Mathematics

The distribution of prime numbers is a central point of study in number theory. This Ulam spiral serves to illustrate it, hinting, in particular, at the conditional independence between being prime and being a value of certain quadratic polynomials.

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 ${\displaystyle (\mathbb {Q} ).}$ Number theory was formerly called arithmetic, but nowadays this term is mostly used for numerical calculations.

One characteristic of number theory is that many problems that can be stated very elementarily are very difficult, and their solutions often require very sophisticated methods from various parts of mathematics. A prominent example of this is Fermat's last theorem. This conjecture was stated in 1637 by Pierre de Fermat, but it was only proved in 1994 by Andrew Wiles, using, among other tools, algebraic geometry (more specifically scheme theory), 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.

Due to the great diversity of the problems studied and the methods of solution used, number theory is presently split into several subareas, including analytic number theory, algebraic number theory, geometry of numbers (method oriented), Diophantine equations and transcendence theory (problem oriented).

Geometry is, with arithmetic, 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 need of surveying and architecture.

A fundamental innovation was the elaboration of proofs by ancient Greeks: it is not sufficient to verify by measurement that, say, two lengths are equal. Such a property must be proved by abstract reasoning from previously proven results (theorems) and basic properties (which are considered as self-evident because they are too basic for being the subject of a proof (postulates)). This principle, which is foundational for all mathematics, was elaborated for the sake of geometry, and was systematized by Euclid around 300 BC in his book Elements.

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 a 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 numbers (their coordinates), and for the use of algebra and later, calculus for solving geometrical problems. This split geometry in two parts that differ only by their methods, synthetic geometry, which uses purely geometrical methods, and analytic geometry, which uses coordinates systemically.

Analytic geometry allows the study of new shapes, in particular curves that are not related to circles and lines; these curves are defined either as graph of functions (whose study led to differential geometry), or by implicit equations, often polynomial equations (which spawned algebraic geometry). Analytic geometry makes it possible to consider spaces dimensions higher than three (it suffices to consider more than three coordinates), which are no longer a model of the physical space.

Geometry expanded quickly during the 19th century. A major event was the discovery (in the second half of the 19th century) of non-Euclidean geometries, which are geometries where the parallel postulate is abandoned. This is, besides Russel's paradox, one of the starting points of the foundational crisis of mathematics, by taking into question the truth of the aforementioned postulate. 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. This results in a number of subareas and generalizations of geometry that include:

• Projective geometry, introduced in the 16th century by Girard Desargues, it extends Euclidean geometry by adding points at infinity at which parallel lines intersect. This simplifies many aspects of classical geometry by avoiding to have a different treatment 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 may be viewed as the art of manipulating equations and formulas. Diophantus (3rd century) and Al-Khwarizmi (9th century) were two main precursors of algebra. The first one solved some relations between unknown natural numbers (that is, equations) by deducing new relations until getting 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 that he used for naming one of these methods in the title of his main treatise.

The quadratic formula expresses concisely the solutions of all quadratic equations

Algebra began to be a specific area only with François Viète (1540–1603), who introduced the use of letters (variables) for representing unknown or unspecified numbers. This allows describing concisely the operations that have to be done on the numbers represented by the variables.

Until the 19th century, algebra consisted mainly of the study of linear equations that is called 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, variables began to represent other things than numbers (such as matrices, modular integers, and geometric transformations), on which some operations can operate, which are often generalizations of arithmetic operations. For dealing with this, the concept of algebraic structure was introduced, which consist of a set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. So, the scope of algebra evolved for becoming essentially the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra, the latter term being still used, mainly in an educational context, in opposition with 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 properties that are useful, and often fundamental, in many areas of mathematics. Their study are nowadays autonomous parts of algebra, which 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 algebraic structures as mathematical objects is the object of universal algebra and category theory. The latter applies to every mathematical structure (not only the 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.

Calculus, formerly called infinitesimal calculus, was introduced in the 17th century by Newton and Leibniz, independently and simultaneously. It is fundamentally the study of the relationship of two changing quantities, called variables, such that one depends on the other. Calculus was largely 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. Presently there are many subareas of analysis, some being shared with other areas of mathematics; they 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 of mathematics.

Discrete mathematics is a recently-emerging wide area of mathematics, which aggregates several previously existing areas, which deal with finite mathematical structures and processes where continuous variations are not allowed. These areas have in common that, because of the discrete aspect, the standard methods of calculus and mathematical analysis do not apply directly.[lower-alpha 3] These areas have also in common that algorithms, their implementation and their computational complexity play a major role. Despite they may have very different objects of study, they share often similar algorithmics methods.

It is generally considered that discrete mathematics include:

• 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 objects of various nature, 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 probabilities
• Game theory (although continuous games are also studied, most common games, such as chess and poker are of discrete nature)
• discrete optimization, including combinatorial optimization, integer programming, constraint programming

Four color theorem and optimal sphere packing are two major problems of discrete mathematics that have been solved since the second half of the 20th century. The open problem P=NP is important for discrete mathematics, since it solution would impact most parts of discrete mathematics, whichever would be the solution.

Combinatorics may be viewed primarily as the art of enumerating a prescribed set of objects. The history of combinatorics is quite long, with combinatorial techniques being developed in a variety of ancient societies. The usage of the term combinatorics in the modern mathematical sense was coined by Leibiniz in the 17th century AD,[11] though it was through the work of Euler that many of its modern tools, such as generating functions, were developed.

The breadth of combinatorics is quite large, and has been used to study enumeration problems arising in pure mathematics within algebra, number theory, probability theory, topology and geometry,[12] as well as many areas of applied math. Due to the wide variety of objects that may be enumerated, the theory is often subdivided into a variety of areas based on either the type of objects under consideration or the methods used, including:

• Algebraic combinatorics;
• Analytic combinatorics;
• Arithmetic combinatorics;
• Combinatorial design theory;
• Enumerative combinatorics;
• Extremal combinatorics;
• Geometric combinatorics;
• Infinitary combinatorics;
• Probabilistic combinatorics;
• Topological combinatorics;
• Ramsey theory;

Combinatorics is also frequently used in graph theory, as well as forming one of the basic tools used in the analysis of algorithms.

These subjects belong to mathematics since the end of the 19th century. Before this period, sets were not considered as mathematical objects, and logic, although used for mathematical proofs, belonged to philosophy, and was not specifically studied by mathematicians.

Before the study of infinite sets by Georg Cantor, mathematicians were reluctant to consider collections that are actually infinite, and considered infinity as the result of an endless enumeration. Cantor's work offended many mathematicians not only by considering actually infinite sets, but also by showing that this implies different sizes of infinity (see Cantor's diagonal argument) and the existence of mathematical objects that cannot be computed, and not even be 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, it appeared in various areas of mathematics that the former intuitive definitions of the basic mathematical objects were insufficient for insuring 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 is the origin of the foundational crisis of mathematics.[13] It has been eventually solved in the mainstream of mathematics by systematize 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 as 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 finding 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 of the foundations of the mathematics was challenged during the first half of the 20th century by mathematicians leaded by L. E. J. Brouwer who promoted an intuitionistic logic that excludes 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 theory), 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.[14]

Applied mathematics concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge. The term applied mathematics also describes the professional specialty in which mathematicians work on practical problems; as a profession focused on practical problems, applied mathematics focuses on the "formulation, study, and use of mathematical models" in science, engineering, and other areas of mathematical practice.

In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics, where mathematics is developed primarily for its own sake. Thus, the activity of applied mathematics is vitally connected with research in pure mathematics.

Applied mathematics has significant overlap with the discipline of statistics, whose theory is formulated mathematically, especially with probability theory. Statisticians (working as part of a research project) "create data that makes sense" with random sampling and with randomized experiments;[15] the design of a statistical sample or experiment specifies the analysis of the data (before the data becomes available). When reconsidering data from experiments and samples or when analyzing data from observational studies, statisticians "make sense of the data" using the art of modelling and the theory of inference—with model selection and estimation; the estimated models and consequential predictions should be tested on 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.[16] Because of its use of optimization, the mathematical theory of statistics shares concerns with other decision sciences, such as operations research, control theory, and mathematical economics.[17]

Computational mathematics proposes and studies methods for solving 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 discretisation 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 word mathematics comes from Ancient Greek máthēma (μάθημα), meaning "that which is learnt,"[32] "what one gets to know," hence also "study" and "science". The word for "mathematics" came to have the narrower and more technical meaning "mathematical study" even in Classical times.[33] Its adjective is mathēmatikós (μαθηματικός), meaning "related to learning" or "studious," which likewise further came to mean "mathematical." In particular, mathēmatikḗ tékhnē (μαθηματικὴ τέχνη; Latin: ars mathematica) meant "the mathematical art."

Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "learners" rather than "mathematicians" in the modern sense.

In Latin, and in English until around 1700, the term mathematics more commonly meant "astrology" (or sometimes "astronomy") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This has resulted in several mistranslations. For example, Saint Augustine's warning that Christians should beware of mathematici, meaning astrologers, is sometimes mistranslated as a condemnation of mathematicians.[34]

The apparent plural form in English, like the French plural form les mathématiques (and the less commonly used singular derivative la mathématique), goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural ta mathēmatiká (τὰ μαθηματικά), used by Aristotle (384–322 BC), and meaning roughly "all things mathematical", although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after the pattern of physics and metaphysics, which were inherited from Greek.[35] In English, the noun mathematics takes a singular verb. It is often shortened to maths or, in North America, math.[36]

The front side of the Fields Medal

The most prestigious award in mathematics is the Fields Medal,[59][60] established in 1936 and awarded every four years (except around World War II) to as many as four individuals.[61][62] It is the mathematical equivalent of the Nobel Prize.[62]

Other prestigious awards include:

• The Abel Prize, instituted in 2002[63] and first awarded in 2003[64]
• The Chern Medal for lifetime achievement,[65] introduced in 2010[66]
• The Wolf Prize in Mathematics, also for lifetime achievement,[67] instituted in 1978[68]

A famous list of 23 open problems, called "Hilbert's problems", was compiled in 1900 by German mathematician David Hilbert.[69] This list achieved great celebrity among mathematicians[70], and at least thirteen of the problems (depending how some are interpreted) have now been solved.[69] A new list of seven important problems, titled the "Millennium Prize Problems", was published in 2000. Only one of them, the Riemann hypothesis, duplicates one of Hilbert's problems. A solution to any of these problems carries a 1 million dollar reward.[71] Currently, only one of these problems, the Poincaré conjecture, has been solved.[72]