human activity, it has a social side, which includes education, careers, recognition, and popularization. Mathematics is a core part of school subjects and of science, technology, engineering, and mathematics (STEM). Professional mathematicians may work as teachers or professors, statistician, actuary, financial analyst, economist, accountant, commodity trader, or computer consultant. Evidence shows that mathematics was taught in ancient Babylonia as early as the second millennium BCE, and similar training was found in the ancient Near East and the Greco-Roman world. The oldest known mathematics textbook is the Rhind papyrus, from Egypt around 1650 BCE. In ancient India, lessons were passed on by oral tradition because books were scarce. In Imperial China during the Tang dynasty, mathematics was part of the civil service exam. Students today learn mathematics in nearly every country, though what they study varies. Some students feel mathematical anxiety, a fear about how they will do in the subject.

Mathematics also involves creativity, puzzles, and beauty. A theorem is valid when its proof is rigorous, and a computer program could theoretically check that automatically. Still, creativity matters, because many important results solve problems other mathematicians could not. One example is Apery's theorem: Roger Apery gave only the ideas for a proof, and the full proof came several months later from three other mathematicians. Some mathematicians see their work as a game of solving puzzles, which is emphasized in recreational mathematics. Mathematicians also find an aesthetic value in their subject, often linked to elegance, such as simplicity, symmetry, and completeness. G. H. Hardy, in A Mathematician's Apology, believed that these aesthetic qualities alone could justify studying pure mathematics. Paul Erdős spoke of "The Book", a supposed collection of the most beautiful proofs, and the book Proofs from THE BOOK was inspired by him. Some people feel that calling mathematics a science downplays its artistry and history within the liberal arts.

Cultural impact

Sounds that fit well together have vibration frequencies in simple ratios. An octave doubles the frequency, and a perfect fifth multiplies it by 3/2. People, and some animals, find symmetrical patterns beautiful. The symmetries of an object form a symmetry group. For mirror symmetry, this is the cyclic group of two elements. A Rorschach test figure has this symmetry, and so do butterfly bodies and most animal bodies on the surface. Fractals show self-similarity.

Main article: Popular mathematics

Popular mathematics means presenting math without technical terms. This can be hard, because many people feel mathematical anxiety, and mathematical objects are very abstract. Popular writing can help by using real-world uses or cultural links, but math is rarely the topic of printed or TV programs. The most prestigious math award is the Fields Medal, set up by Canadian John Charles Fields in 1936. It is given every four years (except around World War II) to up to four people and is seen as the math version of the Nobel Prize. Other major awards include the Abel Prize, the Chern Medal, the American Mathematical Society (AMS) Leroy P. Steele Prize, and the Wolf Prize in Mathematics. In 1900, German mathematician David Hilbert made a famous list of 23 open problems, called "Hilbert's problems," and at least 13 have been solved. A newer list of seven problems, the "Millennium Prize Problems," was published in 2000, with a reward of 1 million dollars for each solution. Only one, the Poincaré conjecture, has been solved so far, by Russian mathematician Grigori Perelman.