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Mathematics (from μάθημα máthēma, "knowledge, study, learning") includes the study of such topics as , structure,

(1963). 9780486417127, Dover.
, and .
(2019). 9781439049570, Cengage Learning.
(2019). 9780070667532, Tata McGraw–Hill Education.
(2019). 9783642195327, Springer.

Mathematicians seek and use to formulate new ; they resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and , mathematics developed from , , , and the systematic study of the and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.

first appeared in Greek mathematics, most notably in 's Elements. Since the pioneering work of (1858–1932), (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing by rigorous deduction from appropriately chosen and . Mathematics developed at a relatively slow pace until the , when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.Eves, p. 306

Mathematics is essential in many fields, including , engineering, medicine, finance, and the . Applied mathematics has led to entirely new mathematical disciplines, such as statistics and . Mathematicians engage in (mathematics for its own sake) without having any application in mind, but practical applications for what began as pure mathematics are often discovered later.Peterson, p. 12


History
The history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, which is shared by many animals, was probably that of numbers: the realization that a collection of two apples and a collection of two oranges (for example) have something in common, namely quantity of their members.

As evidenced by found on bone, in addition to recognizing how to physical objects, peoples may have also recognized how to count abstract quantities, like time – days, seasons, years.See, for example, Raymond L. Wilder, Evolution of Mathematical Concepts; an Elementary Study, passim

Evidence for more complex mathematics does not appear until around 3000 , when the and Egyptians began using , and for taxation and other financial calculations, for building and construction, and for .Kline 1990, Chapter 1. The most ancient mathematical texts from and are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry. It is in Babylonian mathematics that elementary arithmetic (, , and division) first appear in the archaeological record. The Babylonians also possessed a place-value system, and used a numeral system, still in use today for measuring angles and time.

Beginning in the 6th century BC with the , the began a systematic study of mathematics as a subject in its own right with Greek mathematics.

(1981). 9780486240732, Dover Publications. .
Around 300 BC, introduced the still used in mathematics today, consisting of definition, axiom, theorem, and proof. His textbook Elements is widely considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is often held to be (c. 287–212 BC) of Syracuse. He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the under the arc of a with the summation of an infinite series, in a manner not too dissimilar from modern calculus. Other notable achievements of Greek mathematics are (Apollonius of Perga, 3rd century BC), (Hipparchus of Nicaea (2nd century BC), and the beginnings of algebra (, 3rd century AD).

The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of and , and an early form of .

During the Golden Age of Islam, especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of . Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, and Sharaf al-Dīn al-Ṭūsī.

During the early modern period, mathematics began to develop at an accelerating pace in . The development of by Newton and Leibniz in the 17th century revolutionized mathematics. was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries. Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as , analysis, differential geometry, , , and . In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system that is consistent will contain unprovable propositions.

Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries 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 and their proofs."


Etymology
The word mathematics comes from μάθημα ( máthēma), meaning "that which is learnt", "what one gets to know", hence also "study" and "science". The word for "mathematics" came to have the narrower and more technical meaning "mathematical study" even in Classical times.Both senses can be found in Plato. Its adjective is μαθηματικός ( mathēmatikós), meaning "related to learning" or "studious", which likewise further came to mean "mathematical". In particular, μαθηματικὴ τέχνη ( mathēmatikḗ tékhnē), ars mathematica, meant "the mathematical art".

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

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

The apparent plural form in English, like the French plural form les mathématiques (and the less commonly used singular derivative la mathématique), goes back to the Latin neuter plural mathematica (), based on the Greek plural τὰ μαθηματικά ( ta mathēmatiká), used by (384–322 BC), and meaning roughly "all things mathematical"; although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after the pattern of and , which were inherited from Greek. The Oxford Dictionary of English Etymology, Oxford English Dictionary, sub "mathematics", "mathematic", "mathematics" In English, the noun mathematics takes a singular verb. It is often shortened to maths or, in North America, math. "maths, n." and "math, n.3". Oxford English Dictionary, on-line version (2012).


Definitions of mathematics
Mathematics has no generally accepted definition. defined mathematics as "the science of quantity", and this definition prevailed until the 18th century.
(2009). 9780080930589 .
(1564–1642) said, "The universe cannot be read until we have learned the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth."Marcus du Sautoy, A Brief History of Mathematics: 1. Newton and Leibniz , BBC Radio 4, September 27, 2010. Carl Friedrich Gauss (1777–1855) referred to mathematics as "the Queen of the Sciences".Waltershausen, p. 79 David Hilbert said of mathematics: "We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise."Hilbert, D. (1919–20), Natur und Mathematisches Erkennen: Vorlesungen, gehalten 1919–1920 in Göttingen. Nach der Ausarbeitung von Paul Bernays (Edited and with an English introduction by David E. Rowe), p. 14, Basel, Birkhäuser (1992). (1879–1955) 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."

Starting in the 19th century, when the study of mathematics increased in rigor and began to address abstract topics such as and projective geometry, which have no clear-cut relation to quantity and measurement, mathematicians and philosophers began to propose a variety of new definitions.

(1893). 9780821821022, American Mathematical Society (1991 reprint).
Some of these definitions emphasize the deductive character of much of mathematics, some emphasize its abstractness, some emphasize certain topics within mathematics. Today, no consensus on the definition of mathematics prevails, even among professionals. There is not even consensus on whether mathematics is an art or a science.
(2019). 9783034802291, Springer.
A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable. Some just say, "Mathematics is what mathematicians do."

Three leading types of definition of mathematics are called , , and formalist, each reflecting a different philosophical school of thought. All have severe problems, none has widespread acceptance, and no reconciliation seems possible.

An early definition of mathematics in terms of logic was 's "the science that draws necessary conclusions" (1870). In the Principia Mathematica, and Alfred North Whitehead advanced the philosophical program known as , and attempted to prove that all mathematical concepts, statements, and principles can be defined and proved entirely in terms of . A logicist definition of mathematics is Russell's "All Mathematics is Symbolic Logic" (1903).

definitions, developing from the philosophy of mathematician L. E. J. Brouwer, identify mathematics with certain mental phenomena. An example of an intuitionist definition is "Mathematics is the mental activity which consists in carrying out constructs one after the other." A peculiarity of intuitionism is that it rejects some mathematical ideas considered valid according to other definitions. In particular, while other philosophies of mathematics allow objects that can be proved to exist even though they cannot be constructed, intuitionism allows only mathematical objects that one can actually construct.

Formalist definitions identify mathematics with its symbols and the rules for operating on them. defined mathematics simply as "the science of formal systems".

(2019). 9780444533685, Elsevier.
A is a set of symbols, or tokens, and some rules telling how the tokens may be combined into formulas. In formal systems, the word axiom has a special meaning, different from the ordinary meaning of "a self-evident truth". In formal systems, an axiom is a combination of tokens that is included in a given formal system without needing to be derived using the rules of the system.


Mathematics as science
The German mathematician Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences". More recently, Marcus du Sautoy has called mathematics "the Queen of Science ... the main driving force behind scientific discovery". In the original Latin Regina Scientiarum, as well as in German Königin der Wissenschaften, the word corresponding to science means a "field of knowledge", and this was the original meaning of "science" in English, also; mathematics is in this sense a field of knowledge. The specialization restricting the meaning of "science" to follows the rise of , which contrasted "natural science" to , the of inquiring from . The role of empirical experimentation and observation is negligible in mathematics, compared to natural sciences such as , , or . 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."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?" This question was inspired by 's paper "The Unreasonable Effectiveness of Mathematics in the Natural Sciences".

Many philosophers believe that mathematics is not experimentally , and thus not a science according to the definition of . However, in the 1930s Gödel's incompleteness theorems convinced many mathematicians that mathematics cannot be reduced to logic alone, and Karl Popper concluded that "most mathematical theories are, like those of and , -: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently."Popper 1995, p. 56 Other thinkers, notably , have applied a version of to mathematics itself. (1976), Proofs and Refutations. Cambridge: Cambridge University Press.

An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. 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 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.

The opinions of mathematicians on this matter are varied. Many mathematiciansSee, for example 's statement "Mathematics, rightly viewed, possesses not only truth, but supreme beauty ..." in his History of Western Philosophy feel that to call their area a science is to downplay the importance of its aesthetic side, and its history in the traditional seven ; 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.


Inspiration, pure and applied mathematics, and aesthetics
Mathematics arises from many different kinds of problems. At first these were found in commerce, , architecture and later ; today, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. For example, the invented the path integral formulation of quantum mechanics using a combination of mathematical reasoning and physical insight, and today's , a still-developing scientific theory which attempts to unify the four fundamental forces of nature, continues to inspire new mathematics.

Some mathematics is relevant only 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 and applied mathematics. However pure mathematics topics often turn out to have applications, e.g. in . This remarkable fact, that even the "purest" mathematics often turns out to have practical applications, is what has called "the unreasonable effectiveness of mathematics". 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 Classification 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 statistics, operations research, and .

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 and inner beauty. and generality are valued. There is beauty in a simple and elegant proof, such as 's proof that there are infinitely many , and in an elegant 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. He identified criteria such as significance, unexpectedness, inevitability, and economy as factors that contribute to a mathematical aesthetic.

(2019). 9780521427067, Cambridge University Press.
Mathematical research often seeks critical features of a mathematical object. A theorem expressed as a characterization of the object by these features is the prize. Examples of particularly succinct and revelatory mathematical arguments has been published in Proofs from THE BOOK.

The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions. And at the other social extreme, philosophers continue to find problems in philosophy of mathematics, such as the nature of mathematical proof.


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, limiting mathematical discovery.Kline, p. 140, on ; p. 261, on . (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. According to , this can be attributed to the fact that mathematical ideas are both more abstract and more encrypted than those of natural language.Oakley 2014, p. 16: "Focused problem solving in math and science is often more effortful than focused-mode thinking involving language and people. This may be because humans haven't evolved over the millennia to manipulate mathematical ideas, which are frequently more abstractly encrypted than those of conventional language." Unlike natural language, where people can often equate a word (such as cow) with the physical object it corresponds to, mathematical symbols are abstract, lacking any physical analog.Oakley 2014, p. 16: "What do I mean by abstractness? You can point to a real live cow chewing its cud in a pasture and equate it with the letters c–o–w on the page. But you can't point to a real live plus sign that the symbol '+' is modeled after – the idea underlying the plus sign is more abstract." Mathematical symbols are also more highly encrypted than regular words, meaning a single symbol can encode a number of different operations or ideas.Oakley 2014, p. 16: "By encryptedness, I mean that one symbol can stand for a number of different operations or ideas, just as the multiplication sign symbolizes repeated addition."

Mathematical language can be difficult to understand for beginners because even common terms, such as or and only, have a more precise meaning than they have in everyday speech, and other terms such as and field refer to specific mathematical ideas, not covered by their laymen's meanings. Mathematical language also includes many technical terms such as and that have no meaning outside of mathematics. Additionally, shorthand phrases such as iff for "if and only if" belong to mathematical jargon. There is a reason for special notation and technical vocabulary: mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as "rigor".

Mathematical proof is fundamentally a matter of . Mathematicians want their theorems to follow from axioms by means of systematic reasoning. This is to avoid mistaken "", 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 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. Misunderstanding the rigor is a cause for some of the common misconceptions of mathematics. 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.Ivars Peterson, The Mathematical Tourist, Freeman, 1988, . 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).

in traditional thought were "self-evident truths", but that conception is problematic."The method of 'postulating' what we want has many advantages; they are the same as the advantages of theft over honest toil." (1919), Introduction to Mathematical Philosophy, New York and London, p. 71. 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 . 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 of mathematics is impossible. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.Patrick Suppes, Axiomatic Set Theory, Dover, 1972, . 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."


Fields of mathematics
Mathematics can, broadly speaking, be subdivided into the study of quantity, structure, space, and change (i.e. , , , 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 (foundations), to the empirical mathematics of the various sciences (applied mathematics), and more recently to the rigorous study of . While some areas might seem unrelated, the Langlands program has found connections between areas previously thought unconnected, such as , and .


Foundations and philosophy
In order to clarify the foundations of mathematics, the fields of mathematical logic and were developed. Mathematical logic includes the mathematical study of 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. , which deals in an abstract way with mathematical structures 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.Luke Howard Hodgkin & Luke Hodgkin, A History of Mathematics, Oxford University Press, 2005. Some disagreement about the foundations of mathematics continues to the 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 within a rigorous framework, and studying the implications of such a framework. As such, it is home to Gödel's incompleteness theorems which (informally) imply that any effective that contains basic arithmetic, if sound (meaning that all theorems that can be proved are true), is necessarily incomplete (meaning that there are true theorems which cannot be proved in that system). Whatever finite collection of number-theoretical axioms is taken as a foundation, Gödel showed how to construct a formal statement that is a true number-theoretical fact, but which does not follow from those axioms. Therefore, no formal system is a complete axiomatization of full number theory. Modern logic is divided into , , and , and is closely linked to theoretical computer science, as well as to . In the context of recursion theory, the impossibility of a full axiomatization of number theory can also be formally demonstrated as a consequence of the MRDP theorem.

Theoretical computer science includes computability theory, computational complexity theory, and information theory. Computability theory examines the limitations of various theoretical models of the computer, including the most well-known model – the . 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 the rapid advancement of computer hardware. A famous problem is the "" problem, one of the Millennium Prize Problems. Clay Mathematics Institute, P=NP, claymath.org 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 and entropy.

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|p \Rightarrow q || || |
Theory of computation


Pure mathematics

Quantity
The study of quantity starts with numbers, first the familiar and ("whole numbers") and arithmetical operations on them, which are characterized in . The deeper properties of integers are studied in , from which come such popular results as Fermat's Last Theorem. The conjecture and Goldbach's conjecture are two unsolved problems in number theory.

As the number system is further developed, the integers are recognized as a of the ("fractions"). These, in turn, are contained within the , which are used to represent continuous quantities. Real numbers are generalized to . These are the first steps of a hierarchy of numbers that goes on to include and . Consideration of the natural numbers also leads to the transfinite numbers, which formalize the concept of "". According to the fundamental theorem of algebra all solutions of equations in one unknown with complex coefficients are complex numbers, regardless of degree. Another area of study is the size of sets, which is described with the . These include the , which allow meaningful comparison of the size of infinitely large sets.

>
|(0), 1, 2, 3,\ldots || \ldots,-2, -1, 0, 1, 2\,\ldots || -2, \frac{2}{3}, 1.21 || -e, \sqrt{2}, 3, \pi | 2, i, -2+3i, 2e^{i\frac{4\pi}{3}}


Structure
Many mathematical objects, such as sets of numbers and functions, exhibit internal structure as a consequence of operations or relations that are defined on the set. Mathematics then studies properties of those sets that can be expressed in terms of that structure; for instance studies properties of the set of that can be expressed in terms of operations. Moreover, it frequently happens that different such structured sets (or structures) exhibit similar properties, which makes it possible, by a further step of , to state for a class of structures, and then study at once the whole class of structures satisfying these axioms. Thus one can study groups, rings, fields and other abstract systems; together such studies (for structures defined by algebraic operations) constitute the domain of .

By its great generality, abstract algebra can often be applied to seemingly unrelated problems; for instance a number of ancient problems concerning compass and straightedge constructions were finally solved using , which involves field theory and group theory. Another example of an algebraic theory is , which is the general study of , whose elements called vectors have both quantity and direction, and can be used to model (relations between) points in space. This is one example of the phenomenon that the originally unrelated areas of and have very strong interactions in modern mathematics. studies ways of enumerating the number of objects that fit a given structure.

>
|\begin{matrix} (1,2,3) & (1,3,2) \\ (2,1,3) & (2,3,1) \\ (3,1,2) & (3,2,1) \end{matrix} || || || || |


Space
The study of space originates with  – in particular, Euclidean geometry, which combines space and numbers, and encompasses the well-known Pythagorean theorem. is the branch of mathematics that deals with relationships between the sides and the angles of triangles and with the trigonometric functions. 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 . Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. and discrete geometry were developed to solve problems in number theory and functional analysis but now are pursued with an eye on applications in optimization and computer science. Within differential geometry are the concepts of and calculus on , in particular, and . Within algebraic geometry is the description of geometric objects as solution sets of equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. are used to study space, structure, and change. in all its many ramifications may have been the greatest growth area in 20th-century mathematics; it includes point-set topology, set-theoretic topology, algebraic topology and differential topology. In particular, instances of modern-day topology are metrizability theory, axiomatic set theory, , and . Topology also includes the now solved Poincaré conjecture, and the still unsolved areas of the . Other results in geometry and topology, including the four color theorem and Kepler conjecture, have been proved only with the help of computers.

>
| || || || || |


Change
Understanding and describing change is a common theme in the , and was developed as a powerful tool to investigate it. Functions arise here, as a central concept describing a changing quantity. The rigorous study of and functions of a real variable is known as , with the equivalent field for the . 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 ; makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior.


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

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


Statistics and other decision sciences
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 and with randomized experiments;Rao, C.R. (1997) Statistics and Truth: Putting Chance to Work, World Scientific. 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 and estimation; the estimated models and consequential predictions should be tested on new data.

Statistical theory studies decision problems such as minimizing the () of a statistical action, such as using a procedure in, for example, parameter estimation, hypothesis testing, and selecting the best. In these traditional areas of mathematical statistics, a statistical-decision problem is formulated by minimizing an objective function, like expected loss or , under specific constraints: For example, designing a survey often involves minimizing the cost of estimating a population mean with a given level of confidence.

(1981). 9780471080732, Wiley.
Because of its use of optimization, the mathematical theory of statistics shares concerns with other , such as operations research, , and mathematical economics.:
(1994). 9780471948292, John Wiley. .


Computational mathematics
Computational mathematics proposes and studies methods for solving mathematical problems that are typically too large for human numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory; numerical analysis includes the study of and broadly with special concern for . Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially matrix and . Other areas of computational mathematics include and symbolic computation.


Mathematical awards
Arguably the most prestigious award in mathematics is the , established in 1936 and awarded every four years (except around World War II) to as many as four individuals. The Fields Medal is often considered a mathematical equivalent to the Nobel Prize.

The Wolf Prize in Mathematics, instituted in 1978, recognizes lifetime achievement, and another major international award, the , was instituted in 2003. The was introduced in 2010 to recognize lifetime achievement. These accolades are awarded in recognition of a particular body of work, which may be innovational, or provide a solution to an outstanding problem in an established field.

A famous list of 23 , called "Hilbert's problems", was compiled in 1900 by German mathematician . 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. Only one of them, the Riemann hypothesis, duplicates one of Hilbert's problems. A solution to any of these problems carries a $1 million reward.


See also
  • International Mathematical Olympiad
  • Lists of mathematics topics
  • Mathematical sciences
  • Mathematics and art
  • Mathematics education
  • National Museum of Mathematics
  • Philosophy of mathematics
  • Relationship between mathematics and physics
  • Science, Technology, Engineering, and Mathematics


Notes

Bibliography


Further reading
  • (2019). 9780195139198, Oxford University Press.
  • (1999). 9780395929681, Mariner Books.
  • (1997). 9780393040029, W. W. Norton & Company.
  •  – A translated and expanded version of a Soviet mathematics encyclopedia, in ten volumes. Also in paperback and on CD-ROM, and online.
  • (2019). 9780486432687, Dover Publications.

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