Mathematics is the
scienceScience is in its broadest sense to any systematic knowledge-base or prescriptive practice that is capable of resulting in a prediction or predictable type of outcome...
and study of
quantityQuantity is a kind of property which exists as magnitude or multitude. It is among the basic classes of things along with quality, substance, change, and relation. Quantity was first introduced as quantum, an entity having quantity. Being a fundamental term, quantity is used to refer to any type of...
,
structureStructure is a fundamental and sometimes intangible notion covering the recognition, observation, nature, and stability of patterns and relationships of entities...
,
spaceSpace is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of the boundless four-dimensional...
, and
changeCalculus is a discipline in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental...
.
MathematicianA mathematician is a person whose primary area of study and/or research is the field of mathematics. Mathematicians are concerned with particular problems related to logic, space, transformations, numbers and more general ideas which encompass these concepts...
s seek out patterns, formulate new
conjectureA conjecture is a proposition which is presumed to be real, true, or genuine, mostly based on inconclusive grounds. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds...
s, and establish truth by rigorous
deductionDeductive reasoning, sometimes called deductive logic, is reasoning which constructs or evaluates deductive arguments.In logic, an argument is said to be deductive when the truth of the conclusion is purported to follow necessarily or be a logical consequence of the premises and its corresponding...
from appropriately chosen
axiomIn traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject to necessary decision...
s and
definitionA definition is a formal passage describing the meaning of a term . The term to be defined is the definiendum . A term may have many subtly different senses or meanings...
s.
There is debate over whether mathematical objects such as
numberA number is a mathematical object used in counting and measuring. A notational symbol which represents a number is called a numeral, but in common usage the word number is used for both the abstract object and the symbol, as well as for the word for the number...
s and points exist naturally or are human creations. The mathematician
Benjamin PeirceBenjamin Peirce Benjamin Peirce Benjamin Peirce( purse, (April 4, 1809 – October 6, 1880) was an American mathematician who taught at Harvard University for forty years. He made contributions to celestial mechanics, number theory, algebra, and the philosophy of mathematics....
called mathematics "the science that draws necessary conclusions".
Albert EinsteinAlbert Einstein was a theoretical physicist. His many contributions to physics include the special and general theories of relativity, the founding of relativistic cosmology, the first post-Newtonian expansion, explaining the perihelion advance of Mercury, prediction of the deflection of...
, on the other hand, stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."
Through the use of
abstractionAbstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalising it so that it has wider applications or matching among other abstract...
and
logicLogic, from the Greek λογική is the art and science of reasoning. More specifically, it is defined by the Penguin Encyclopedia to be "The formal systematic study of the principles of valid inference and correct reasoning". As a discipline, logic dates back to Aristotle, who established its...
al
reasoningReasoning is the cognitive process of looking for reasons for beliefs, conclusions, actions or feelings.Humans have the ability to engage in reasoning about their own reasoning. Different forms of such reflection on reasoning occur in different fields...
, mathematics evolved from
countingCounting is the mathematical action of repeatedly adding one, usually to find out how many objects there are or to set aside a desired number of objects , or for well-ordered objects, to find the ordinal number of a...
,
calculationA calculation is a deliberate process for transforming one or more inputs into one or more results, with variable change.The term is used in a variety of senses, from the very definite arithmetical calculation of using an algorithm to the vague heuristics of calculating a strategy in a competition...
,
measurementIn science, measurement is the process of obtaining the magnitude of a quantity, such as length or mass, relative to a unit of measurement, such as a meter or a kilogram...
, and the systematic study of the
shapeThe shape of an object located in some space is the part of that space occupied by the object, as determined by its external boundary – abstracting from other properties such as colour, content, and material composition, as well as from the object's other spatial properties The shape (from...
s and
motionsIn physics, motion means a change in the location of a body. Change in motion is the result of applied force. Motion is typically described in terms of velocity, acceleration, displacement, and time. An object's velocity cannot change unless it is acted upon by a force, as described by Newton's...
of physical objects. Practical mathematics has been a human activity for as far back as
written recordsThe area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past....
exist.
Rigorous argumentsLogic, from the Greek λογική is the art and science of reasoning. More specifically, it is defined by the Penguin Encyclopedia to be "The formal systematic study of the principles of valid inference and correct reasoning". As a discipline, logic dates back to Aristotle, who established its...
first appeared in
Greek mathematicsGreek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 6th century BC to the 5th century AD around the Eastern shores of the Mediterranean. The word "mathematics" itself derives from the ancient Greek μάθημα , meaning "subject of...
, most notably in
EuclidEuclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician and is often referred to as the "Father of Geometry." He was active in Hellenistic Alexandria during the reign of Ptolemy I...
's
ElementsEuclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria circa 300 BC. It comprises a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...
. Mathematics continued to develop, in fitful bursts, until the
RenaissanceThe Renaissance was a cultural movement that spanned roughly the 14th to the 17th century, beginning in Florence in the Late Middle Ages and later spreading to the rest of Europe...
, when mathematical innovations interacted with new
scientific discoveriesThe timeline below shows the date of publication of major scientific theories and discoveries, along with the discoverer. In many cases, the discovery spanned several years.-BC:...
, leading to an acceleration in research that continues to the present day.
Today, mathematics is used throughout the world as an essential tool in many fields, including
natural scienceIn Science, the term natural science refers to a naturalistic approach to the study of the universe, which is understood as obeying rules or laws of natural origin...
,
engineeringEngineering is the discipline, art and profession of acquiring and applying technical, scientific and mathematical knowledge to design and implement materials, structures, machines, devices, systems, and processes that safely realize a desired objective or inventions.The American Engineers' Council...
,
medicineMedicine is the art and science of healing. It encompasses a range of health care practices evolved to maintain and restore health by the prevention and treatment of illness....
, and the
social sciencesThe social sciences are the fields of scientific knowledge and academic scholarship that study social groups and, more generally, human society. The social sciences initially were constituted of five fields: Jurisprudence and Amendment of the Law; Education; Health; Economy and Trade; Art...
.
Applied mathematicsApplied mathematics is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains.-Divisions of applied mathematics:...
, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new disciplines. Mathematicians also engage in
pure mathematicsBroadly speaking, pure mathematics is mathematics motivated entirely for reasons other than application. It is distinguished by its rigour, abstraction, and beauty...
, or mathematics for its own sake, without having any application in mind, although practical applications for what began as pure mathematics are often discovered later.
Etymology
The word "mathematics" comes from the Greek μάθημα (
máthēma), which means
learning,
study,
science, and additionally came to have the narrower and more technical meaning "mathematical study", even in Classical times. Its adjective is μαθηματικός (
mathēmatikós),
related to learning, or
studious, which likewise further came to mean
mathematical. In particular, (
mathēmatikḗ tékhnē), in
LatinLatin is an Italic language originally spoken in Latium and Ancient Rome. Through the Roman conquest, Latin spread throughout the Mediterranean and a large part of Europe...
ars mathematica, meant
the mathematical art.
The apparent plural form in
EnglishEnglish is a West Germanic language that developed in England during the Anglo-Saxon era. As a result of the military, economic, scientific, political, and cultural influence of the British Empire during the 18th, 19th, and early 20th centuries, and of the United States since the mid 20th century,...
, like the
FrenchFrench is a Romance language globally spoken by about 65 million people as a first language , by 50 million as a second language, and by about another 200 million people as an acquired foreign language, with significant speakers in 57 countries. Most native speakers of the language live in France,...
plural form
les mathématiques (and the less commonly used singular derivative
la mathématique), goes back to the Latin neuter plural
mathematica (
CiceroMarcus Tullius Cicero was a Roman philosopher, statesman, lawyer, political theorist, and Roman constitutionalist. Cicero is widely considered one of Rome's greatest orators and prose stylists.Cicero is generally perceived to be one of the most versatile minds of ancient Rome...
), based on the Greek plural τα μαθηματικά (
ta mathēmatiká), used by
AristotleAristotle was a Greek philosopher, a student of Plato and teacher of Alexander the Great. He wrote on many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, politics, government, ethics, biology, and zoology.Together with Plato and Socrates , Aristotle is one of...
, 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
physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
and
metaphysicsMetaphysics investigates principles of reality transcending those of any particular science. Cosmology and ontology are traditional branches of metaphysics. It is concerned with explaining the fundamental nature of being and the world...
, which were inherited from the Greek. In English, the noun
mathematics takes singular verb forms. It is often shortened to
maths, or
math in English-speaking North America.
History
The evolution of mathematics might be seen as an ever-increasing series of
abstractionsAbstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalising it so that it has wider applications or matching among other abstract...
, or alternatively an expansion of subject matter. The first abstraction, which is shared by many animals, was probably that of
numberA number is a mathematical object used in counting and measuring. A notational symbol which represents a number is called a numeral, but in common usage the word number is used for both the abstract object and the symbol, as well as for the word for the number...
s: the realization that two apples and two oranges (for example) have something in common.
In addition to recognizing how to
countCounting is the mathematical action of repeatedly adding one, usually to find out how many objects there are or to set aside a desired number of objects , or for well-ordered objects, to find the ordinal number of a...
physical objects,
prehistoricPrehistory is a term used to describe the period before recorded history. Paul Tournal originally coined the term Pré-historique in describing the finds he had made in the caves of southern France...
peoples also recognized how to count
abstract quantities, like
timeTime is a component of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify the motions of objects...
–
dayA day is a unit of time equivalent to approximately 24 hours. It is not an SI unit but it is accepted for use with SI. The SI unit of time is the second....
s,
seasonA season is a division of the year, marked by changes in weather.Seasons result from the yearly revolution of the Earth around the Sun and the tilt of the Earth's axis relative to the plane of revolution...
s,
yearA year is the amount of time it takes the Earth to make one revolution around the Sun...
s.
Elementary arithmeticElementary arithmetic is the most basic kind of mathematics: it concerns the operations of addition, subtraction, multiplication, and division. Most people learn elementary arithmetic in elementary school....
(
additionAddition is the mathematical process of combining quantities. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples. Therefore, 3 + 2 = 5...
,
subtractionSubtraction is one of the four basic arithmetic operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with...
,
multiplicationMultiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....
and
divisionright|thumb|200px|In mathematics, especially in elementary arithmetic, division is the arithmetic operation that is the inverse of multiplication...
) naturally followed.
Further steps needed
writingWriting is the representation of language in a textual medium through the use of a set of signs or symbols . It is distinguished from illustration, such as cave drawing and painting, and the recording of language via a non-textual medium such as magnetic tape audio.In Eurasia writing began as a...
or some other system for recording numbers such as
talliesA tally was an ancient memory aid device to record and document numbers, quantities, or even messages.Tally sticks first appear as notches carved on animal bones, in the Upper Paleolithic...
or the knotted strings called
quipuQuipus or khipus were recording devices used in the Inca Empire and its predecessor societies in the Andean region. A quipu usually consisted of colored spun and plied thread or strings from llama or alpaca hair. It could also be made of cotton cords...
used by the
IncaThe Inca civilization began as a tribe in the Cuzco area, where the legendary first Sapa Inca, Manco Capac founded the Kingdom of Cuzco around 1200. Under the leadership of the descendants of Manco Capac, the Inca state grew to absorb other Andean communities. In 1442, the Incas began a...
to store numerical data.
Numeral systemA numeral system is a writing system for expressing numbers, that is a mathematical notation for representing numbers of a given set, using graphemes or symbols in a consistent manner....
s have been many and diverse, with the first known written numerals created by
EgyptiansAncient Egypt was an ancient civilization of eastern North Africa, concentrated along the lower reaches of the Nile River in what is now the modern country of Egypt. The civilization coalesced around 3150 BC with the political unification of Upper and Lower Egypt under the first pharaoh, and...
in
Middle KingdomThe Middle Kingdom is the period in the history of ancient Egypt stretching from the establishment of the Eleventh Dynasty to the end of the Fourteenth Dynasty, between 2080 BC and 1640 BC....
texts such as the
Rhind Mathematical PapyrusThe Rhind Mathematical Papyrus , is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum. It dates to around 1650 B.C...
. The
Indus Valley civilizationThe Indus Valley Civilization was a Bronze Age civilization which centred mostly in the western part of the Indian Subcontinent and flourished around the Indus river basin....
developed the modern
decimalThe decimal numeral system has ten as its base. It is the most widely used numeral base.- Decimal notation :...
system, including the concept of zero.
The earliest uses of mathematics were in
tradingTrading can refer to:*Trade, the voluntary exchange of goods, services, or both**International trade, importing and exporting*Trader , a buyer and seller of financial instruments...
,
land measurementLand measurement is the general concept describing the application and theory of measurement of land. Land measurement is an integral quantitative element of Surveying....
,
paintingPainting is the practice of applying paint, pigment, color or other medium to a surface . In art, the term describes both the act and the result, which is called a painting. Paintings may have for their support such surfaces as walls, paper, canvas, wood, glass, lacquer, clay or concrete...
and
weavingWeaving is the textile art in which two distinct sets of yarns or threads, called the warp and the filling or weft , are interlaced with each other to form a fabric or cloth...
patterns and the recording of
timeTime is a component of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify the motions of objects...
and nothing much more advanced until around 3000BC onwards when the Babylonians and
EgyptiansAncient Egypt was an ancient civilization of eastern North Africa, concentrated along the lower reaches of the Nile River in what is now the modern country of Egypt. The civilization coalesced around 3150 BC with the political unification of Upper and Lower Egypt under the first pharaoh, and...
began using arithmetic, algebra and geometry for taxation and other financial calculations, building and construction and
astronomyAstronomy is the scientific study of celestial objects and phenomena that originate outside the Earth's atmosphere...
. The systematic study of mathematics in its own right began with the Ancient Greeks between 600 and 300BC.
Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and
scienceScience is in its broadest sense to any systematic knowledge-base or prescriptive practice that is capable of resulting in a prediction or predictable type of outcome...
, to the benefit of both. Mathematical discoveries have been made throughout history and continue to be made today. According to Mikhail B. Sevryuk, in the January 2006 issue of the
Bulletin of the American Mathematical SocietyBulletin of the American Mathematical Society is a quarterly mathematical journal published by the American Mathematical Society...
, "The number of papers and books included in the
Mathematical ReviewsMathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses of many articles in mathematics, statistics and theoretical computer science....
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
theoremIn mathematics, a theorem is a statement proved on the basis of previously accepted or established statements such as axioms. In formal mathematical logic, the concept of a theorem may be taken to mean a formula that can be derived according to the derivation rules of a fixed formal system.In...
s and their
proofsIn mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
."
Inspiration, pure and applied mathematics, and aesthetics
Mathematics arises from many different kinds of problems. At first these were found in
commerceCommerce is a division of trade or production which deals with the exchange of goods and services from producer to final consumer. It comprises the trading of something of economic value such as goods, services, information, or money between two or more entities...
,
land measurementLand measurement is the general concept describing the application and theory of measurement of land. Land measurement is an integral quantitative element of Surveying....
,
architectureFor a topical guide to this subject, see Outline of architecture. Architecture is the art and science of designing and constructing buildings and other physical structures for human shelter or use....
and later
astronomyAstronomy is the scientific study of celestial objects and phenomena that originate outside the Earth's atmosphere...
; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. For example, the
physicistA physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from sub-atomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...
Richard FeynmanRichard Phillips Feynman was an American physicist known for the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as work in particle physics...
invented the
path integral formulationThe path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics...
of
quantum mechanicsQuantum mechanics is a set of principles describing the physical reality at the atomic level of matter and the subatomic . These descriptions include the simultaneous wave-like and particle-like behavior of both matter and radiation...
using a combination of mathematical reasoning and physical insight, and today's
string theoryString theory is a developing branch of theoretical physics that combines quantum mechanics and general relativity into a quantum theory of gravity...
, a still-developing scientific theory which attempts to unify the four
fundamental forces of natureIn physics, fundamental interactions are the ways that the simplest particles in the universe interact with one other...
, continues to inspire new mathematics. Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. A distinction is often made between
pure mathematicsBroadly speaking, pure mathematics is mathematics motivated entirely for reasons other than application. It is distinguished by its rigour, abstraction, and beauty...
and
applied mathematicsApplied mathematics is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains.-Divisions of applied mathematics:...
. However pure mathematics topics often turn out to have applications, e.g.
number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
in
cryptographyCryptography is the practice and study of hiding information. Modern cryptography intersects the disciplines of mathematics, computer science, and engineering...
. This remarkable fact that even the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "
the unreasonable effectiveness of mathematicsIn 1960, the physicist Eugene Wigner published an article titled "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". In it, he observed that the mathematical structure of a physics theory often points the way to further advances in that theory and even to empirical...
."
As in most areas of study, the explosion of knowledge in the scientific age has led to specialization: there are now hundreds of specialized areas in mathematics and the latest
Mathematics Subject ClassificationThe Mathematics Subject Classification is an alphanumerical classification scheme formulated by the American Mathematical Society based on the coverage of two major reviewing databases Mathematical Reviews and Zentralblatt MATH...
runs to 46 pages. Several areas of applied mathematics have merged with related traditions outside of mathematics and become disciplines in their own right, including
statisticsStatistics is a branch of mathematics concerned with collecting and interpreting data. According to other definitions, it is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. Statisticians improve the quality of data with the...
,
operations researchOperations research or Quantitative management, as termed in the USA, Canada, South Africa and Australia, and operational research, as termed in Europe, is an interdisciplinary branch of applied mathematics that uses methods such as mathematical modeling, statistics, and algorithms to arrive at...
, and
computer scienceComputer science is the study of the theoretical foundations of information and computation, and of practical techniques for their implementation and application in computer systems. It is frequently described as the systematic study of algorithmic processes that create, describe and transform...
.
For those who are mathematically inclined, there is often a definite aesthetic aspect to much of mathematics. Many mathematicians talk about the
elegance of mathematics, its intrinsic
aestheticsAesthetics is commonly known as the study of sensory or sensori-emotional values, sometimes called judgments of sentiment and taste...
and inner
beautyBeauty is a characteristic of a person, animal, place, object, or idea that provides a perceptual experience of pleasure, meaning, or satisfaction. Beauty is studied as part of aesthetics, sociology, social psychology, and culture. As a cultural creation, beauty has been extremely commercialized...
.
SimplicitySimplicity is being simple. It is a property, condition, or quality which things can be judged to have. It usually relates to the burden which a thing puts on someone trying to explain or understand it. Something which is easy to understand or explain is simple, in contrast to something...
and generality are valued. There is beauty in a simple and elegant proof, such as
EuclidEuclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician and is often referred to as the "Father of Geometry." He was active in Hellenistic Alexandria during the reign of Ptolemy I...
's proof that there are infinitely many
prime numberIn mathematics, a prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. The first twenty-six prime numbers are:An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC. The number 1 is by definition not a prime number...
s, and in an elegant numerical method that speeds calculation, such as the
fast Fourier transformA fast Fourier transform is an efficient algorithm to compute the discrete Fourier transform and its inverse. There are many distinct FFT algorithms involving a wide range of mathematics, from simple complex-number arithmetic to group theory and number theory; this article gives an overview of...
.
G. H. HardyG. H. Hardy FRS was a prominent English mathematician, known for his achievements in number theory and mathematical analysis....
in
A Mathematician's ApologyA Mathematician's Apology is a 1940 essay by British mathematician G. H. Hardy. It concerns the aesthetics of mathematics with some personal content, and gives the layman an insight into the mind of a working mathematician.-Summary:...
expressed the belief that these aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. He identified criteria such as significance, unexpectedness, inevitability, and economy as factors that contribute to a mathematical aesthetic. Mathematicians often strive to find proofs of theorems that are particularly elegant, a quest
Paul ErdősPaul Erdős was an immensely prolific and famously eccentric Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators...
often referred to as finding proofs from "The Book" in which God had written down his favorite proofs. The popularity of
recreational mathematicsRecreational mathematics is an umbrella term, referring to mathematical puzzles and mathematical games.Not all problems in this field require a knowledge of advanced mathematics, and thus, recreational mathematics often piques the curiosity of non-mathematicians, and inspires their further study of...
is another sign of the pleasure many find in solving mathematical questions.
Notation, language, and rigor
Most of the mathematical notation in use today was not invented until the 16th century. Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery.
EulerLeonhard Paul Euler was a pioneering Swiss mathematician and physicist who spent most of his life in Russia and Germany. His surname is in English ; the common English pronunciation is incorrect....
(1707–1783) was responsible for many of the notations in use today. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. It is extremely compressed: a few symbols contain a great deal of information. Like
musical notationMusic notation or musical notation is any system which represents aurally perceived music, through the use of written symbols.-Western history:...
, modern mathematical notation has a strict syntax and encodes information that would be difficult to write in any other way.
Mathematical
languageA language is a system for encoding and decoding information. In its most common use, the term refers to so-called "natural languages" — the forms of communication considered peculiar to humankind. In linguistics the term is extended to refer to the human cognitive facility of creating and using...
can also be hard for beginners. Words such as
or and
only have more precise meanings than in everyday speech. Additionally, words such as
openIn mathematics, more specifically point-set topology and metric topology, the notion of an open set provides a fundamental way to speak of distance in a topological space, without explicitly defining a metric on the space...
and
fieldIn abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
have been given specialized mathematical meanings.
Mathematical jargonThe language of mathematics has a vast vocabulary of specialist and technical terms. It also has a certain amount of jargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject. Jargon often appears in lectures, and sometimes in print, as informal...
includes technical terms such as
homeomorphismIn the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between two topological spaces that has a continuous inverse function...
and
integrableIntegrability may refer to:* Riemann integrability; see Riemann integral* Lebesgue integrability; see Lebesgue integral* System integration * Interoperability * Integrable system...
. But there is a reason for special notation and technical jargon: mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as "rigor".
Mathematical proofIn mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
is fundamentally a matter of rigor. Mathematicians want their theorems to follow from axioms by means of systematic reasoning. This is to avoid mistaken "
theoremIn mathematics, a theorem is a statement proved on the basis of previously accepted or established statements such as axioms. In formal mathematical logic, the concept of a theorem may be taken to mean a formula that can be derived according to the derivation rules of a fixed formal system.In...
s", based on fallible intuitions, of which many instances have occurred in the history of the subject. The level of rigor expected in mathematics has varied over time: the Greeks expected detailed arguments, but at the time of
Isaac NewtonSir Isaac Newton FRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian who is perceived and considered by a substantial number of scholars and the general public as one of the most influential men in history...
the methods employed were less rigorous. Problems inherent in the definitions used by Newton would lead to a resurgence of careful analysis and formal proof in the 19th century. Today, mathematicians continue to argue among themselves about
computer-assisted proofA computer-assisted proof is a mathematical proof that has been at least partially generated by computer.Most computer-aided proofs to date have been implementations of large proofs-by-exhaustion of a mathematical theorem. The idea is to use a computer program to perform lengthy computations, and...
s. Since large computations are hard to verify, such proofs may not be sufficiently rigorous.
AxiomIn traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject to necessary decision...
s in traditional thought were "self-evident truths", but that conception is problematic. At a formal level, an axiom is just a string of
symbolsSymbolic logic is the area of mathematics which studies the purely formal properties of strings of symbols. The interest in this area springs from two sources. First, the symbols used in symbolic logic can be seen as representing the words used in philosophical logic...
, which has an intrinsic meaning only in the context of all derivable formulas of an
axiomatic systemIn mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A mathematical theory consists of an axiomatic system and all its derived theorems...
. It was the goal of
Hilbert's programIn mathematics, Hilbert's program, formulated by German mathematician David Hilbert in the 1920s, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies...
to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (sufficiently powerful) axiomatic system has
undecidableIn mathematical logic, independence refers to the unprovability of a sentence from other sentences.A sentence σ is independent of a given first-order theory T if T neither proves nor refutes σ; that is, it is impossible to prove σ from T, and it is also impossible to prove from T...
formulas; and so a final axiomatization of mathematics is impossible. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but
set theoryThe modern study of set theory was initiated by Cantor and Dedekind in the 1870s. After the discovery of paradoxes in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.The...
in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.
Mathematics as science
Carl Friedrich GaussJohann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics...
referred to mathematics as "the Queen of the Sciences". In the original Latin
Regina Scientiarum, as well as in
GermanGerman is a West Germanic language, thus related to and classified alongside English and Dutch. It is one of the world's major languages and the most widely spoken first language in the European Union. Around the world, German is spoken by approximately 105 million native speakers and also by...
Königin der Wissenschaften, the word corresponding to
science means (field of) knowledge. Indeed, this is also the original meaning in English, and there is no doubt that mathematics is in this sense a science. The specialization restricting the meaning to
natural science is of later date. If one considers
scienceScience is in its broadest sense to any systematic knowledge-base or prescriptive practice that is capable of resulting in a prediction or predictable type of outcome...
to be strictly about the physical world, then mathematics, or at least
pure mathematicsBroadly speaking, pure mathematics is mathematics motivated entirely for reasons other than application. It is distinguished by its rigour, abstraction, and beauty...
, is not a science.
Albert EinsteinAlbert Einstein was a theoretical physicist. His many contributions to physics include the special and general theories of relativity, the founding of relativistic cosmology, the first post-Newtonian expansion, explaining the perihelion advance of Mercury, prediction of the deflection of...
stated that
"as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."
Many philosophers believe that mathematics is not experimentally
falsifiableFalsifiability is the logical possibility that an assertion can be shown false by an observation or a physical experiment. That something is "falsifiable" does not mean it is false; rather, that if it is false, then this can be shown by observation or experiment. Falsifiability is an important...
, and thus not a science according to the definition of
Karl PopperSir Karl Raimund Popper, CH, FRS, FBA was an Austrian and British philosopher and a professor at the London School of Economics. He is considered one of the most influential philosophers of science of the 20th century, and also wrote extensively on social and political philosophy...
. However, in the 1930s important work in mathematical logic showed that mathematics cannot be reduced to logic, and Karl Popper concluded that "most mathematical theories are, like those of
physicsPhysics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...
and
biologyBiology is the natural science concerned with the study of life and living organisms, including their structure, function, growth, origin, evolution, distribution, and taxonomy...
,
hypotheticoA hypothesis is a proposed explanation for an observable phenomenon. The term derives from the Greek, ὑποτιθέναι - hypotithenai meaning "to put under" or "to suppose." For a hypothesis to be put forward as a scientific hypothesis, the scientific method requires that one can test it...
-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently." Other thinkers, notably
Imre LakatosImre Lakatos was a philosopher of mathematics and science, known for his thesis of the fallibility of mathematics and its 'methodology of proofs and refutations', and also for introducing the concept of the 'research programme' in his methodology of scientific research programmes.-Life:Lakatos was...
, have applied a version of falsificationism to mathematics itself.
An alternative view is that certain scientific fields (such as
theoretical physicsTheoretical physics is a branch of physics which employs mathematical models and abstractions of physics in an attempt to explain natural phenomena. Its central core is mathematical physics,[Sometimes mathematical physics and theoretical physics are used synonymously to refer to the...]
) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is
public knowledge and thus includes mathematics. In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions.
IntuitionIntuition is the apparent ability to acquire knowledge without inference or the use of reason. “The word ‘intuition’ comes from the Latin word 'intueri', which is often roughly translated as meaning ‘to look inside’ or ‘to contemplate’." Intuition provides us with beliefs that we cannot necessarily...
and
experimentIn scientific research, an experiment is a method of investigating causal relationships among variables, or to test a hypothesis. An experiment is a cornerstone of the empirical approach to acquiring data about the world and is used in both natural sciences and social sciences...
ation also play a role in the formulation of
conjectureA conjecture is a proposition which is presumed to be real, true, or genuine, mostly based on inconclusive grounds. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds...
s in both mathematics and the (other) sciences.
Experimental mathematicsExperimental mathematics is an approach to mathematics in which numerical computation is used to investigate mathematical objects and identify properties and patterns...
continues to grow in importance within mathematics, and computation and simulation are playing an increasing role in both the sciences and mathematics, weakening the objection that mathematics does not use the
scientific methodScientific method refers to a body of techniques for investigating phenomena, acquiring new knowledge, or correcting and integrating previous knowledge. To be termed scientific, a method of inquiry must be based on gathering observable, empirical and measurable evidence subject to specific...
. In his 2002 book
A New Kind of ScienceA New Kind of Science is a book by Stephen Wolfram, published in 2002. It contains an empirical and systematic study of computational systems such as cellular automata...
,
Stephen WolframStephen Wolfram is a British physicist, software developer, mathematician, computer programmer, author and businessman, known for his work in theoretical particle physics, cosmology, cellular automata, complexity theory, computer algebra and the Wolfram Alpha computational knowledge engine.-...
argues that computational mathematics deserves to be explored empirically as a scientific field in its own right.
The opinions of mathematicians on this matter are varied. Many mathematicians feel that to call their area a science is to downplay the importance of its aesthetic side, and its history in the traditional seven
liberal artsLiberal arts are the skills derived from the Classical education curriculum.-Definition:The term liberal arts denotes a curriculum that imparts general knowledge and develops the student’s rational thought and intellectual capabilities, unlike the professional, vocational, technical curricula...
; others feel that to ignore its connection to the sciences is to turn a blind eye to the fact that the interface between mathematics and its applications in science and
engineeringEngineering is the discipline, art and profession of acquiring and applying technical, scientific and mathematical knowledge to design and implement materials, structures, machines, devices, systems, and processes that safely realize a desired objective or inventions.The American Engineers' Council...
has driven much development in mathematics. One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematics is
created (as in art) or
discovered (as in science). It is common to see
universitiesA university is an institution of higher education and research, which grants academic degrees in a variety of subjects. A university provides both undergraduate education and postgraduate education...
divided into sections that include a division of
Science and Mathematics, indicating that the fields are seen as being allied but that they do not coincide. In practice, mathematicians are typically grouped with scientists at the gross level but separated at finer levels. This is one of many issues considered in the
philosophy of mathematicsThe philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of...
.
Mathematical awards are generally kept separate from their equivalents in science. The most prestigious award in mathematics is the , established in 1936 and now awarded every 4 years. It is often considered the equivalent of science's
Nobel PrizeThe Nobel Prize is a Sweden-based international monetary prize. The award was established by the 1895 will and estate of Swedish chemist and inventor Alfred Nobel. It was first awarded in Physics, Chemistry, Physiology or Medicine, Literature, and Peace in 1901...
s. The
Wolf Prize in MathematicsThe Wolf Prize in Mathematics is awarded almost annually by the Wolf Foundation. It is one of the six Wolf Prizes established by the Foundation and awarded since 1978; the others are in Agriculture, Chemistry, Medicine, Physics and Arts...
, instituted in 1978, recognizes lifetime achievement, and another major international award, the
Abel PrizeThe Abel Prize is an international prize presented annually by the King of Norway to one or more outstanding mathematicians. The prize is named after Norwegian mathematician Niels Henrik Abel . It has been often described as the "mathematician's Nobel" prize and is among the most prestigious...
, was introduced in 2003. These are awarded for a particular body of work, which may be innovation, or resolution of an outstanding problem in an established field. A famous list of 23 such
open problemIn science and mathematics, an open problem or an open question is a known problem that can be accurately stated, and has not yet been solved...
s, called "
Hilbert's problemsHilbert's problems are a list of twenty-three problems in mathematics published by German mathematician David Hilbert during 1900. The problems were all unsolved at the time, and several of them were very influential for 20th century mathematics...
", was compiled in 1900 by German mathematician
David HilbertDavid Hilbert was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. He discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry...
. This list achieved great celebrity among mathematicians, and at least nine of the problems have now been solved. A new list of seven important problems, titled the "
Millennium Prize ProblemsThe Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. Currently, six of the problems remain unsolved. A correct solution to any of the problems results in a US$1,000,000 prize being awarded by the institute...
", was published in 2000. Solution of each of these problems carries a $1 million reward, and only one (the
Riemann hypothesisIn mathematics, the Riemann hypothesis, proposed by , is a conjecture about the distribution of the zeros of the Riemann zeta-function stating that all non-trivial zeros of the Riemann zeta function have real part 1/2...
) is duplicated in Hilbert's problems.
Fields of mathematics
Mathematics can, broadly speaking, be subdivided into the study of quantity, structure, space, and change (i.e.
arithmeticArithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations, such as addition, subtraction, multiplication and division...
,
algebraAlgebra is the branch of mathematics concerning the study of the rules of operations and the things which can be constructed from them, including terms, polynomials, equations and algebraic structures...
,
geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, and
analysisMathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of pure mathematics most explicitly concerned with the notion of a limit, whether the limit of a sequence or the limit of a function...
). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to
logicMathematical logic is a subfield of mathematics with close connections to computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
, to
set theoryThe modern study of set theory was initiated by Cantor and Dedekind in the 1870s. After the discovery of paradoxes in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.The...
(
foundationsFoundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...
), to the empirical mathematics of the various sciences (
applied mathematicsApplied mathematics is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains.-Divisions of applied mathematics:...
), and more recently to the rigorous study of
uncertaintyUncertainty is a term used in subtly different ways in a number of fields, including philosophy, physics, statistics, economics, finance, insurance, psychology, sociology, engineering, and information science...
.
Quantity
The study of quantity starts with
numberA number is a mathematical object used in counting and measuring. A notational symbol which represents a number is called a numeral, but in common usage the word number is used for both the abstract object and the symbol, as well as for the word for the number...
s, first the familiar
natural numberIn mathematics, there are two conventions for the set of natural numbers: it is either the set of positive integers {, , , ...} according to the traditional definition or the set of non-negative integers {, 1, 2, ...} according to...
s and
integerThe integers are natural numbers including 0 and their negatives . They are numbers that can be written without a fractional or decimal component, and fall within the set {.....
s ("whole numbers") and arithmetical operations on them, which are characterized in
arithmeticArithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations, such as addition, subtraction, multiplication and division...
. The deeper properties of integers are studied in
number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
, from which come such popular results as
Fermat's Last TheoremIn number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two...
. Number theory also holds two problems widely considered unsolved: the
twin prime conjectureThe twin prime conjecture is a famous unsolved problem in number theory that involves prime numbers. It states:Such a pair of prime numbers is called a prime twin. The conjecture has been researched by many number theorists...
and
Goldbach's conjectureGoldbach's conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. It states:Expressing a given even number as a sum of two primes is called a Goldbach partition of the number...
.
As the number system is further developed, the integers are recognized as a
subsetIn mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide...
of the
rational numberIn mathematics, a rational number is any number that can be expressed as the quotient a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer corresponds to a rational number. The set of all rational numbers is usually denoted .Formally each rational...
s ("
fractionsA fraction is a number that can represent part of a whole.The earliest fractions were reciprocals of integers, symbols representing one half, one third, one quarter, and so on...
"). These, in turn, are contained within the
real numberIn mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real...
s, which are used to represent
continuousIn mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous. A continuous function with a continuous inverse function is called bicontinuous...
quantities. Real numbers are generalized to
complex numberA complex number, in mathematics, is a number comprising a real number and an imaginary number; it can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit, having the property that i
2 = −1...
s. These are the first steps of a hierarchy of numbers that goes on to include quarternions and
octonionIn mathematics, the octonions are a nonassociative and noncommutative extension of the quaternions. Their 8-dimensional normed division algebra over the real numbers is the widest possible that can be obtained from the Cayley-Dickson construction...
s. Consideration of the natural numbers also leads to the
transfinite numberTransfinite numbers are cardinal numbers or ordinal numbers that are larger than all finite numbers, yet not necessarily absolutely infinite. The term transfinite was coined by Georg Cantor, who wished to avoid some of the implications of the word infinite in connection with these objects, which...
s, which formalize the concept of "
infinityInfinity refers to several distinct concepts – usually linked to the idea of "without end" – which arise in philosophy, mathematics, and theology...
". Another area of study is size, which leads to the
cardinal numberIn mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
s and then to another conception of infinity: the
aleph numberIn set theory, the aleph numbers are a sequence of numbers used to represent the cardinality of infinite sets. They are named after the symbol used to denote them, the Hebrew letter aleph ....
s, which allow meaningful comparison of the size of infinitely large sets.
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s |
IntegerThe integers are natural numbers including 0 and their negatives . They are numbers that can be written without a fractional or decimal component, and fall within the set {.....
s |
Rational number In mathematics, a rational number is any number that can be expressed as the quotient a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer corresponds to a rational number. The set of all rational numbers is usually denoted .Formally each rational...
s |
Real number In mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real...
s |
Complex numberA complex number, in mathematics, is a number comprising a real number and an imaginary number; it can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit, having the property that i 2 = −1...
s |
Space
The study of space originates with
geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
– in particular,
Euclidean geometryEuclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry. It has been one of the most influential books in history, as much for its method as for its mathematical content...
.
TrigonometryTrigonometry is a branch of mathematics that deals with triangles, particularly those plane triangles in which one angle has 90 degrees...
is the branch of mathematics that deals with relationships between the sides and the angles of triangles and with the trigonometric functions; it combines space and numbers, and encompasses the well-known
Pythagorean theoremIn mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
. The modern study of space generalizes these ideas to include higher-dimensional geometry,
non-Euclidean geometriesA non-Euclidean geometry is characterized by a non-vanishing Riemann curvature tensor. Examples of non-Euclidean geometries include the hyperbolic and elliptic geometry, which are contrasted with a Euclidean geometry. The essential difference between Euclidean and non-Euclidean geometry is the...
(which play a central role in
general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics. It unifies special relativity and Newton's law of universal gravitation, and describes gravity as a...
) and
topologyTopology is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example deformations that involve stretching, but no tearing or gluing...
. Quantity and space both play a role in
analytic geometryAnalytic geometry, also known as coordinate geometry, analytical geometry, or Cartesian geometry, is the study of geometry using a coordinate system and the principles of algebra and analysis...
, differential geometry, and
algebraic geometryAlgebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such...
. Within differential geometry are the concepts of fiber bundles and calculus on
manifoldIn mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a certain dimension, called the dimension of the manifold....
s, in particular,
vectorVector calculus is a branch of mathematics concerned with differentiation and integration of vector fields. The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial differentiation and multiple...
and tensor calculus. Within algebraic geometry is the description of geometric objects as solution sets of
polynomialIn mathematics, a polynomial is a finite length expression constructed from variables and constants, by using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents...
equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space.
Lie groupIn mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s are used to study space, structure, and change.
TopologyTopology is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example deformations that involve stretching, but no tearing or gluing...
in all its many ramifications may have been the greatest growth area in 20th century mathematics, and includes the long-standing
Poincaré conjectureIn mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere among three-dimensional manifolds. It began as a popular, important conjecture, but is now considered a theorem to the satisfaction of the awarders of the Fields medal...
and the controversial
four color theoremIn mathematics, the four color theorem, or the four color map theorem, states that given any separation of a plane into contiguous regions, called a map, the regions can be colored using at most four colors so that no two adjacent regions have the same color...
, whose only proof, by computer, has never been verified by a human.
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|
TrigonometryTrigonometry is a branch of mathematics that deals with triangles, particularly those plane triangles in which one angle has 90 degrees...
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Differential geometry |
TopologyTopology is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example deformations that involve stretching, but no tearing or gluing...
|
Fractal geometry A fractal is "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
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Measure Theory |
Change
Understanding and describing change is a common theme in the
natural scienceIn Science, the term natural science refers to a naturalistic approach to the study of the universe, which is understood as obeying rules or laws of natural origin...
s, and
calculusCalculus is a discipline in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental...
was developed as a powerful tool to investigate it.
FunctionsIn mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...
arise here, as a central concept describing a changing quantity. The rigorous study of
real numberIn mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real...
s and functions of a real variable is known as
real analysisReal analysis, or theory of functions of a real variable is a branch of mathematical analysis dealing with the set of real numbers. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of...
, with
complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers...
the equivalent field for the
complex numberA complex number, in mathematics, is a number comprising a real number and an imaginary number; it can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit, having the property that i
2 = −1...
s.
Functional analysisFunctional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well...
focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is
quantum mechanicsQuantum mechanics is a set of principles describing the physical reality at the atomic level of matter and the subatomic . These descriptions include the simultaneous wave-like and particle-like behavior of both matter and radiation...
. Many problems lead naturally to relationships between a quantity and its rate of change, and these are studied as
differential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s. Many phenomena in nature can be described by
dynamical systemThe dynamical system concept is a mathematical formalization for any fixed "rule" which describes the time dependence of a point's position in its ambient space...
s;
chaos theoryChaos theory is a branch of mathematics which studies the behavior of certain dynamical systems that may be highly sensitive to initial conditions. This sensitivity is popularly referred to as the butterfly effect. As a result of this sensitivity, which manifests itself as an exponential growth of...
makes precise the ways in which many of these systems exhibit unpredictable yet still
deterministicIn mathematics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. Deterministic models thus produce the same output for a given starting condition...
behavior.
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| Calculus Calculus is a discipline in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental...
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Vector calculus Vector calculus is a branch of mathematics concerned with differentiation and integration of vector fields. The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial differentiation and multiple...
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Differential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
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Dynamical systemThe dynamical system concept is a mathematical formalization for any fixed "rule" which describes the time dependence of a point's position in its ambient space...
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Chaos theoryChaos theory is a branch of mathematics which studies the behavior of certain dynamical systems that may be highly sensitive to initial conditions. This sensitivity is popularly referred to as the butterfly effect. As a result of this sensitivity, which manifests itself as an exponential growth of...
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Structure
Many mathematical objects, such as sets of numbers and
functionsIn mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...
, exhibit internal structure. The structural properties of these objects are investigated in the study of
groupsIn mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
,
ringsIn mathematics, a ring is an algebraic structure consisting of a set together with two binary operations , where each operation combines two elements to form a third element...
,
fieldsIn abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
and other abstract systems, which are themselves such objects. This is the field of
abstract algebraAbstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
. An important concept here is that of vectors, generalized to
vector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s, and studied in
linear algebraLinear algebra is a branch of mathematics concerned with the study of vectors, vector spaces , linear maps , and systems of linear equations. Vector spaces are a central theme in modern mathematics; thus, linear algebra is widely used in both abstract algebra and functional analysis...
. The study of vectors combines three of the fundamental areas of mathematics: quantity, structure, and space. A number of ancient problems concerning Compass and straightedge constructions were finally solved using
Galois theoryIn mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
.
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Number theoryNumber theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study....
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Abstract algebra Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
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Group theoryIn mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
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Order theoryOrder theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of ordering, providing a framework for saying when one thing is "less than" or "precedes" another. This article gives a detailed introduction to the field and includes some of...
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Foundations and philosophy
In order to clarify the
foundations of mathematicsFoundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...
, the fields of
mathematical logicMathematical logic is a subfield of mathematics with close connections to computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
and
set theoryThe modern study of set theory was initiated by Cantor and Dedekind in the 1870s. After the discovery of paradoxes in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.The...
were developed. Mathematical logic includes the mathematical study of
logicLogic, from the Greek λογική is the art and science of reasoning. More specifically, it is defined by the Penguin Encyclopedia to be "The formal systematic study of the principles of valid inference and correct reasoning". As a discipline, logic dates back to Aristotle, who established its...
and the applications of formal logic to other areas of mathematics; set theory is the branch of mathematics that studies sets or collections of objects.
Category theoryIn mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions to objects linked in diagrams by morphisms or arrows....
, which deals in an abstract way with
mathematical structureIn mathematics, a structure on a set, or more generally a type, consists of additional mathematical objects that in some manner attach to the set, making it easier to visualize or work with, or endowing the collection with meaning or significance.A partial list of possible structures are measures,...
s and relationships between them, is still in development. The phrase "crisis of foundations" describes the search for a rigorous foundation for mathematics that took place from approximately 1900 to 1930. Some disagreement about the foundations of mathematics continues to present day. The crisis of foundations was stimulated by a number of controversies at the time, including the
controversy over Cantor's set theoryIn mathematical logic, the theory of infinite sets was first developed by Georg Cantor. Although this work has found near-universal acceptance in the mathematics community, it has been criticized in several areas by mathematicians and philosophers....
and the Brouwer-Hilbert controversy.
Mathematical logic is concerned with setting mathematics on a rigorous
axiomIn traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject to necessary decision...
atic framework, and studying the results of such a framework. As such, it is home to Gödel's second incompleteness theorem, perhaps the most widely celebrated result in logic, which (informally) implies that any
formal systemIn logic, a formal system consists of a formal language together with a deductive system which consists of...
that contains basic arithmetic, if
sound (meaning that all theorems that can be proven are true), is necessarily
incomplete (meaning that there are true theorems which cannot be proved
in that system). Gödel showed how to construct, whatever the given collection of number-theoretical axioms, a formal statement in the logic that is a true number-theoretical fact, but which does not follow from those axioms. Therefore no formal system is a true axiomatization of full number theory. Modern logic is divided into
recursion theoryRecursion theory, also called computability theory, is a branch of mathematical logic that originated in the 1930s with the study of computable functions and Turing degrees. The field has grown to include the study of generalized computability and definability...
,
model theoryIn mathematics, model theory is the study of mathematical structures such as groups, fields, graphs, or even universes of set theory, using tools from mathematical logic. A structure that gives meaning to the sentences of a formal language is called a model for the language...
, and
proof theoryProof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed...
, and is closely linked to
theoreticalTheoretical computer science is the collection of topics of computer science that focuses on the more abstract, logical and mathematical aspects of computing, such as the theory of computation, analysis of algorithms, and semantics of programming languages...
computer scienceComputer science is the study of the theoretical foundations of information and computation, and of practical techniques for their implementation and application in computer systems. It is frequently described as the systematic study of algorithmic processes that create, describe and transform...
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Set theory The modern study of set theory was initiated by Cantor and Dedekind in the 1870s. After the discovery of paradoxes in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.The...
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Category theory In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions to objects linked in diagrams by morphisms or arrows....
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Discrete mathematics
Discrete mathematicsDiscrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. Real numbers and rational numbers have the property that between any two numbers a third can be found, and consequently these numbers vary "smoothly"...
is the common name for the fields of mathematics most generally useful in
theoretical computer scienceTheoretical computer science is the collection of topics of computer science that focuses on the more abstract, logical and mathematical aspects of computing, such as the theory of computation, analysis of algorithms, and semantics of programming languages...
. This includes, on the computer science side, computability theory,
computational complexity theoryComputational complexity theory is a branch of the theory of computation in computer science that focuses on classifying computational problems according to their inherent difficulty. In this context, a computational problem is understood to be a task that is in principle amenable to being solved...
, and
information theoryInformation theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Historically, information theory was developed by Claude E. Shannon to find fundamental limits on compressing and reliably storing and communicating data...
. Computability theory examines the limitations of various theoretical models of the computer, including the most powerful known model – the
Turing machineA Turing machine is a theoretical device that manipulates symbols contained on a strip of tape. Despite its simplicity, a Turing machine can be adapted to simulate the logic of any computer algorithm, and is particularly useful in explaining the functions of a CPU inside of a computer. The Turing...
. Complexity theory is the study of tractability by computer; some problems, although theoretically solvable by computer, are so expensive in terms of time or space that solving them is likely to remain practically unfeasible, even with rapid advance of computer hardware. Finally, information theory is concerned with the amount of data that can be stored on a given medium, and hence deals with concepts such as
compressionIn computer science and information theory, data compression or source coding is the process of encoding information using fewer bits than an unencoded representation would use, through use of specific encoding schemes.As with any communication, compressed data communication only works when both...
and entropy.
On the purely mathematical side, this field includes
combinatoricsCombinatorics is a branch of pure mathematics concerning the study of discrete objects. It is related to many other areas of mathematics, such as algebra, probability theory, ergodic theory and geometry, as well as to applied subjects in computer science and statistical physics...
and
graph theoryIn mathematics and computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
.
As a relatively new field, discrete mathematics has a number of fundamental open problems. The most famous of these is the "
P=NP?The relationship between the complexity classes P and NP is an unsolved question in theoretical computer science and it is considered to be the most important problem in the field....
" problem, one of the
Millennium Prize ProblemsThe Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. Currently, six of the problems remain unsolved. A correct solution to any of the problems results in a US$1,000,000 prize being awarded by the institute...
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CombinatoricsCombinatorics is a branch of pure mathematics concerning the study of discrete objects. It is related to many other areas of mathematics, such as algebra, probability theory, ergodic theory and geometry, as well as to applied subjects in computer science and statistical physics...
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Theory of computation The theory of computation is the branch of computer science and mathematics that deals with whether and how efficiently problems can be solved on a model of computation, using an algorithm...
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CryptographyCryptography is the practice and study of hiding information. Modern cryptography intersects the disciplines of mathematics, computer science, and engineering...
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Graph theory In mathematics and computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
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Applied mathematics
Applied mathematicsApplied mathematics is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains.-Divisions of applied mathematics:...
considers the use of abstract mathematical tools in solving concrete problems in the
scienceScience is in its broadest sense to any systematic knowledge-base or prescriptive practice that is capable of resulting in a prediction or predictable type of outcome...
s,
businessA business is a legally recognized organization designed to provide goods and/or services to consumers. Businesses are predominant in capitalist economies, most being privately owned and formed to earn profit that will increase the wealth of its owners and grow the business itself...
, and other areas.
Applied mathematics has significant overlap with the discipline of
statisticsStatistics is a branch of mathematics concerned with collecting and interpreting data. According to other definitions, it is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. Statisticians improve the quality of data with the...
, whose theory is formulated mathematically, especially with
probability theoryProbability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
. Statisticians (working as part of a research project) "create data that makes sense" with random sampling and with randomized
experimentsDesign of experiments, or experimental design, is the design of all information-gathering exercises where variation is present, whether under the full control of the experimenter or not...
; the design of a statistical sample or experiment specifies the analysis of the data (before the data be available). When reconsidering data from experiments and samples or when analyzing data from
observational studiesIn statistics, an observational study draws inferences about the possible effect of a treatment on subjects, where the assignment of subjects into a treated group versus a control group is outside the control of the investigator...
, statisticians "make sense of the data" using the art of
modellingA statistical model is a set of mathematical equations which describe the behavior of an object of study in terms of random variables and their associated probability distributions...
and the theory of
inferenceStatistical inference or statistical induction comprises the use of statistics and random sampling to make inferences concerning some unknown aspect of a population...
– with
modelA statistical model is a set of mathematical equations which describe the behavior of an object of study in terms of random variables and their associated probability distributions...
selectionModel selection is the task of selecting a statistical model from a set of potential models, given data. In its most basic forms, this is one of the fundamental tasks of scientific inquiry. Determining the principle behind a series of observations is often linked directly to a mathematical model...
and
estimationEstimation is the calculated approximation of a result which is usable even if input data may be incomplete or uncertain.In statistics, see estimation theory, estimator....
; the estimated models and consequential predictions should be
testedA statistical hypothesis test is a method of making statistical decisions using experimental data. In statistics, a result is called statistically significant if it is unlikely to have occurred by chance...
on new data.
Computational mathematicsComputational mathematics involves mathematical research in areas of science where computing plays a central and essential role, emphasizing algorithms, numerical methods, and symbolic methods. Computation in the research is prominent. Computational mathematics emerged as a distinct part of applied...
proposes and studies methods for solving mathematical problems that are typically too large for human numerical capacity.
Numerical analysisNumerical analysis is the study of algorithms for the problems of continuous mathematics .One of the earliest mathematical writings is the Babylonian tablet YBC 7289, which gives a sexagesimal numerical approximation of , the length of the diagonal in a unit square.Being able to compute the sides...
studies methods for problems in analysis using ideas of
functional analysisFunctional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well...
and techniques of
approximation theoryIn mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby...
; numerical analysis includes the study of
approximationAn approximation is an inexact representation of something that is still close enough to be useful. Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws.Approximations may be used because...
and
discretizationIn mathematics, discretization concerns the process of transferring continuous models and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers...
broadly with special concern for rounding errors. Other areas of computational mathematics include computer algebra and
symbolic computationSymbolic computation or algebraic computation, relates to the use of machines, such as computers, to manipulate mathematical equations and expressions in symbolic form, as opposed to manipulating the approximations of specific numerical quantities represented by those symbols...
.
Common misconceptions
Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of
open problemsThis article lists some unsolved problems in mathematics. See individual articles for details and sources.- Millennium Prize Problems :Of the seven Millennium Prize Problems set by the Clay Mathematics Institute, the six yet to be solved are:* P versus NP...
. Every month, mathematicians publish many thousands of papers that embody new discoveries in the field.
Mathematics is not
numerologyNumerology is any of many systems, traditions or beliefs in a mystical or esoteric relationship between numbers and physical objects or living things....
; it is not concerned with "supernatural" properties of numbers. It is not
accountancyAccountancy or accounting is the art of communicating financial information about a business entity to users such as shareholders and managers...
; nor is it restricted to
arithmeticArithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations, such as addition, subtraction, multiplication and division...
.
PseudomathematicsPseudomathematics is a form of mathematics-like activity that does not work within the framework, definitions, rules, or rigor of formal mathematical models...
is a form of mathematics-like activity undertaken outside
academiaAcademia, Acadème, or the Academy are collective terms for the community of students and scholars engaged in higher education and research....
, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally accepted mathematics is similar to that between
pseudosciencePseudoscience is a methodology, belief, or practice that is claimed to be scientific, or that is made to appear to be scientific, but which does not adhere to an appropriate scientific methodology, lacks supporting evidence or plausibility, or otherwise lacks scientific status...
and real science. The misconceptions involved are normally based on:
- misunderstanding of the implications of mathematical rigor;
- attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review
Peer review is the process of subjecting an author's scholarly work, research, or ideas to the scrutiny of others who are experts in the same field. Peer review requires a community of experts in a given field, who are qualified and able to perform impartial review...
, often in the belief that the journal is biased against the author;
- lack of familiarity with, and therefore underestimation of, the existing literature.
Like
astronomyAstronomy is the scientific study of celestial objects and phenomena that originate outside the Earth's atmosphere...
, mathematics owes much to
amateur contributors such as
FermatPierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to modern calculus...
and
MersenneMarin Mersenne, Marin Mersennus or le Père Mersenne was a French theologian, philosopher, mathematician and music theorist, often referred to as the "father of acoustics" .-Life:...
.
See also