Mathematics is the study of quantity Quantity 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 quantitative properties or attributes of things, structure Structure is a fundamental and sometimes intangible notion covering the recognition, observation, nature, and stability of patterns and relationships of entities. From a child's verbal description of a snowflake, to the detailed scientific analysis of the properties of magnetic fields, the concept of structure is an essential foundation of nearly, space Space 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 continuum known as spacetime. In mathematics spaces with, and change Calculus is a discipline in mathematics focused on limits, functions, derivatives, integrals, and infinite series, and which constitutes a major part of modern university education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus is the study of change, in. Mathematicians The publication of new discoveries in mathematics continues at an immense rate in hundreds of scientific journals. One of the most exciting recent developments was the proof of Fermat's Last Theorem by Andrew Wiles, following 350 years of the brightest mathematical minds attempting to settle the problem seek out patterns.[2][3]They formulate new conjectures In mathematics, a conjecture is a proposition that is unproven but appears correct and has not been disproven and establish truth by rigorous Rigour or rigor has a number of meanings in relation to intellectual life and discourse. These are separate from public and political applications with their suggestion of laws enforced to the letter, or political absolutism. A religion, too, may be worn lightly, or applied with rigour deduction Deductive 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 conditional is a necessary truth. Deductive from appropriately chosen axioms In 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. Therefore, its truth is taken for granted, and serves as a starting point for deducing and inferring other truths and definitions A definition is a passage describing the meaning of a word or phrase. The term to be defined is known as the definiendum (Latin: what is to be defined). The words which define it are known as the definiens (Latin: what defines). Definitions also occur in more formal languages (like mathematics), often for the sake of discussion within the text of.[4]

There is debate over whether mathematical objects such as numbers and points really exist or whether they are manmade. The mathematician Benjamin Peirce After graduating from Harvard, he remained as a tutor , and was subsequently appointed professor of mathematics in 1831. He added astronomy to his portfolio in 1842, and remained as Harvard professor until his death. In addition, he was instrumental in the development of Harvard's science curriculum, served as the college librarian, and was called mathematics "the science that draws necessary conclusions".[5] Albert Einstein Albert Einstein was an ethnically Jewish, German-born theoretical physicist. He is best known for his theories of special relativity and general relativity. Einstein received the 1921 Nobel Prize in Physics "for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect.", 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."[6]

Through the use of abstraction Abstraction 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 descriptions of equivalent phenomena and logical Logic, from the Greek λογική 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 fundamental place in philosophy. It became part of the classical trivium, a fundamental part of a reasoning Humans have the ability to engage in reasoning about their own reasoning. Different forms of such reflection on reasoning occur in different fields. In philosophy, the study of reasoning typically focuses on what makes reasoning efficient or inefficient, appropriate or inappropriate, good or bad. Philosophers do this by either examining the form, mathematics evolved from counting Counting 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 (starting with one for the first object and proceeding with an injective function from the remaining objects to the natural numbers starting from two), or for well-ordered objects, to find the, calculation The term is used in a variety of senses, from the very definite arithmetical calculation using an algorithm to the vague heuristics of calculating a strategy in a competition or calculating the chance of a successful relationship between two people, measurement The metre was standardized as the unit for length after the French revolution, and has since been adopted throughout most of the world, and the systematic study of the shapes The 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 (position and orientation in space; size) and motions In 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 first law also known as Inertia. An object's momentum of physical objects. Practical mathematics has been a human activity for as far back as written records go (see: History of Mathematics The 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). Rigorous arguments Logic, from the Greek λογική 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 fundamental place in philosophy. It became part of the classical trivium, a fundamental part of a first appeared in Greek mathematics Greek 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 instruction".. The study of, most notably in Euclid Euclid , 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 (323–283 BC). His Elements is the most successful textbook and one of the most influential works in the history of mathematics,'s Elements Euclid'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 (axioms), propositions (theorems and constructions), and mathematical proofs of the propositions. The thirteen books cover Euclidean geometry. Mathematics continued to develop, in fitful bursts, until the Renaissance The 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. The term is also used more loosely to refer to the historic era, but since the changes of the Renaissance were not uniform across Europe, this is a general use of the, when mathematical innovations interacted with new scientific discoveries, leading to an acceleration in research that continues to the present day.[7]

Today, mathematics is used throughout the world as an essential tool in many fields, including natural science In 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. The term natural science is also used to distinguish those fields that use the scientific method to study nature from the social sciences and the humanities, which use the scientific, engineering Engineering 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 for Professional Development has defined, medicine Medicine 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 sciences The social sciences comprise academic disciplines concerned with the study of the social life of human groups, animals and individuals including anthropology, archeology, communication studies, cultural studies, demography, economics, human geography, history, linguistics, media studies, political science, psychology, social work, and sociology. Applied mathematics Historically, applied mathematics consisted principally of applied analysis, most notably differential equations, approximation theory , and applied probability. These areas of mathematics were intimately tied to the development of Newtonian Physics, and in fact the distinction between mathematicians and physicists was not sharply drawn before the, 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 mathematics Broadly speaking, pure mathematics is mathematics motivated entirely for reasons other than application. It is distinguished by its rigour, abstraction, and beauty. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards, 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.[8]

Contents

Etymology

The word "mathematics" comes from the Greek Ancient Greek is the historical stage in the development of the Greek language spanning across the Archaic , Classical (c. 5th–4th centuries BC), and Hellenistic (c. 3rd century BC–6th century AD) periods of ancient Greece and the ancient world. It is predated in the 2nd millennium BC by Mycenaean Greek. Its Hellenistic phase is known as Koine 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, μαθηματικὴ τέχνη (mathēmatikḗ tékhnē), in Latin Latin is an Italic language historically spoken in Latium and Ancient Rome. Through the Roman conquest, Latin spread throughout the Mediterranean and a large part of Europe. Romance languages such as Italian, French, Catalan, Romanian, Spanish, and Portuguese are descended from Latin, while many others, especially European languages, including ars mathematica, meant the mathematical art.

The apparent plural form in English English is a West Germanic language that originated in Anglo-Saxon England. 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, it has become the lingua franca in many parts of the world. It is, like the French French is a Romance language spoken, around the world, by more than 100 million people as a first language (mother tongue), by 190 million as a second language, and by about another 200 million people as an acquired foreign language, with significant speakers in 54 countries. Most native speakers of the language live in France, where the language plural form les mathématiques (and the less commonly used singular derivative la mathématique), goes back to the Latin neuter plural mathematica (Cicero Marcus 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), based on the Greek plural τα μαθηματικά (ta mathēmatiká), used by Aristotle Aristotle (384 BC – 322 BC) 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, and meaning roughly "all things mathematical".[9] In English, however, the noun mathematics takes singular verb forms. It is often shortened to math in English-speaking North America and maths elsewhere.

History

Main article: History of mathematics The 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 A quipu Quipu or khipu 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 also consisted of cotton cords with numeric and other values encoded by knots in a base ten positional system. Quipus may have just a, used by the Inca The Inca Empire was the largest empire in pre-Columbian America. The administrative, political and military center of the empire was located in Cusco in modern-day Peru. The Inca Empire arose from the highlands of Peru sometime in early 13th century. From 1438 to 1533, the Incas used a variety of methods, from conquest to peaceful assimilation, to to record numbers.

The evolution of mathematics might be seen as an ever-increasing series of abstractions Abstraction 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 descriptions of equivalent phenomena, or alternatively an expansion of subject matter. The first abstraction, which is shared by many animals[10], was probably that of numbers A 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. In addition to their use in counting and measuring, numerals are often used for labels ,: the realization that two apples and two oranges (for example) have something in common.

In addition to recognizing how to count Counting 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 (starting with one for the first object and proceeding with an injective function from the remaining objects to the natural numbers starting from two), or for well-ordered objects, to find the physical objects, prehistoric Prehistory is a term used to describe the period before written history. Paul Tournal originally coined the term Pré-historique in describing the finds he had made in the caves of southern France.[citation needed] It came into use in French in the 1830s to describe the time before writing, and the word "prehistoric" was introduced into peoples also recognized how to count abstract quantities, like time Time 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. Time has been a major subject of religion, philosophy, and science, but defining it in a non-controversial manner applicable to all fields of study has consistently eluded — days A 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, seasons 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. In temperate and polar regions, the seasons are marked by changes in the intensity of sunlight that reaches the Earth's surface, variations of which may cause animals to go into hibernation or to migrate, and, years A year is the amount of time it takes the Earth to make one revolution around the Sun. By extension, this can be applied to any planet: for example, a "Martian year" is the time in which Mars completes its own orbit. Elementary arithmetic Elementary 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 (addition, subtraction, multiplication and division) naturally followed.

Further steps needed writing or some other system for recording numbers such as tallies or the knotted strings called quipu used by the Inca to store numerical data. Numeral systems have been many and diverse, with the first known written numerals created by Egyptians in Middle Kingdom texts such as the Rhind Mathematical Papyrus. The Indus Valley civilization developed the modern decimal system, including the concept of zero.

Mayan numerals

From the beginning of recorded history, the major disciplines within mathematics arose out of the need to do calculations relating to taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics into the studies of quantity, structure, space, and change.

Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, 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 Society, "The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs."[11]

Inspiration, pure and applied mathematics, and aesthetics

Main article: Mathematical beauty Sir Isaac Newton (1643-1727), an inventor of infinitesimal calculus.

Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. For example, the physicist Richard Feynman invented the path integral formulation of quantum mechanics using a combination of mathematical reasoning and physical insight, and today's string theory, a still-developing scientific theory which attempts to unify the four fundamental forces of nature, continues to inspire new mathematics.[12] 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. The remarkable fact that even the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics."[13]

As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics: most mathematicians focus their research solely on one of these areas, and sometimes the choice is made as early as their undergraduate studies. Several areas of applied mathematics have merged with related traditions outside of mathematics and become disciplines in their own right, including statistics, operations research, and computer science.

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 aesthetics and inner beauty. Simplicity and generality are valued. There is beauty in a simple and elegant proof, such as Euclid's proof that there are infinitely many prime numbers, and in an elegant numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in A Mathematician's Apology expressed the belief that these aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics.[14] Mathematicians often strive to find proofs of theorems that are particularly elegant, a quest Paul Erdős often referred to as finding proofs from "The Book" in which God had written down his favorite proofs.[15][16] The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions.

Notation, language, and rigor

Leonhard Euler. Probably the most prolific mathematician of all times Main article: Mathematical notation

Most of the mathematical notation in use today was not invented until the 16th century.[17] Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery.[citation needed] In the 18th century, Euler 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 notation, modern mathematical notation has a strict syntax and encodes information that would be difficult to write in any other way.

Mathematical language can also be hard for beginners. Words such as or and only have more precise meanings than in everyday speech. Additionally, words such as open and field have been given specialized mathematical meanings. Mathematical jargon includes technical terms such as homeomorphism and integrable. 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".

The infinity symbol ∞ in several typefaces.

Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow from axioms by means of systematic reasoning. This is to avoid mistaken "theorems", based on fallible intuitions, of which many instances have occurred in the history of the subject.[18] The level of rigor expected in mathematics has varied over time: the Greeks expected detailed arguments, but at the time of Isaac Newton 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 proofs. Since large computations are hard to verify, such proofs may not be sufficiently rigorous.[19]

Axioms in traditional thought were "self-evident truths", but that conception is problematic. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (sufficiently powerful) axiomatic system has undecidable 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 theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.[20]

Mathematics as science

Carl Friedrich Gauss, himself known as the "prince of mathematicians", referred to mathematics as "the Queen of the Sciences".

Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences".[21] In the original Latin Regina Scientiarum, as well as in German 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 science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. Albert Einstein has 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."[6]

Many philosophers believe that mathematics is not experimentally falsifiable, and thus not a science according to the definition of Karl Popper.[22] 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 physics and biology, hypothetico-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently."[23] Other thinkers, notably Imre Lakatos, have applied a version of falsificationism to mathematics itself.

An alternative view is that certain scientific fields (such as theoretical physics) 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.[24] In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences. Experimental mathematics 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 method. In his 2002 book A New Kind of Science, Stephen Wolfram 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 arts; 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 engineering 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 universities 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 mathematics.

Mathematical awards are generally kept separate from their equivalents in science. The most prestigious award in mathematics is the Fields Medal,[25][26] established in 1936 and now awarded every 4 years. It is often considered the equivalent of science's Nobel Prizes. The Wolf Prize in Mathematics, instituted in 1978, recognizes lifetime achievement, and another major international award, the Abel Prize, 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 problems, called "Hilbert's problems", was compiled in 1900 by German mathematician David Hilbert. 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 Problems", was published in 2000. Solution of each of these problems carries a $1 million reward, and only one (the Riemann hypothesis) is duplicated in Hilbert's problems.

Fields of mathematics

An abacus, a simple calculating tool used since ancient times.

The major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry, and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations), to the empirical mathematics of the various sciences (applied mathematics), and more recently to the rigorous study of uncertainty.

Quantity

The study of quantity starts with numbers, first the familiar natural numbers and integers ("whole numbers") and arithmetical operations on them, which are characterized in arithmetic. The deeper properties of integers are studied in number theory, from which come such popular results as Fermat's Last Theorem. Number theory also holds two widely-considered unsolved problems: the twin prime conjecture and Goldbach's conjecture.

As the number system is further developed, the integers are recognized as a subset of the rational numbers ("fractions"). These, in turn, are contained within the real numbers, which are used to represent continuous quantities. Real numbers are generalized to complex numbers. These are the first steps of a hierarchy of numbers that goes on to include quarternions and octonions. Consideration of the natural numbers also leads to the transfinite numbers, which formalize the concept of counting to infinity. Another area of study is size, which leads to the cardinal numbers and then to another conception of infinity: the aleph numbers, which allow meaningful comparison of the size of infinitely large sets.

Natural numbers Integers Rational numbers Real numbers Complex numbers

Space

The study of space originates with geometry – in particular, Euclidean geometry. Trigonometry combines space and numbers, and encompasses the well-known Pythagorean theorem. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles and calculus on manifolds. Within algebraic geometry is the description of geometric objects as solution sets of polynomial equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may have been the greatest growth area in 20th century mathematics, and includes the long-standing Poincaré conjecture and the controversial four color theorem, whose only proof, by computer, has never been verified by a human.

Geometry Trigonometry Differential geometry Topology Fractal geometry

Change

Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a powerful tool to investigate it. Functions arise here, as a central concept describing a changing quantity. The rigorous study of real numbers and functions of a real variable is known as real analysis, with complex analysis the equivalent field for the complex numbers. The Riemann hypothesis, one of the most fundamental open questions in mathematics, is drawn from complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many problems lead naturally to relationships between a quantity and its rate of change, and these are studied as differential equations. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior.

Calculus Vector calculus Differential equations Dynamical systems Chaos theory

Structure

Many mathematical objects, such as sets of numbers and functions, exhibit internal structure. The structural properties of these objects are investigated in the study of groups, rings, fields and other abstract systems, which are themselves such objects. This is the field of abstract algebra. An important concept here is that of vectors, generalized to vector spaces, and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics: quantity, structure, and space. Vector calculus expands the field into a fourth fundamental area, that of change. Tensor calculus studies symmetry and the behavior of vectors under rotation. A number of ancient problems concerning Compass and straightedge constructions were finally solved using Galois theory.

Number theory Abstract algebra Group theory Order theory

Foundations and philosophy

In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed, as well as category theory which 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.[27] 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 theory and the Brouwer-Hilbert controversy.

Mathematical logic is concerned with setting mathematics on a rigorous axiomatic 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 system 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 theory, model theory, and proof theory, and is closely linked to theoretical computer science.

Mathematical logic Set theory Category theory

Discrete mathematics

Discrete mathematics is the common name for the fields of mathematics most generally useful in theoretical computer science. This includes computability theory, computational complexity theory, and information theory. Computability theory examines the limitations of various theoretical models of the computer, including the most powerful known model - the Turing machine. 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 compression and entropy.

As a relatively new field, discrete mathematics has a number of fundamental open problems. The most famous of these is the "P=NP?" problem, one of the Millennium Prize Problems.[28]

Combinatorics Theory of computation Cryptography Graph theory

Applied mathematics

Applied mathematics considers the use of abstract mathematical tools in solving concrete problems in the sciences, business, and other areas.

Applied mathematics has significant overlap with the discipline of statistics, whose theory is formulated mathematically, especially with probability theory. Statisticians (working as part of a research project) "create data that makes sense" with random sampling and with randomized experiments; 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 studies, statisticians "make sense of the data" using the art of modelling and the theory of inference—with model selection and estimation; the estimated models and consequential predictions should be tested on new data.[29]

Computational mathematics proposes and studies methods for solving mathematical problems that are typically too large for human numerical capacity. Numerical analysis studies methods for problems in analysis using ideas of functional analysis and techniques of approximation theory; numerical analysis includes the study of approximation and discretization broadly with special concern for rounding errors. Other areas of computational mathematics include computer algebra and symbolic computation.

Mathematical physics Mathematical fluid dynamics Numerical analysis Optimization
Probability theory Statistics Financial mathematics Game theory

Common misconceptions

Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. Every month, mathematicians publish many thousands of papers that embody new discoveries in the field.

Mathematics is not numerology; it is not concerned with "supernatural" properties of numbers. It is not accountancy; nor is it restricted to arithmetic.

Pseudomathematics is a form of mathematics-like activity undertaken outside academia, 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 pseudoscience and real science. The misconceptions involved are normally based on:

Like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne.

See also

Mathematics portal
Wikipedia:Books has a book on: Mathematics

Notes

  1. ^ No likeness or description of Euclid's physical appearance made during his lifetime survived antiquity. Therefore, Euclid's depiction in works of art depends on the artist's imagination (see Euclid).
  2. ^ Steen, L.A. (April 29, 1988). The Science of Patterns. Science, 240: 611–616. and summarized at
  3. ^ Devlin, Keith, Mathematics: The Science of Patterns: The Search for Order in Life, Mind and the Universe (Scientific American Paperback Library) 1996, ISBN 9780716750475
  4. ^ Jourdain
  5. ^ Peirce, p.97
  6. ^ a b Einstein, p. 28. The quote is Einstein's answer to the question: "how can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?" He, too, is concerned with The Unreasonable Effectiveness of Mathematics in the Natural Sciences.
  7. ^ Eves
  8. ^ Peterson
  9. ^ The Oxford Dictionary of English Etymology, Oxford English Dictionary
  10. ^ S. Dehaene, G. Dehaene-Lambertz and L. Cohen, Abstract representations of numbers in the animal and human brain, Trends in Neuroscience, Vol. 21 (8), Aug 1998, 355-361. http://dx.doi.org/10.1016/S0166-2236(98)01263-6.
  11. ^ Sevryuk
  12. ^ Johnson, Gerald W.; Lapidus, Michel L. (2002). The Feynman Integral and Feynman's Operational Calculus. Oxford University Press.
  13. ^ Eugene Wigner, 1960, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," Communications on Pure and Applied Mathematics 13(1): 1–14.
  14. ^ Hardy, G. H. (1940). A Mathematician's Apology. Cambridge University Press.
  15. ^ Gold, Bonnie; Simons, Rogers A. (2008). Proof and Other Dilemmas: Mathematics and Philosophy. MAA.
  16. ^ Aigner, Martin; Ziegler, Gunter M. (2001). Proofs from the Book. Springer.
  17. ^ Earliest Uses of Various Mathematical Symbols (Contains many further references)
  18. ^ See false proof for simple examples of what can go wrong in a formal proof. The history of the Four Color Theorem contains examples of false proofs accidentally accepted by other mathematicians at the time.
  19. ^ Ivars Peterson, The Mathematical Tourist, Freeman, 1988, ISBN 0-7167-1953-3. p. 4 "A few complain that the computer program can't be verified properly," (in reference to the Haken-Apple proof of the Four Color Theorem).
  20. ^ Patrick Suppes, Axiomatic Set Theory, Dover, 1972, ISBN 0-486-61630-4. p. 1, "Among the many branches of modern mathematics set theory occupies a unique place: with a few rare exceptions the entities which are studied and analyzed in mathematics may be regarded as certain particular sets or classes of objects."
  21. ^ Waltershausen
  22. ^ Shasha, Dennis Elliot; Lazere, Cathy A. (1998). Out of Their Minds: The Lives and Discoveries of 15 Great Computer Scientists. Springer. p. 228.
  23. ^ Popper 1995, p. 56
  24. ^ Ziman
  25. ^ "The Fields Medal is now indisputably the best known and most influential award in mathematics." Monastyrsky
  26. ^ Riehm
  27. ^ Luke Howard Hodgkin & Luke Hodgkin, A History of Mathematics, Oxford University Press, 2005.
  28. ^ Clay Mathematics Institute P=NP
  29. ^ Like other mathematical sciences such as physics and computer science, statistics is an autonomous discipline rather than a branch of applied mathematics. Like research physicists and computer scientists, research statisticians are mathematical scientists. Many statisticians have a degree in mathematics, and some statisticians are also mathematicians.

References

External links

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Learning resources from Wikiversity At Wikiversity you can learn more and teach others about Mathematics at: The School of Mathematics
Major fields of mathematics

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