Mathematics

This is the Ulam spiral, which illustrates the distribution of prime numbers. The dark diagonal lines in the spiral hint at the hypothesized approximate independence between being prime and being a value of a quadratic polynomial, a conjecture now known as Hardy and Littlewood's Conjecture F.

Number theory began with the manipulation of numbers, that is, natural numbers ${\displaystyle (\mathbb {N} ),}$ and later expanded to integers ${\displaystyle (\mathbb {Z} )}$ and rational numbers ${\displaystyle (\mathbb {Q} ).}$ Formerly, number theory was called arithmetic, but nowadays this term is mostly used for numerical calculations.

Many easily-stated number problems have solutions that require sophisticated methods from across mathematics. One prominent example is Fermat's last theorem. This conjecture was stated in 1637 by Pierre de Fermat, but it was proved only in 1994 by Andrew Wiles, who used tools including scheme theory from algebraic geometry, category theory and homological algebra. Another example is Goldbach's conjecture, which asserts that every even integer greater than 2 is the sum of two prime numbers. Stated in 1742 by Christian Goldbach, it remains unproven to this day despite considerable effort.

Number theory includes several subareas, including analytic number theory, algebraic number theory, geometry of numbers (method oriented), diophantine equations, and transcendence theory (problem oriented).

Geometry is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as lines, angles and circles, which were developed mainly for the needs of surveying and architecture, but has since blossomed out into many other subfields.

A fundamental innovation was the introduction of the concept of proofs by ancient Greeks, with the requirement that every assertion must be proved. For example, it is not sufficient to verify by measurement that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results (theorems) and a few basic statements. The basic statements are not subject to proof because they are self-evident (postulates), or they are a part of the definition of the subject of study (axioms). This principle, which is foundational for all mathematics, was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements.[22][23]

The resulting Euclidean geometry is the study of shapes and their arrangements constructed from lines, planes and circles in the Euclidean plane (plane geometry) and the (three-dimensional) Euclidean space.[lower-alpha 2]

Euclidean geometry was developed without change of methods or scope until the 17th century, when René Descartes introduced what is now called Cartesian coordinates. This was a major change of paradigm, since instead of defining real numbers as lengths of line segments (see number line), it allowed the representation of points using their coordinates (which are numbers). This allows one to use algebra (and later, calculus) to solve geometrical problems. This split geometry into two new subfields: synthetic geometry, which uses purely geometrical methods, and analytic geometry, which uses coordinates systemically.[24]

Analytic geometry allows the study of curves that are not related to circles and lines. Such curves can be defined as graph of functions (whose study led to differential geometry). They can also be defined as implicit equations, often polynomial equations (which spawned algebraic geometry). Analytic geometry also makes it possible to consider spaces of higher than three dimensions.

In the 19th century, mathematicians discovered non-Euclidean geometries, which do not follow the parallel postulate. By questioning the truth of that postulate, this discovery joins Russel's paradox as revealing the foundational crisis of mathematics. This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not a mathematical problem. In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that are invariant under specific transformations of the space.

In the present day, the subareas of geometry include:

• Projective geometry, introduced in the 16th century by Girard Desargues, extends Euclidean geometry by adding points at infinity at which parallel lines intersect. This simplifies many aspects of classical geometry by unifying the treatments for intersecting and parallel lines.
• Affine geometry, the study of properties relative to parallelism and independent from the concept of length.
• Differential geometry, the study of curves, surfaces, and their generalizations, which are defined using differentiable functions
• Manifold theory, the study of shapes that are not necessarily embedded in a larger space
• Riemannian geometry, the study of distance properties in curved spaces
• Algebraic geometry, the study of curves, surfaces, and their generalizations, which are defined using polynomials
• Topology, the study of properties that are kept under continuous deformations
  • Algebraic topology, the use in topology of algebraic methods, mainly homological algebra
• Discrete geometry, the study of finite configurations in geometry
• Convex geometry, the study of convex sets, which takes its importance from its applications in optimization
• Complex geometry, the geometry obtained by replacing real numbers with complex numbers

Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were the two main precursors of algebra. The first one solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. The second one introduced systematic methods for transforming equations (such as moving a term from a side of an equation into the other side). The term algebra is derived from the Arabic word al-jabr meaning "the reunion of broken parts"[25] that he used for naming one of these methods in the title of his main treatise.

The quadratic formula, which concisely expresses the solutions of all quadratic equations

Algebra became an area in its own right only with François Viète (1540–1603), who introduced the use of letters (variables) for representing unknown or unspecified numbers. This allows mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas.

Until the 19th century, algebra consisted mainly of the study of linear equations (presently linear algebra), and polynomial equations in a single unknown, which were called algebraic equations (a term that is still in use, although it may be ambiguous). During the 19th century, mathematicians began to use variables to represent things other than numbers (such as matrices, modular integers, and geometric transformations), on which generalizations of arithmetic operations are often valid. The concept of algebraic structure addresses this, consisting of a set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. Due to this change, the scope of algebra grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra. (The latter term appears mainly in an educational context, in opposition to elementary algebra, which is concerned with the older way of manipulating formulas.)

Rubik's cube: the study of its possible moves is a concrete application of group theory

Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, and include:

• group theory;
• field theory;
• vector spaces, whose study is essentially the same as linear algebra;
• ring theory;
• commutative algebra, which is the study of commutative rings, includes the study of polynomials, and is a foundational part of algebraic geometry;
• homological algebra
• Lie algebra and Lie group theory;
• Boolean algebra, which is widely used for the study of the logical structure of computers.

The study of types of algebraic structures as mathematical objects is the object of universal algebra and category theory. The latter applies to every mathematical structure (not only algebraic ones). At its origin, it was introduced, together with homological algebra for allowing the algebraic study of non-algebraic objects such as topological spaces; this particular area of application is called algebraic topology.

A Cauchy sequence consists of elements that become arbitrarily close to each other as the sequence progresses (from left to right).

Calculus, formerly called infinitesimal calculus, was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz. It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many other results. Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis" is commonly used for advanced parts.

Analysis is further subdivided into real analysis, where variables represent real numbers, and complex analysis, where variables represent complex numbers. Analysis includes many subareas shared by other areas of mathematics which include:

• Multivariable calculus
• Functional analysis, where variables represent varying functions;
• Integration, measure theory and potential theory, all strongly related with Probability theory;
• Ordinary differential equations;
• Partial differential equations;
• Numerical analysis, mainly devoted to the computation on computers of solutions of ordinary and partial differential equations that arise in many applications.

Discrete mathematics, broadly speaking, is the study of finite mathematical objects. Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply.[lower-alpha 3] Algorithms – especially their implementation and computational complexity – play a major role in discrete mathematics.

Discrete mathematics includes:

• Combinatorics, the art of enumerating mathematical objects that satisfy some given constraints. Originally, these objects were elements or subsets of a given set; this has been extended to various objects, which establishes a strong link between combinatorics and other parts of discrete mathematics. For example, discrete geometry includes counting configurations of geometric shapes
• Graph theory and hypergraphs
• Coding theory, including error correcting codes and a part of cryptography
• Matroid theory
• Discrete geometry
• Discrete probability distributions
• Game theory (although continuous games are also studied, most common games, such as chess and poker are discrete)
• Discrete optimization, including combinatorial optimization, integer programming, constraint programming

The four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second half of the 20th century. The P versus NP problem, which remains open to this day, is also important for discrete mathematics, since its solution would impact much of it.

The Venn diagram is a commonly used method to illustrate the relations between sets.

The two subjects of mathematical logic and set theory have both belonged to mathematics since the end of the 19th century.[26][27] Before this period, sets were not considered to be mathematical objects, and logic, although used for mathematical proofs, belonged to philosophy, and was not specifically studied by mathematicians.

Before Cantor's study of infinite sets, mathematicians were reluctant to consider actually infinite collections, and considered infinity to be the result of endless enumeration. Cantor's work offended many mathematicians not only by considering actually infinite sets,[28] but by showing that this implies different sizes of infinity (see Cantor's diagonal argument) and the existence of mathematical objects that cannot be computed, or even explicitly described (for example, Hamel bases of the real numbers over the rational numbers). This led to the controversy over Cantor's set theory.

In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour. Examples of such intuitive definitions are "a set is a collection of objects", "natural number is what is used for counting", "a point is a shape with a zero length in every direction", "a curve is a trace left by a moving point", etc.

This became the foundational crisis of mathematics.[29] It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory. Roughly speaking, each mathematical object is defined by the set of all similar objects and the properties that these objects must have. For example, in Peano arithmetic, the natural numbers are defined by "zero is a number", "each number has a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning. The "nature" of the objects defined this way is a philosophical problem that mathematicians leave to philosophers, even if many mathematicians have opinions on this nature, and use their opinion—sometimes called "intuition"—to guide their study and proofs.

This approach allows considering "logics" (that is, sets of allowed deducing rules), theorems, proofs, etc. as mathematical objects, and to prove theorems about them. For example, Gödel's incompleteness theorems assert, roughly speaking that, in every theory that contains the natural numbers, there are theorems that are true (that is provable in a larger theory), but not provable inside the theory.

This approach to the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led by Brouwer, who promoted intuitionistic logic, which explicitly lacks the law of excluded middle.

These problems and debates led to a wide expansion of mathematical logic, with subareas such as model theory (modeling some logical theories inside other theories), proof theory, type theory, computability theory and computational complexity theory. Although these aspects of mathematical logic were introduced before the rise of computers, their use in compiler design, program certification, proof assistants and other aspects of computer science, contributed in turn to the expansion of these logical theories.[30]

Whatever the form of a random population distribution (μ), the sampling mean (x̄) tends to a Gaussian distribution and its variance (σ) is given by the central limit theorem of probability theory.[31]

The field of statistics is a type of mathematical application that is employed for the collection and processing of data samples, using procedures based on mathematical methods especially probability theory. Statisticians generate data with random sampling or randomized experiments.[32] The design of a statistical sample or experiment determines the analytical methods that will be used. Analysis of data from observational studies is done using statistical models and the theory of inference, using model selection and estimation. The models and consequential predictions should then be tested against new data.[lower-alpha 4]

Statistical theory studies decision problems such as minimizing the risk (expected loss) of a statistical action, such as using a procedure in, for example, parameter estimation, hypothesis testing, and selecting the best. In these traditional areas of mathematical statistics, a statistical-decision problem is formulated by minimizing an objective function, like expected loss or cost, under specific constraints: For example, designing a survey often involves minimizing the cost of estimating a population mean with a given level of confidence.[33] Because of its use of optimization, the mathematical theory of statistics overlaps with other decision sciences, such as operations research, control theory, and mathematical economics.[34]

Computational mathematics is the study of mathematical problems that are typically too large for human, numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory; numerical analysis broadly includes the study of approximation and discretization with special focus on rounding errors. Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic-matrix-and-graph theory. Other areas of computational mathematics include computer algebra and symbolic computation.

The history of mathematics is an ever-growing series of abstractions. Evolutionarily speaking, the first abstraction to ever be discovered, one shared by many animals,[35] was probably that of numbers: the realization that, for example, a collection of two apples and a collection of two oranges (say) have something in common, namely that there are two of them. As evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have also known how to count abstract quantities, like time—days, seasons, or years.[36][37]

The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC

Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy.[38] The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It is in Babylonian mathematics that elementary arithmetic (addition, subtraction, multiplication, and division) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a sexagesimal numeral system which is still in use today for measuring angles and time.[39]

Euclid holding a compass, as imagined by Raphael in this detail from The School of Athens[lower-alpha 5]

In the 6th century BC, Greek mathematics began to emerge as a distinct discipline and some Ancient Greeks such as the Pythagoreans appeared to have considered it a subject in its own right.[40] Around 300 BC, Euclid organized mathematical knowledge by way of postulates and first principles, which evolved into the axiomatic method that is used in mathematics today, consisting of definition, axiom, theorem, and proof.[41] His book, Elements, is widely considered the most successful and influential textbook of all time.[42] The greatest mathematician of antiquity is often held to be Archimedes (c. 287–212 BC) of Syracuse.[43] He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.[44] Other notable achievements of Greek mathematics are conic sections (Apollonius of Perga, 3rd century BC),[45] trigonometry (Hipparchus of Nicaea, 2nd century BC),[46] and the beginnings of algebra (Diophantus, 3rd century AD).[47]

The numerals used in the Bakhshali manuscript, dated between the 2nd century BC and the 2nd century AD

The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine, and an early form of infinite series.

A page from al-Khwārizmī's Algebra

Leonardo Fibonacci, the Italian mathematician who introduced the Hindu–Arabic numeral system invented between the 1st and 4th centuries by Indian mathematicians, to the Western World.

During the Golden Age of Islam, especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of algebra. Other achievements of the Islamic period include advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system.[48] Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī.

During the early modern period, mathematics began to develop at an accelerating pace in Western Europe. The development of calculus by Isaac Newton and Gottfried Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries. Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Gauss, who made numerous contributions to fields such as algebra, analysis, differential geometry, matrix theory, number theory, and statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show in part that any consistent axiomatic system—if powerful enough to describe arithmetic—will contain true propositions that cannot be proved.

Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made to this very day. According to Mikhail B. Sevryuk, in the January 2006 issue of the Bulletin of the American Mathematical Society, "The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs."[49]

Isaac Newton (left) and Gottfried Wilhelm Leibniz developed infinitesimal calculus.

Until the end of the 19th century, the development of mathematics was mainly motivated by the needs of technology and science, and there was no clear distinction between pure and applied mathematics. For example, the natural numbers and arithmetic were introduced for the need of counting, and geometry was motivated by surveying, architecture and astronomy. Later, Isaac Newton introduced infinitesimal calculus for explaining the movement of the planets with his law of gravitation. Moreover, most mathematicians were also scientists, and many scientists were also mathematicians. However, a notable exception occurred in Ancient Greece; see Pure mathematics § Ancient Greece.

In the second half ot the 19th century, new mathematical theories were introduced which were not related with the physical world (at least at that time), in particular, non-Euclidean geometries and Cantor's theory of transfinite numbers. This was one of the starting points of the foundational crisis of mathematics, which was eventually solved by the systematization of the axiomatic method for defining mathematical structures.

So, many mathematicians focused their research on internal problems, that is, pure mathematics, and this led to split mathematics into pure mathematics and applied mathematics, the latter being often considered as having a lower value.

During the second half of the 20th century, it appeared that many theories issued from applications are also interesting from the point of view of pure mathematics, and that many results of pure mathematics have applications outside mathematics (see next section); in turn, the study of these applications may give new insights on the "pure theory". An example of the first case is the theory of distributions, introduced by Laurent Schwartz for validating computations done in quantum mechanics, which became immediately an important tool of (pure) mathematical analysis. An example of the second case is the decidability of the first-order theory of the real numbers, a problem of pure mathematics that was proved true by Alfred Tarski, with an algorithm that is definitely impossible to implement, because of a computational complexity that is much too high. For getting an algorithm that can be implemented and can solve systems of polynomial equations and inequalities, George Collins introduced the cylindrical algebraic decomposition that became a fundamental tool in real algebraic geometry.

So, the distinction between pure and applied mathematics is presently more a question of personal research aim of mathematicians than a division of mathematics into broad areas. The Mathematics Subject Classification does not mention "pure mathematics" nor "applied mathematics". However, these terms are still used in names of some university departments, such as at the Faculty of Mathematics at the University of Cambridge.

The unreasonable effectiveness of mathematics[9] is a phenomenon that was named and first made explicit by physicist Eugene Wigner. It is the fact that many mathematical theories, even the "purest" have applications outside their initial object. These applications may be completely outside their initial area of mathematics, and may concern physical phenomena that were completely unknown when the mathematical theory was introduced.

A famous example is the prime factorization of natural numbers that was discovered more than 2,000 years before its common use for secure internet communications through the RSA cryptosystem.

Another historical example is the theory of ellipses. They were studied by the ancient Greek mathematicians as conic sections (that is, intersections of cones with planes). It is almost 2,000 years later that Johannes Kepler discovered that the trajectories of the planets are ellipses.

In the 19th century, the internal development of geometry (pure mathematics) lead to define and study non-Euclidean geometries, spaces of dimension higher than three and manifolds. At this time, these concepts seemed totally disconnected from the physical reality, but at the beginning of the 20th century, Albert Einstein developed the theory of relativity that uses fundamentally these concepts. In particular, spacetime of the special relativity is a non-Euclidean space of dimension four, and spacetime of the general relativity is a (curved) manifold of dimension four.

Similar examples of unexpected applications of mathematical theories can be found in many areas of mathematics.

Another striking aspect of the interaction between mathematics and physics is when mathematics drives research in physics. This is illustrated by the discoveries of the positron and the baryon ${\displaystyle \Omega ^{-}.}$ In both cases, the equations of the theories had unexplained solutions, which led to conjecture the existence of a unknown particle, and to search these particles. In both cases, these particles were discovered a few years later by specific experiments.[56]

The connection between mathematics and material reality has led to philosophical debates since at least the time of Pythagoras. The ancient philosopher Plato argued that abstractions that reflect material reality have themselves a reality that exists outside space and time. As a result, the philosophical view that mathematical objects somehow exist on their own in abstraction is often referred to as Platonism. Independently of their possible philosophical opinions, modern mathematicians may be generally considered as Platonists, since they think of and talk of their objects of study as real objects.[57]

Armand Borel summarized this view of mathematics reality as follows, and provided quotations of G. H. Hardy, Charles Hermite, Henri Poincaré and Albert Einstein that support his views.[56]

Something becomes objective (as opposed to "subjective") as soon as we are convinced that it exists in the minds of others in the same form as it does in ours and that we can think about it and discuss it together[58]. Because the language of mathematics is so precise, it is ideally suited to defining concepts for which such a consensus exists. In my opinion, that is sufficient to provide us with a feeling of an objective existence, of a reality of mathematics ...

Nevertheless, Platonism and the concurrent views on abstraction do not explain the unreasonable effectiveness of mathematics.

There is no general consensus about a definition of mathematics or its epistemological status—that is, its place among other human activities.[59][60]

A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable.[59] There is not even consensus on whether mathematics is an art or a science.[60] Some just say, "mathematics is what mathematicians do".[59] This makes sense, as there is a strong consensus among them about what is mathematics and what is not.

Most proposed definitions try to define mathematics by its object of study.

Aristotle defined mathematics as "the science of quantity" and this definition prevailed until the 18th century. However, Aristotle also noted a focus on quantity alone may not distinguish mathematics from sciences like physics; in his view, abstraction and studying quantity as a property "separable in thought" from real instances set mathematics apart.[61]

In the 19th century, when mathematicians began to address topics—such as infinite sets—which have no clear-cut relation to physical reality, a variety of new definitions were given.[62] With the large number of new areas of mathematics that appeared since the beginning of the 20th century and continue to appear, defining mathematics by this object of study becomes an impossible task.

Another approach for defining mathematics is to use its methods. So, an area of study can be qualified as mathematics as soon as one can prove theorems —assertions whose validity relies on a proof, that is, a purely-logical deduction.

Mathematicians strive to develop their results with systematic reasoning in order to avoid mistaken "theorems". These false proofs often arise from fallible intuitions and have been common in mathematics' history. To allow deductive reasoning, some basic assumptions need to be admitted explicitly as axioms. Traditionally, these axioms were selected on the grounds of common-sense, but modern axioms typically express formal guarantees for primitive notions, such as simple objects and relations.

The validity of a mathematical proof is fundamentally a matter of rigour, and misunderstanding rigor is a notable cause for some common misconceptions about mathematics. Mathematical language may give more precision than in everyday speech to ordinary words like or and only. Other words such as open and field are given new meanings for specific mathematical concepts. Sometimes, mathematicians even coin entirely new words (e.g. homeomorphism). This technical vocabulary is both precise and compact, making it possible to mentally process complex ideas. Mathematicians refer to this precision of language and logic as "rigor".

The rigor expected in mathematics has varied over time: the ancient Greeks expected detailed arguments, but in Isaac Newton's time, the methods employed were less rigorous (not because of a different conception of mathematics, but because of the lack of the mathematical methods that are required for reaching rigor). Problems inherent in Newton's approach were solved only in the second half of the 19th century, with the formal definitions of real numbers, limits and integrals. Later in the early 20th century, Bertrand Russell and Alfred North Whitehead would publish their Principia Mathematica, an attempt to show that all mathematical concepts and statements could be defined, then proven entirely through symbolic logic. This was part of a wider philosophical program known as logicism, which sees mathematics as primarily an extension of logic.

Despite mathematics' concision, many proofs require hundreds of pages to express. The emergence of computer-assisted proofs has allowed proof lengths to further expand. Assisted proofs may be erroneous if the proving software has flaws.[lower-alpha 6][63] On the other hand, proof assistants allow for the verification of details that cannot be given in a hand-written proof, and provide certainty of the correctness of long proofs such as that of the 255-page Feit–Thompson theorem.[lower-alpha 7]