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Broadly speaking, pure mathematics is mathematics motivated entirely for reasons other than application. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and It is distinguished by its rigour, abstraction and beauty. Rigour or rigor (see spelling differences) has a number of meanings in relation to intellectual life and discourse 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 Many Mathematicians derive aesthetic pleasure from their work and from Mathematics in general From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterised as speculative mathematics,[1] and at variance with the trend towards meeting the needs of navigation, astronomy, physics, engineering, and so on. The 18th century lasted from 1701 to 1800 in the Gregorian calendar, in accordance with the Anno Domini / Common Era numbering system Navigation is the process of reading and controlling the movement of a craft or vehicle from one place to another Astronomy (from the Greek words astron (ἄστρον "star" and nomos (νόμος "law" is the scientific study Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Engineering is the Discipline and Profession of applying technical and scientific Knowledge and

Contents

History

Ancient Greece

Ancient Greek mathematicians were among the earliest to make a distinction between pure and applied mathematics. Plato helped to create the gap between "arithmetic", now called Number Theory, and "logistic", now called arithmetic. 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 Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone Plato regarded logistic as appropriate for business men and men of war who "must learn the art of numbers or he will not know how to array his troops," while arithmetic was appropriate for philosophers "because he has to arise out of the sea of change and lay hold of true being. Biography Early life Birth and family Plato was born in Athens Greece "[2] Euclid of Alexandria, when asked by one of his students of what use was the study of geometry, asked his slave to give the student threepence, "since he must needs make gain of what he learns. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry "[3] The Greek mathematician Apollonius of Perga was asked about the usefulness of some of his theorems in Book IV of Conics to which he proudly asserted,[4]

They are worthy of acceptance for the sake of the demonstrations themselves, in the same way as we accept many other things in mathematics for this and for no other reason.

And since many of his results were not applicable to the science or engineering of his day, Apollonius further argued in the preface of the fifth book of Conics that "the subject is one of those which seems worthy of study for their own sake. "[4]

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 Sadleirian Chair is a Professorship in Pure mathematics at the University of Cambridge. The 19th century of the Common Era began on January 1, 1801 and ended on December 31, 1900, according to the Gregorian calendar 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. Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German In the following years, specialisation and professionalisation (particularly in the Weierstrass approach to mathematical analysis) started to make a rift more apparent. Karl Theodor Wilhelm Weierstrass ( Weierstraß) ( October 31, 1815 &ndash February 19, 1897) was a German mathematician Analysis has its beginnings in the rigorous formulation of Calculus.

20th century

At the start of the twentieth century mathematicians took up the axiomatic method, strongly influenced by David Hilbert's example. The twentieth century of the Common Era began on 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 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most 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. Bertrand Arthur William Russell 3rd Earl Russell, OM, FRS (18 May 1872 – 2 February 1970 was a British Philosopher, Historian Quantification has two distinct meanings In Mathematics and Empirical science, it refers to human acts known as Counting and Measuring In Logic and Philosophy, proposition refers to either (a the content or Meaning of a meaningful Declarative sentence Rigour or rigor (see spelling differences) 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 can be ascribed to the Bourbaki group, is what is proved. Nicolas Bourbaki is the collective Pseudonym under which a group of (mainly French) 20th-century Mathematicians wrote a series of books presenting an exposition Pure mathematician became a recognized vocation, to be achieved through training.

Generality and abstraction

One central concept in pure mathematics is the idea of generality; pure mathematics often exhibits a trend towards increased generality. Generality has many different manifestations, such as proving theorems under weaker assumptions, or defining mathematical structures using fewer assumptions. Although generality is sometimes pursued or valued for its own sake, it has certain benefits, including:

Generality's impact on intuition is both dependent on the subject and a matter of personal preference or learning style. Intuition is apparent ability to acquire knowledge without a clear inference or the use of reason Often generality is seen as a hindrance to intuition, although it can certainly function as an aid to it, especially when it provides analogies to material for which one already has good intuition.

As a prime example of generality, the Erlangen program involved an expansion of Geometry to accommodate Non-euclidean geometries as well as the field of topology, and other forms of geometry, by viewing geometry as the study of a space together with a group of transformations. An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element 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. A number is an Abstract object, tokens of which are Symbols used in Counting and measuring. Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives Analysis has its beginnings in the rigorous formulation of Calculus. For functional analysis as used in psychology see the Functional analysis (psychology article 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 20th century. --> Abstraction is the process or result of generalization by reducing the information

In practice, however, these developments led to a sharp divergence from physics, particularly from 1950 to 1980. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Later this was criticised, for example by Vladimir Arnold, as too much Hilbert, not enough Poincaré. Vladimir Igorevich Arnol'd or Arnold (Влади́мир И́горевич Арно́льд born June 12, 1937 in Odessa, Ukrainian SSR David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most Jules Henri Poincaré ( 29 April 1854 &ndash 17 July 1912) (ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe was a French Mathematician 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. String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings 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

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 this debate can be found in G.H. Hardy's A Mathematician's Apology. Godfrey Harold Hardy FRS ( February 7, 1877 Cranleigh, Surrey, England &ndash December 1, 1947 A Mathematician's Apology is a 1940 essay by British mathematician G

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. Painting (pān'tīng in Art, is the practice of applying Color to a Surface (support base such as e Hardy made a separate 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. Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons 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.

Subfields in pure mathematics

Analysis is concerned with the properties of functions. Analysis has its beginnings in the rigorous formulation of Calculus. 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. 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 Calculus and analysis concerning the behavior of that function near a particular In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space Real analysis studies functions of real numbers, while complex analysis extends the aforementioned concepts to functions of complex numbers. Real analysis is a branch of Mathematical analysis dealing with the set of Real numbers In particular it deals with the analytic properties of real Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex Functional analysis is a branch of analysis that studies infinite-dimensional vector spaces and views functions as points in these spaces. For functional analysis as used in psychology see the Functional analysis (psychology article

Abstract algebra is not to be confused with the manipulation of formulae that is covered in secondary education. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules It studies sets together with binary operations defined on them. In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two Sets and their binary operations may be classified according to their properties: for instance, if an operation is associative on a set which contains an identity element and inverses for each member of the set, the set and operation is considered to be a group. In Mathematics, associativity is a property that a Binary operation can have In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element Other structures include rings, fields and vector spaces. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added

Geometry is the study of shapes and space, in particular, groups of transformations that act on spaces. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position For example, projective geometry is about the group of projective transformations that act on the real projective plane, whereas inversive geometry is concerned with the group of inversive transformations acting on the extended complex plane. Projective geometry is a non- metrical form of Geometry, notable for its principle of duality. In Geometry, inversive geometry is the study of a type of transformations of the Euclidean plane, called inversions. Geometry has been extended to topology, which deals with objects known as topological spaces and continuous maps between them. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Topology is concerned with the way in which a space is connected and ignores precise measurements of distance or angle.

Number theory is the theory of the positive integers. 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 The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French It is based on ideas such as divisibility and congruence. In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers Its fundamental theorem states that each positive integer has a unique prime factorization. In Number theory, the fundamental theorem of arithmetic (or unique-prime-factorization theorem) states that every Natural number greater than 1 can be written In some ways it is 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). Goldbach's conjecture is one of the oldest unsolved problems in Number theory and in all of Mathematics. In other ways it is the least accessible discipline; for example, Wiles' proof that Fermat's equation has no nontrivial solutions requires understanding automorphic forms, which though intrinsic to nature have not found a place in Physics or in public discourse.

Quotes

Notes

  1. ^ See for example titles of works by Thomas Simpson from the mid-18th century: Essays on Several Curious and Useful Subjects in Speculative and Mixed Mathematicks, Miscellaneous Tracts on Some Curious and Very Interesting Subjects in Mechanics, Physical Astronomy and Speculative Mathematics. Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical Thomas Simpson ( August 20, 1710 &ndash May 14, 1761) was a British Mathematician, Inventor and Eponym [1]
  2. ^ Boyer, Carl B. (1991). Carl Benjamin Boyer ( November 3, 1906 – April 26, 1976) has been called the " Gibbon of math history"he "The age of Plato and Aristotle", A History of Mathematics, Second Edition, John Wiley & Sons, Inc. , 86. ISBN 0471543977.  “Plato is important in the history of mathematics largely for his role as inspirer and director of others, and perhaps to him is due the sharp distinction in ancient Greece between arithmetic (in the sense of the theory of numbers) and logistic (the technique of computation). Plato regarded logistic as appropriate for the businessman and for the man of war, who "must learn the art of numbers or he will not know how to array his troops. " The philosopher, on the other hand, must be an arithmetician "because he has to arise out of the sea of change and lay hold of true being. "” 
  3. ^ Boyer, Carl B. (1991). Carl Benjamin Boyer ( November 3, 1906 – April 26, 1976) has been called the " Gibbon of math history"he "Euclid of Alexandria", A History of Mathematics, Second Edition, John Wiley & Sons, Inc. , 101. ISBN 0471543977.  “Evidently Euclid did not stress the practical aspects of his subject, for there is a tale told of him that when one of his students asked of what use was the study of geometry, Euclid asked his slave to hive the student threepence, "since he must needs make gain of what he learns. "” 
  4. ^ a b Boyer, Carl B. (1991). Carl Benjamin Boyer ( November 3, 1906 – April 26, 1976) has been called the " Gibbon of math history"he "Apollonius of Perga", A History of Mathematics, Second Edition, John Wiley & Sons, Inc. , 152. ISBN 0471543977.  “It is in connection with the theorems in this book that Apollonius makes a statement implying that in his day, as in ours, there were narrow-minded opponents of pure mathematics who pejoratively inquired about the usefulness of such results. The author proudly asserted: "They are worthy of acceptance for the sake of the demonstrations themselves, in the same way as we accept many other things in mathematics for this and for no other reason. " (Heath 1961, p. lxxiv).
    The preface to Book V, relating to maximum and minimum straight lines drawn to a conic, again argues that "the subject is one of those which seems worthy of study for their own sake. " While one must admire the author for his lofty intellectual attitude, it may be pertinently pointed out that s day was beautiful theory, with no prospect of applicability to the science or engineering of his time, has since become fundamental in such fields as terrestrial dynamics and celestial mechanics. ”     

See also

External links

Dictionary

pure mathematics

-noun

  1. Mathematics which is done for its own sake rather than being motivated by other sciences.
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