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 characterised as *speculative mathematics*, and at variance with the trend towards meeting the needs of navigation, astronomy, physics, engineering, and so on. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ...
(17th century - 18th century - 19th century - more centuries) As a means of recording the passage of time, the 18th century refers to the century that lasted from 1701 through 1800. ...
Table of geography, hydrography, and navigation, from the 1728 Cyclopaedia. ...
A giant Hubble mosaic of the Crab Nebula, a supernova remnant. ...
Physics (from the Greek, (phÃºsis), nature and (phusikÃ©), knowledge of nature) is the science concerned with the discovery and understanding of the fundamental laws which govern matter, energy, space, and time. ...
Engineering is the design, analysis, and/or construction of works for practical purposes. ...
## History
### 19th century The term itself is enshrined in the full title of the Sadleirian Chair, founded (as a professorship) in the mid-nineteenth century. The idea of a separate discipline of *pure* mathematics may have emerged at that time. The generation of Gauss made no sweeping distinction of the kind, between *pure* and *applied*. In the following years, specialisation and professionalisation (particularly in the Weierstrass approach to mathematical analysis) started to make a rift more apparent. The Sadleirian Chair is a Professorship in pure mathematics at Cambridge University. ...
Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ...
(23 April 1776 â€“ 29 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including integral number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. ...
Karl Theodor Wilhelm Weierstraß (October 31, 1815 – February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. (The letter ß may be transliterated as ss; one often writes Weierstrass. ...
Analysis is the branch of mathematics most explicitly concerned with the notion of a limit, either the limit of a sequence or the limit of a function. ...
### 20th century At the start of the twentieth century mathematicians took up the axiomatic method, strongly influenced by David Hilbert's example. The logical formulation of **pure mathematics** suggested by Bertrand Russell in terms of a quantifier structure of propositions seemed more and more plausible, as large parts of mathematics became axiomatised and thus subject to the simple criteria of *rigorous proof*. (19th century - 20th century - 21st century - more centuries) Decades: 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s The 20th century lasted from 1901 to 2000 in the Gregorian calendar (often from (1900 to 1999 in common usage). ...
In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. ...
David Hilbert (January 23, 1862, KÃ¶nigsberg, East Prussia â€“ February 14, 1943, GÃ¶ttingen, Germany) was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. ...
Bertrand Arthur William Russell, 3rd Earl Russell OM FRS (18 May 1872 â€“ 2 February 1970), was a British philosopher, logician, and mathematician. ...
In language and logic, quantification is a construct that specifies the extent of validity of a predicate, that is the extent to which a predicate holds over a range of things. ...
Proposition is a term used in logic to describe the content of assertions. ...
For the medical term see rigor (medicine) Rigour (American English: rigor) has a number of meanings in relation to intellectual life and discourse. ...
In fact in an axiomatic setting *rigorous* adds nothing to the idea of *proof*. Pure mathematics, according to a view that continued to and through the Bourbaki group, is what is proved. **Pure mathematician** began to be a recognisable vocation, with access through a training. Nicolas Bourbaki is the pseudonym under which a group of mainly French 20th-century mathematicians wrote a series of books of exposition of modern advanced mathematics, beginning in 1935. ...
## Generality and abstraction Geometry has expanded to accommodate topology. The study of numbers, called algebra at the beginning undergraduate level, extends to abstract algebra at a more advanced level; and the study of functions, called calculus at the college freshman level becomes mathematical analysis and functional analysis at a more advanced level. Each of these branches of more *abstract* mathematics have many sub-specialties, and there are in fact many connections between pure mathematics and applied mathematics disciplines. Undeniably, though, a steep rise in abstraction was seen mid-century. Table of Geometry, from the 1728 Cyclopaedia. ...
A MÃ¶bius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ...
A number is an abstract entity that represents a count or measurement. ...
Algebra is a branch of mathematics concerning the study of structure, relation and quantity. ...
Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ...
Partial plot of a function f. ...
Calculus is the name given to a group of systematic methods of calculation, computation, and analysis in mathematics which use a common and specialized algebraic notation. ...
Analysis is the branch of mathematics most explicitly concerned with the notion of a limit, either the limit of a sequence or the limit of a function. ...
Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ...
Abstraction is the process of reducing the information content of a concept, typically in order to retain only information which is relevant for a particular purpose. ...
In practice, however, these developments led to a sharp divergence from physics, particularly from 1950 to 1980. Later this was criticised, for example by Vladimir Arnold, as too much Hilbert, not enough PoincarĂ©. The point does not yet seem to be settled (unlike the foundational controversies over set theory), in that string theory pulls one way, while discrete mathematics pulls back towards proof as central. Physics (from the Greek, (phÃºsis), nature and (phusikÃ©), knowledge of nature) is the science concerned with the discovery and understanding of the fundamental laws which govern matter, energy, space, and time. ...
Vladimir I. Arnold (Moscow, December 2001). ...
David Hilbert David Hilbert (January 23, 1862 – February 14, 1943) was a German mathematician born in Wehlau, near Königsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ...
Jules Henri PoincarÃ© (April 29, 1854 â€“ July 17, 1912) (IPA: [][1]), was one of Frances greatest mathematicians and theoretical physicists, and a philosopher of science. ...
Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...
Interaction in the subatomic world: world lines of pointlike particles in the Standard Model or a world sheet swept up by closed strings in string theory Whoever reads this and they arent asian (any Jason or An Ting will count as a non-asian) will be jinxed and cursed...
Discrete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally discrete, in the sense of not supporting or requiring the notion of continuity. ...
## Purism Mathematicians have always had differing opinions regarding the distinction between pure and applied mathematics. One of the most famous (but perhaps misunderstood) modern examples of the debate between the subfields can be found in G.H. Hardy's *A Mathematician's Apology*. It is widely believed that Hardy considered applied mathematics to be "ugly" and "dull". Although it is true that Hardy preferred pure mathematics, which he often compared to painting and poetry, Hardy saw the distinction between pure and applied mathematics to be simply that applied mathematics sought to express physical truth in a mathematical framework, whereas pure mathematics expressed truths that were independent of the physical world. However, Hardy made another distinction in mathematics, between what he called "real" mathematics, "which has permanent aesthetic value", and "the dull and elementary parts of mathematics" that have practical use. Hardy considered some physicists, such as Einstein and Dirac, to be among the "real" mathematicians, but at the time that he was writing the *Apology* he also considered general relativity and quantum mechanics to be "useless", which allowed him to hold the opinion that only "dull" mathematics was useful. Moreover, Hardy briefly admitted that--just as the application of matrix theory and group theory to physics had come unexpectedly--the time may come where some kinds of beautiful, "real" mathematics may be useful as well. G. H. Hardy Godfrey Harold Hardy (February 7, 1877 – December 1, 1947) was a prominent British mathematician, known for his achievements in number theory and mathematical analysis. ...
A Mathematicians Apology is a 1940 essay by British mathematician G. H. Hardy (ISBN 0521427061). ...
For building painting, see painter and decorator. ...
The Chinese poem Quatrain on Heavenly Mountain by Emperor Gaozong (Song Dynasty) Poetry (from the Greek , poiesis, making or creating) is a form of art in which language is used for its aesthetic qualities in addition to, or in lieu of, its ostensible meaning. ...
## Subfields in pure mathematics Analysis is concerned with the properties of functions. It deals with concepts such as continuity, limits, differentiation and integration, thus providing a rigorous foundation for the calculus of infinitesimals introduced by Newton and Leibniz in the 17th century. Real analysis studies functions of real numbers, while complex analysis extends the aforementioned concepts to functions of complex numbers. Analysis is the branch of mathematics most explicitly concerned with the notion of a limit, either the limit of a sequence or the limit of a function. ...
In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...
In mathematics, the limit of a function is a fundamental concept in mathematical analysis. ...
In mathematics, a derivative is the rate of change of a quantity. ...
In calculus, the integral of a function is an extension of the concept of a sum. ...
Real analysis is a branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ...
Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ...
Abstract algebra is not to be confused with the manipulation of formulae that is covered in secondary education. It studies sets together with binary operations defined on them. Sets and binary operations may be classified according to their properties: For instance, if a a operation is associative on a set which contains an identity element and inverses for each member of the set, the set and operation is concerned to be a group. Other structures include rings, fields and vector spaces. In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...
In mathematics, a binary operation, or binary operator, is a calculation involving two input quantities and one kind of a specific operation. ...
In mathematics, associativity is a property that a binary operation can have. ...
This picture illustrates how the hours in a clock form a group. ...
In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...
In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
The fundamental concept in linear algebra is that of a vector space or linear space. ...
Geometry is the study of shapes and space, in particular, groups of transformations that act on spaces. For example, projective geometry is about the group of projective transformations that act on the real projective plane. Geometry has been extended to topology, which deals with objects known as topological spaces and continuous maps between them. Topology is concerned with the way in which a space is connected than precise distances and angles. Table of Geometry, from the 1728 Cyclopaedia. ...
Projective geometry is a non-metrical form of geometry that emerged in the early 19th century. ...
A MÃ¶bius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ...
Number theory is the theory of the positive integers. It is based on ideas such as divisibility and congruence. Its fundamental theorem states that each positive integer has a unique prime factorization. It is perhaps the most accessible discipline in pure mathematics for the general public: for instance the Goldbach conjecture is easily stated (but is yet to be proved or disproved). Number 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. ...
The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ...
Modular arithmetic (sometimes called modulo arithmetic) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value â€” the modulus. ...
In number theory, the fundamental theorem of arithmetic (or unique factorization theorem) states that every natural number either is itself a prime number, or can be written as a unique product of prime numbers. ...
In mathematics, Goldbachs conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. ...
## Quotes - "There is no branch of mathematics, however abstract, which may not someday be applied to the phenomena of the real world."
- Nikolai Lobachevsky
- "God does not care about our mathematical difficulties - he integrates empirically"
- Albert Einstein
Nikolay Ivanovich Lobachevsky Nikolai Ivanovich Lobachevsky (Никола́й Ива́нович Лобаче́вский) (December 1, 1792 - February 24, 1856) was a Russian mathematician. ...
Albert Einstein ( ) (March 14, 1879 â€“ April 18, 1955) was a German-born theoretical physicist who is widely considered one of the greatest physicists of all time. ...
## See also Applied mathematics is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains. ...
## External links *What is Pure Mathematics?* by Lis D'Alessio, University of Waterloo *What is Pure Mathematics?* by Professor P.J. Giblin The University of Liverpool |