Mathematics
Mathematics
Mathematics
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related
structures, shapes and the spaces in which they are contained, and quantities and their changes.
These topics are represented in modern mathematics with the major subdisciplines of number
theory,[1] algebra,[2] geometry,[1] and analysis,[3][4] respectively. There is no general consensus
among mathematicians about a common definition for their academic discipline.
Most mathematical activity involves the discovery of properties of abstract objects and the use of
pure reason to prove them. These objects consist of either abstractions from nature or—in modern
mathematics—entities that are stipulated to have certain properties, called axioms. A proof
consists of a succession of applications of deductive rules to already established results. These
results include previously proved theorems, axioms, and—in case of abstraction from nature—
some basic properties that are considered true starting points of the theory under consideration.[5]
Mathematics is essential in the natural sciences, engineering, medicine, finance, computer science
and the social sciences. Although mathematics is extensively used for modeling phenomena, the
fundamental truths of mathematics are independent from any scientific experimentation. Some
areas of mathematics, such as statistics and game theory, are developed in close correlation with
their applications and are often grouped under applied mathematics. Other areas are developed
independently from any application (and are therefore called pure mathematics), but often later
find practical applications.[6][7] The problem of integer factorization, for example, which goes back
to Euclid in 300 BC, had no practical application before its use in the RSA cryptosystem, now
widely used for the security of computer networks.
Historically, the concept of a proof and its associated mathematical rigour first appeared in Greek
mathematics, most notably in Euclid's Elements.[8] Since its beginning, mathematics was
essentially divided into geometry and arithmetic (the manipulation of natural numbers and
fractions), until the 16th and 17th centuries, when algebra[a] and infinitesimal calculus were
introduced as new areas. Since then, the interaction between mathematical innovations and
scientific discoveries has led to a rapid lockstep increase in the development of both.[9] At the end
of the 19th century, the foundational crisis of mathematics led to the systematization of the
axiomatic method,[10] which heralded a dramatic increase in the number of mathematical areas
and their fields of application. The contemporary Mathematics Subject Classification lists more
than 60 first-level areas of mathematics.
Etymology
The word mathematics comes from Ancient Greek máthēma (μάθημα), meaning "that which is
learnt",[11] "what one gets to know", hence also "study" and "science". The word came to have the
narrower and more technical meaning of "mathematical study" even in Classical times.[12] Its
adjective is mathēmatikós (μαθηματικός), meaning "related to learning" or "studious", which
likewise further came to mean "mathematical".[13] In particular, mathēmatikḗ tékhnē
(μαθηματικὴ τέχνη; Latin: ars mathematica) meant "the mathematical art".[11]
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. The Pythagoreans were likely the first to constrain the use of the word to just
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the study of arithmetic and geometry. By the time of Aristotle (384–322 BC) this meaning was
fully established.[14]
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 change 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.[15]
The apparent plural form in English goes back to the Latin neuter plural mathematica (Cicero),
based on the Greek plural ta mathēmatiká (τὰ μαθηματικά) and means 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, inherited
from Greek.[16] In English, the noun mathematics takes a singular verb. It is often shortened to
maths or, in North America, math.[17]
Areas of mathematics
Before the Renaissance, mathematics was divided into two main areas: arithmetic, regarding the
manipulation of numbers, and geometry, regarding the study of shapes.[18] Some types of
pseudoscience, such as numerology and astrology, were not then clearly distinguished from
mathematics.[19]
During the Renaissance, two more areas appeared. Mathematical notation led to algebra which,
roughly speaking, consists of the study and the manipulation of formulas. Calculus, consisting of
the two subfields differential calculus and integral calculus, is the study of continuous functions,
which model the typically nonlinear relationships between varying quantities, as represented by
variables. This division into four main areas–arithmetic, geometry, algebra, calculus[20]–endured
until the end of the 19th century. Areas such as celestial mechanics and solid mechanics were then
studied by mathematicians, but now are considered as belonging to physics.[21] The subject of
combinatorics has been studied for much of recorded history, yet did not become a separate
branch of mathematics until the seventeenth century.[22]
At the end of the 19th century, the foundational crisis in mathematics and the resulting
systematization of the axiomatic method led to an explosion of new areas of mathematics.[23][10]
The 2020 Mathematics Subject Classification contains no less than sixty-three first-level areas.[24]
Some of these areas correspond to the older division, as is true regarding number theory (the
modern name for higher arithmetic) and geometry. Several other first-level areas have "geometry"
in their names or are otherwise commonly considered part of geometry. Algebra and calculus do
not appear as first-level areas but are respectively split into several first-level areas. Other first-
level areas emerged during the 20th century or had not previously been considered as
mathematics, such as mathematical logic and foundations.[25]
Number theory
Number theory began with the manipulation of numbers, that is, natural numbers and later
expanded to integers and rational numbers Number theory was once called arithmetic,
but nowadays this term is mostly used for numerical calculations.[26] Number theory dates back to
ancient Babylon and probably China. Two prominent early number theorists were Euclid of
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Geometry
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.[b][31]
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 constituted a
major change of paradigm: 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.
Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry was split
into two new subfields: synthetic geometry, which uses purely geometrical methods, and analytic
geometry, which uses coordinates systemically.[34]
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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.[31]
In the 19th century, mathematicians discovered non-Euclidean geometries, which do not follow
the parallel postulate. By questioning that postulate's truth, this discovery has been viewed as
joining Russell's paradox in 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.[35][10] In turn, the axiomatic method allows for the study of
various geometries obtained either by changing the axioms or by considering properties that do
not change under specific transformations of the space.[36]
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
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.[41] Variables allow
mathematicians to describe the operations that have to be done on the numbers represented using
mathematical formulas.
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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:[25]
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 purpose of universal
algebra and category theory.[44] 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.[45]
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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:[25]
Multivariable calculus
Functional analysis, where variables represent varying functions;
Integration, measure theory and potential theory, all strongly related with probability theory on
a continuum;
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
The four color theorem and optimal sphere packing were two
major problems of discrete mathematics solved in the second
half of the 20th century.[50] The P versus NP problem, which
remains open to this day, is also important for discrete A diagram representing a two-state
mathematics, since its solution would potentially impact a large Markov chain. The states are
number of computationally difficult problems.[51] represented by 'A' and 'E'. The
numbers are the probability of
Discrete mathematics includes:[25] flipping the state.
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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.[57] 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.[23] 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.[58] This mathematical abstraction from reality
is embodied in the modern philosophy of formalism, as founded by David Hilbert around 1910.[59]
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. 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.[60] 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.[61][62]
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.[25] 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.[63]
The field of statistics is a 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
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Computational mathematics
Computational mathematics is the study of mathematical problems that are typically too large for
human, numerical capacity.[68][69] 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.[70] 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
Ancient
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.[74] 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
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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.[87] Many notable mathematicians
from this period were Persian, such as Al-Khwarismi, Omar
Khayyam and Sharaf al-Dīn al-Ṭūsī.[88] The Greek and Arabic
mathematical texts were in turn translated to Latin during the
Middle Ages and made available in Europe.[89]
geometry to algebra, and the development of calculus by Isaac Newton (1642–1726/27) and
Gottfried Leibniz (1646–1716). Leonhard Euler (1707–1783), the most notable mathematician of
the 18th century, unified these innovations into a single corpus with a standardized terminology,
and completed them with the discovery and the proof of numerous theorems.
Mathematics has developed a rich terminology covering a broad range of fields that study the
properties of various abstract, idealized objects and how they interact. It is based on rigorous
definitions that provide a standard foundation for communication. An axiom or postulate is a
mathematical statement that is taken to be true without need of proof. If a mathematical statement
has yet to be proven (or disproven), it is termed a conjecture. Through a series of rigorous
arguments employing deductive reasoning, a statement that is proven to be true becomes a
theorem. A specialized theorem that is mainly used to prove another theorem is called a lemma. A
proven instance that forms part of a more general finding is termed a corollary.[96]
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Numerous technical terms used in mathematics are neologisms, such as polynomial and
homeomorphism.[97] Other technical terms are words of the common language that are used in an
accurate meaning that may differ slightly from their common meaning. For example, in
mathematics, "or" means "one, the other or both", while, in common language, it is either
ambiguous or means "one or the other but not both" (in mathematics, the latter is called "exclusive
or"). Finally, many mathematical terms are common words that are used with a completely
different meaning.[98] This may lead to sentences that are correct and true mathematical
assertions, but appear to be nonsense to people who do not have the required background. For
example, "every free module is flat" and "a field is always a ring".
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his law of gravitation. Moreover, most mathematicians were also scientists, and many scientists
were also mathematicians.[110] However, a notable exception occurred with the tradition of pure
mathematics in Ancient Greece.[111]
In the 19th century, mathematicians such as Karl Weierstrass and Richard Dedekind increasingly
focused their research on internal problems, that is, pure mathematics.[109][112] This led to split
mathematics into pure mathematics and applied mathematics, the latter being often considered
as having a lower value among mathematical purists. However, the lines between the two are
frequently blurred.[113]
The aftermath of World War II led to a surge in the development of applied mathematics in the US
and elsewhere.[114][115] Many of the theories developed for applications were found interesting
from the point of view of pure mathematics, and many results of pure mathematics were shown to
have applications outside mathematics; in turn, the study of these applications may give new
insights on the "pure theory".[116][117]
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.[118] 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 impossible to implement because of a computational
complexity that is much too high.[119] 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.[120]
In the present day, the distinction between pure and applied mathematics is more a question of
personal research aim of mathematicians than a division of mathematics into broad areas.[121][122]
The Mathematics Subject Classification has a section for "general applied mathematics" but does
not mention "pure mathematics".[25] However, these terms are still used in names of some
university departments, such as at the Faculty of Mathematics at the University of Cambridge.
Unreasonable effectiveness
The unreasonable effectiveness of mathematics is a phenomenon that was named and first made
explicit by physicist Eugene Wigner.[7] 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.[123] Examples of unexpected
applications of mathematical theories can be found in many areas of mathematics.
A notable 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.[124] A second 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.[125]
In the 19th century, the internal development of geometry (pure mathematics) led to definition
and study of 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
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A 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 In
both cases, the equations of the theories had unexplained solutions, which led to conjecture of the
existence of an unknown particle, and the search for these particles. In both cases, these particles
were discovered a few years later by specific experiments.[128][129][130]
Specific sciences
Physics
Computing
In return, computing has also become essential for obtaining new results. This is a group of
techniques known as experimental mathematics, which is the use of experimentation to discover
mathematical insights.[133] The most well-known example is the four-color theorem, which was
proven in 1976 with the help of a computer. This revolutionized traditional mathematics, where
the rule was that the mathematician should verify each part of the proof. In 1998, the Kepler
conjecture on sphere packing seemed to also be partially proven by computer. An international
team had since worked on writing a formal proof; it was finished (and verified) in 2015.[134]
Once written formally, a proof can be verified using a program called a proof assistant.[135] These
programs are useful in situations where one is uncertain about a proof's correctness.[135]
A major open problem in theoretical computer science is P versus NP. It is one of the seven
Millennium Prize Problems.[136]
Biology uses probability extensively – for example, in ecology or neurobiology.[137] Most of the
discussion of probability in biology, however, centers on the concept of evolutionary fitness.[137]
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Since the start of the 20th century, chemistry has used computing to model molecules in three
dimensions. It turns out that the form of macromolecules in biology is variable and determines the
action. Such modeling uses Euclidean geometry; neighboring atoms form a polyhedron whose
distances and angles are fixed by the laws of interaction.
Earth sciences
Structural geology and climatology use probabilistic models to predict the risk of natural
catastrophes. Similarly, meteorology, oceanography, and planetology also use mathematics due to
their heavy use of models.
Social sciences
Areas of mathematics used in the social sciences include probability/statistics and differential
equations (stochastic or deterministic). These areas are used in fields such as sociology,
psychology, economics, finance, and linguistics.
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choices.[144] Also, as shown in laboratory experiments, people care about fairness and sometimes
altruism, not just personal gain.[144] According to critics, mathematization is a veneer that allows
for the material's scientific valorization.
At the start of the 20th century, there was a movement to express historical movements in
formulas. In 1922, Nikolai Kondratiev discerned the ~50-year-long Kondratiev cycle, which
explains phases of economic growth or crisis.[145] Towards the end of the 19th century, Nicolas-
Remi Brück and Charles Henri Lagrange had extended their analysis into geopolitics. They wanted
to establish the historical existence of vast movements that took peoples to their apogee, then to
their decline.[146] More recently, Peter Turchin has been working on developing cliodynamics
since the 1990s.[147] (In particular, he discovered the Turchin cycle, which predicts that violence
spikes in a short cycle of ~50-year intervals, superimposed over a longer cycle of ~200–300
years.[148])
Even so, mathematization of the social sciences is not without danger. In the controversial book
Fashionable Nonsense (1997), Sokal and Bricmont denounced the unfounded or abusive use of
scientific terminology, particularly from mathematics or physics, in the social sciences. The study
of complex systems (evolution of unemployment, business capital, demographic evolution of a
population, etc.) uses elementary mathematical knowledge. However, the choice of counting
criteria, particularly for unemployment, or of models can be subject to controversy.
Philosophy
Reality
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.[151]
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.[128]
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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.[153]
Proposed definitions
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.[157] 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.[158] 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.[159] Others take the perspective that mathematics is an
investigation of axiomatic set theory, as this study is now a foundational discipline for much of
modern mathematics.[160]
Rigor
Mathematical reasoning requires rigor. This means that the definitions must be absolutely
unambiguous and the proofs must be reducible to a succession of applications of inference rules,[f]
without any use of empirical evidence and intuition.[g][161] Rigorous reasoning is not specific to
mathematics, but, in mathematics, the standard of rigor is much higher than elsewhere. Despite
mathematics' concision, rigorous proofs can require hundreds of pages to express. The emergence
of computer-assisted proofs has allowed proof lengths to further expand,[h][162] such as the 255-
page Feit–Thompson theorem.[i] The result of this trend is a philosophy of the quasi-empiricist
proof that can not be considered infallible, but has a probability attached to it.[10]
The concept of rigor in mathematics dates back to ancient Greece, where their society encouraged
logical, deductive reasoning. However, this rigorous approach would tend to discourage
exploration of new approaches, such as irrational numbers and concepts of infinity. The method of
demonstrating rigorous proof was enhanced in the sixteenth century through the use of symbolic
notation. In the 18th century, social transition led to mathematicians earning their keep through
teaching, which led to more careful thinking about the underlying concepts of mathematics. This
produced more rigorous approaches, while transitioning from geometric methods to algebraic and
then arithmetic proofs.[10]
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At the end of the 19th century, it appeared that the definitions of the basic concepts of
mathematics were not accurate enough for avoiding paradoxes (non-Euclidean geometries and
Weierstrass function) and contradictions (Russell's paradox). This was solved by the inclusion of
axioms with the apodictic inference rules of mathematical theories; the re-introduction of
axiomatic method pioneered by the ancient Greeks.[10] It results that "rigor" is no more a relevant
concept in mathematics, as a proof is either correct or erroneous, and a "rigorous proof" is simply
a pleonasm. Where a special concept of rigor comes into play is in the socialized aspects of a proof,
wherein it may be demonstrably refuted by other mathematicians. After a proof has been accepted
for many years or even decades, it can then be considered as reliable.[163]
Nevertheless, the concept of "rigor" may remain useful for teaching to beginners what is a
mathematical proof.[164]
Education
Mathematics has a remarkable ability to cross cultural boundaries and time periods. As a human
activity, the practice of mathematics has a social side, which includes education, careers,
recognition, popularization, and so on. In education, mathematics is a core part of the curriculum
and forms an important element of the STEM academic disciplines. Prominent careers for
professional mathematicians include math teacher or professor, statistician, actuary, financial
analyst, economist, accountant, commodity trader, or computer consultant.[165]
Archaeological evidence shows that instruction in mathematics occurred as early as the second
millennium BCE in ancient Babylonia.[166] Comparable evidence has been unearthed for scribal
mathematics training in the ancient Near East and then for the Greco-Roman world starting
around 300 BCE.[167] The oldest known mathematics textbook is the Rhind papyrus, dated from
c. 1650 BCE in Egypt.[168] Due to a scarcity of books, mathematical teachings in ancient India were
communicated using memorized oral tradition since the Vedic period (c. 1500 – c. 500 BCE).[169]
In Imperial China during the Tang dynasty (618–907 CE), a mathematics curriculum was adopted
for the civil service exam to join the state bureaucracy.[170]
Following the Dark Ages, mathematics education in Europe was provided by religious schools as
part of the Quadrivium. Formal instruction in pedagogy began with Jesuit schools in the 16th and
17th century. Most mathematical curriculum remained at a basic and practical level until the
nineteenth century, when it began to flourish in France and Germany. The oldest journal
addressing instruction in mathematics was L'Enseignement Mathématique, which began
publication in 1899.[171] The Western advancements in science and technology led to the
establishment of centralized education systems in many nation-states, with mathematics as a core
component—initially for its military applications.[172] While the content of courses varies, in the
present day nearly all countries teach mathematics to students for significant amounts of time.[173]
During school, mathematical capabilities and positive expectations have a strong association with
career interest in the field. Extrinsic factors such as feedback motivation by teachers, parents, and
peer groups can influence the level of interest in mathematics.[174] Some students studying math
may develop an apprehension or fear about their performance in the subject. This is known as
math anxiety or math phobia, and is considered the most prominent of the disorders impacting
academic performance. Math anxiety can develop due to various factors such as parental and
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teacher attitudes, social stereotypes, and personal traits. Help to counteract the anxiety can come
from changes in instructional approaches, by interactions with parents and teachers, and by
tailored treatments for the individual.[175]
The validity of a mathematical theorem relies only on the rigor of its proof, which could
theoretically be done automatically by a computer program. This does not mean that there is no
place for creativity in a mathematical work. On the contrary, many important mathematical results
(theorems) are solutions of problems that other mathematicians failed to solve, and the invention
of a way for solving them may be a fundamental way of the solving process.[176][177] An extreme
example is Apery's theorem: Roger Apery provided only the ideas for a proof, and the formal proof
was given only several months later by three other mathematicians.[178]
Creativity and rigor are not the only psychological aspects of the activity of mathematicians. Some
mathematicians can see their activity as a game, more specifically as solving puzzles.[179] This
aspect of mathematical activity is emphasized in recreational mathematics.
Mathematicians can find an aesthetic value to mathematics. Like beauty, it is hard to define, it is
commonly related to elegance, which involves qualities like simplicity, symmetry, completeness,
and generality. G. H. Hardy in A Mathematician's Apology expressed the belief that the aesthetic
considerations are, in themselves, sufficient to justify the study of pure mathematics. He also
identified other criteria such as significance, unexpectedness, and inevitability, which contribute to
mathematical aesthetic.[180] Paul Erdős expressed this sentiment more ironically by speaking of
"The Book", a supposed divine collection of the most beautiful proofs. The 1998 book Proofs from
THE BOOK, inspired by Erdős, is a collection of particularly succinct and revelatory mathematical
arguments. Some examples of particularly elegant results included are Euclid's proof that there are
infinitely many prime numbers and the fast Fourier transform for harmonic analysis.[181]
Some feel that to consider mathematics a science is to downplay its artistry and history in the
seven traditional liberal arts.[182] One way this difference of viewpoint plays out is in the
philosophical debate as to whether mathematical results are created (as in art) or discovered (as in
science).[128] The popularity of recreational mathematics is another sign of the pleasure many find
in solving mathematical questions.
In the 20th century, the mathematician L. E. J. Brouwer even initiated a philosophical perspective
known as intuitionism, which primarily identifies mathematics with certain creative processes in
the mind.[59] Intuitionism is in turn one flavor of a stance known as constructivism, which only
considers a mathematical object valid if it can be directly constructed, not merely guaranteed by
logic indirectly. This leads committed constructivists to reject certain results, particularly
arguments like existential proofs based on the law of excluded middle.[183]
In the end, neither constructivism nor intuitionism displaced classical mathematics or achieved
mainstream acceptance. However, these programs have motivated specific developments, such as
intuitionistic logic and other foundational insights, which are appreciated in their own right.[183]
Cultural impact
Artistic expression
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Popularization
Popular mathematics is the act of presenting mathematics without technical terms.[191] Presenting
mathematics may be hard since the general public suffers from mathematical anxiety and
mathematical objects are highly abstract.[192] However, popular mathematics writing can
overcome this by using applications or cultural links.[193] Despite this, mathematics is rarely the
topic of popularization in printed or televised media.
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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.[208] To date, only one of these problems,
the Poincaré conjecture, has been solved.[209]
See also
Mathematics portal
References
Notes
a. Here, algebra is taken in its modern sense, which is, roughly speaking, the art of manipulating
formulas.
b. This includes conic sections, which are intersections of circular cylinders and planes.
c. However, some advanced methods of analysis are sometimes used; for example, methods of
complex analysis applied to generating series.
d. Like other mathematical sciences such as physics and computer science, statistics is an
autonomous discipline rather than a branch of applied mathematics. Like research physicists
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and computer scientists, research statisticians are mathematical scientists. Many statisticians
have a degree in mathematics, and some statisticians are also mathematicians.
e. Ada Lovelace, in the 1840s, is known for having written the first computer program ever in
collaboration with Charles Babbage
f. This does not mean to make explicit all inference rules that are used. On the contrary, this is
generally impossible, without computers and proof assistants. Even with this modern
technology, it may take years of human work for writing down a completely detailed proof.
g. This does not mean that empirical evidence and intuition are not needed for choosing the
theorems to be proved and to prove them.
h. For considering as reliable a large computation occurring in a proof, one generally requires two
computations using independent software
i. The book containing the complete proof has more than 1,000 pages.
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doi:10.1090/noti1223 (https://doi.org/10.1090%2Fnoti1223).
182. See, for example Bertrand Russell's statement "Mathematics, rightly viewed, possesses not
only truth, but supreme beauty ..." in his History of Western Philosophy. 1919. p. 60.
183. Iemhoff, Rosalie (2020). "Intuitionism in the Philosophy of Mathematics" (https://plato.stanford.
edu/archives/fall2020/entries/intuitionism). In Zalta, Edward N. (ed.). The Stanford
Encyclopedia of Philosophy (Fall 2020 ed.). Metaphysics Research Lab, Stanford University.
Archived (https://web.archive.org/web/20220421162527/https://plato.stanford.edu/archives/fall
2020/entries/intuitionism/) from the original on April 21, 2022. Retrieved April 2, 2022.
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Bibliography
Bouleau, Nicolas (1999). Philosophie des mathématiques et de la modélisation: Du chercheur
à l'ingénieur. L'Harmattan. ISBN 978-2-7384-8125-2.
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Further reading
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