Before the
Renaissance, mathematics was divided into two main areas:
arithmetic, regarding the study and manipulation of numbers, and
geometry, regarding the study of shapes. Some types of
pseudoscience, such as
numerology and
astrology, were not then clearly distinguished from mathematics. Beginning with the Renaissance, two more areas became predominant. New
mathematical notation led to
modern algebra which, roughly speaking, begins with the study and manipulation of
algebraic expressions.
Calculus, consisting of the two subfields
differential calculus and
integral calculus, originated with geometry but evolved into the study of
continuous functions, which model the typically
nonlinear relationships between varying quantities, as represented by
variables. This division into four main areasarithmetic, geometry, algebra, and calculusendured until the end of the 19th century. Other areas that were previously studied by mathematicians, such as
celestial mechanics and
solid mechanics, are now considered as belonging to physics. The subject of
combinatorics has been studied for much of recorded history, yet did not become a separate branch of mathematics until the 17th century. At the end of the 19th century, the
foundational crisis in mathematics and the systematic use of the
axiomatic method led to an explosion of new areas of mathematics. The study of numbers arguably dates back to ancient
Babylon and probably
China, but developed into a distinct discipline in
Ancient Greece. Two prominent early number theorists were
Euclid and
Diophantus of
Alexandria. The modern study of number theory in its abstract form is largely attributed to
Pierre de Fermat and
Leonhard Euler. The field came to full fruition with the contributions of
Adrien-Marie Legendre and
Carl Friedrich Gauss. Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics. A 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 despite considerable effort. Number theory includes several subareas, including
analytic number theory,
algebraic number theory,
geometry of numbers (method oriented),
Diophantine analysis, and
transcendence theory (problem oriented). A fundamental innovation was the ancient Greeks' introduction of the concept of
proofs, which require 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 are part of the definition of the subject of study (
axioms). This principle, foundational for all mathematics, was first elaborated for 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. Analytic geometry allows the study of
curves unrelated to circles and lines. Such curves can be defined as the
graph of functions, the study of which 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 Euclidean spaces of higher than three dimensions. Today's subareas of geometry include: Algebra is the art of manipulating
equations and formulas.
Diophantus (3rd century) and
al-Khwarizmi (9th century) were the two main precursors of algebra. Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one 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' that he used for naming one of these methods in the title of
his main treatise. Algebra became an area in its own right only with
François Viète (1540–1603), who introduced the use of variables for representing unknown or unspecified numbers. Variables allow 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 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. The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called
modern algebra or
abstract algebra, as established by the influence and works of
Emmy Noether, and popularized by
Van der Waerden's book
Moderne Algebra. 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: 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.
Calculus and analysis consists of elements such that all subsequent terms of a term 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 between variables that depend continuously 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: Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply.
Algorithmsespecially their
implementation and
computational complexityplay a major role in discrete mathematics. 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 potentially impact a large number of
computationally difficult problems. Discrete mathematics includes: 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 but by showing that this implies different sizes of infinity, per
Cantor's diagonal argument. 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 rigor. This became the foundational crisis of mathematics. 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. This
mathematical abstraction from reality is embodied in the modern philosophy of
formalism, as founded by
David Hilbert around 1910. 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 opinionsometimes called "intuition"to guide their study and proofs. The 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
consistent formal system that contains the natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system. 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.
Computational mathematics Computational mathematics is the study of
mathematical problems that are typically too large for human, numerical capacity. Part of computational mathematics involves
numerical analysis, which is the study of 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. == History ==