|
|
| Line 1: |
Line 1: |
| {{semiprotected|small=yes}}{{pp-move-indef}}
| | Greetings. The writer's title is Phebe and she feels comfy when people use the full title. South Dakota is her beginning place but she needs to transfer simply because of her family. Doing ceramics is what love doing. Bookkeeping is her day job now.<br><br>Review my web blog - [http://minihttp.com/fooddeliveryservices36486 healthy food delivery] |
| {{dablink|"Maths" and "Math" redirect here. For other uses see [[Mathematics (disambiguation)]] and [[Math (disambiguation)]].}}
| |
| [[File:Euclid.jpg|thumb|[[Euclid]], Greek mathematician, 3rd century BC, as imagined by [[Raphael]] in this detail from ''[[The School of Athens]]''.<ref>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]]'').</ref>]]
| |
| | |
| '''Mathematics''' is the abstract study of topics such as [[quantity]] ([[number]]s),<ref name="OED">{{cite web |url=http://oed.com/view/Entry/114974 |title=mathematics, ''n.'' |publisher=Oxford University Press |work=Oxford English Dictionary |year=2012 |accessdate=June 16, 2012 |quote=The science of space, number, quantity, and arrangement, whose methods involve logical reasoning and usually the use of symbolic notation, and which includes geometry, arithmetic, algebra, and analysis.}}</ref> [[structure]],<ref name="Kneebone">{{cite book |title=Mathematical Logic and the Foundations of Mathematics: An Introductory Survey |publisher=Dover |author=Kneebone, G.T. |year=1963 |pages=[http://books.google.com/books?id=tCXxf4vbXCcC&pg=PA4 4] |isbn=0-486-41712-3 |quote=Mathematics ... is simply the study of abstract structures, or formal patterns of connectedness.}}</ref> [[space]],<ref name=OED/> and [[calculus|change]].<ref name="LaTorre">{{cite book |title=Calculus Concepts: An Informal Approach to the Mathematics of Change |publisher=Cengage Learning |author=LaTorre, Donald R., John W. Kenelly, Iris B. Reed, Laurel R. Carpenter, and Cynthia R Harris |year=2011 |pages=[http://books.google.com/books?id=1Ebu2Tij4QsC&pg=PA2 2] |isbn=1-4390-4957-2 |quote=Calculus is the study of change—how things change, and how quickly they change.}}</ref><ref name="Ramana">{{cite book |title=Applied Mathematics |publisher=Tata McGraw–Hill Education |author=Ramana |year=2007 |page=[http://books.google.com/books?id=XCRC6BeKhIIC&pg=SA2-PA10 2.10] |isbn=0-07-066753-5 |quote=The mathematical study of change, motion, growth or decay is calculus.}}</ref><ref name="Ziegler">{{cite book |title=An Invitation to Mathematics: From Competitions to Research |publisher=Springer |author=Ziegler, Günter M. |authorlink=Günter M. Ziegler |year=2011 |pages=[http://books.google.com/books?id=9TATfteVeVYC&pg=PR7 7] |isbn=3-642-19532-6 |chapter=What Is Mathematics?}}</ref> There is a range of views among mathematicians and philosophers as to the exact scope and [[definition of mathematics]].<ref name=Mura/><ref name=Runge/><!--<<< Please do NOT change the opening sentence without discussion; much time and discussion have been invested in its current form.-->
| |
| | |
| [[Mathematician]]s seek out [[patterns]]<ref name=future/><ref name=devlin/> and use them to formulate new [[conjecture]]s. Mathematicians 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 (mathematics)|abstraction]] and [[logic]], mathematics developed from [[counting]], [[calculation]], [[measurement]], and the systematic study of the [[shape]]s and [[motion (physics)|motions]] of physical objects. Practical mathematics has been a human activity for as far back as [[History of Mathematics|written records]] exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.
| |
| | |
| [[Logic|Rigorous arguments]] first appeared in [[Greek mathematics]], most notably in [[Euclid]]'s ''[[Euclid's Elements|Elements]]''. Since the pioneering work of [[Giuseppe Peano]] (1858–1932), [[David Hilbert]] (1862–1943), and others [[Foundations of mathematics|on axiomatic systems in the late 19th century]], it has become customary to view mathematical research as establishing [[truth]] by [[Mathematical rigour|rigorous]] [[deductive reasoning|deduction]] from appropriately chosen [[axiom]]s and [[definition]]s. Mathematics developed at a relatively slow pace until the [[Renaissance]], when mathematical innovations interacting with new [[timeline of scientific discoveries|scientific discoveries]] led to a rapid increase in the rate of mathematical discovery that has continued to the present day.<ref>Eves</ref>
| |
| | |
| [[Galileo Galilei]] (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."<ref>[[Marcus du Sautoy]], ''[http://www.bbc.co.uk/programmes/b00sr3fm A Brief History of Mathematics: 1. Newton and Leibniz]'', [[BBC Radio 4]], 27 September 2010.</ref> [[Carl Friedrich Gauss]] (1777–1855) referred to mathematics as "the Queen of the Sciences".<ref name="Waltershausen">Waltershausen</ref> [[Benjamin Peirce]] (1809–1880) called mathematics "the science that draws necessary conclusions".<ref>Peirce, p. 97.</ref> 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."<ref>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), Basel, Birkhäuser (1992).</ref> [[Albert Einstein]] (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."<ref name=certain/> French mathematician [[Claire Voisin]] states "There is creative drive in mathematics, it's all about movement trying to express itself." <ref>{{cite web|url=http://www2.cnrs.fr/en/1402.htm |title=Claire Voisin, Artist of the Abstract |publisher=.cnrs.fr |date= |accessdate=2013-10-13}}</ref>
| |
| | |
| Mathematics is used throughout the world as an essential tool in many fields, including [[natural science]], [[engineering]], [[medicine]], [[finance]] and the [[social sciences]]. [[Applied mathematics]], the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries, which has led to the development of entirely new mathematical disciplines, such as [[statistics]] and [[game theory]]. Mathematicians also engage in [[pure mathematics]], or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered.<ref>Peterson</ref>
| |
| | |
| {{TOC limit|3}}
| |
| | |
| ==History==
| |
| | |
| ===Evolution===
| |
| {{main|History of mathematics}}
| |
| | |
| [[File:Kapitolinischer Pythagoras adjusted.jpg|thumb|left|upright|Greek mathematician [[Pythagoras]] ({{nowrap|c. 570 – c. 495 BC}}), commonly credited with discovering the [[Pythagorean theorem]]]]
| |
| The evolution of mathematics might be seen as an ever-increasing series of [[abstraction (mathematics)|abstractions]], or alternatively an expansion of subject matter. The first abstraction, which is shared by many animals,<ref>{{cite journal |title=Abstract representations of numbers in the animal and human brain |journal=Trends in Neuroscience |volume=21 |issue=8|date=Aug 1998 |pages=355–361 |doi=10.1016/S0166-2236(98)01263-6 |pmid=9720604 |ref=harv |last1=Dehaene |first1=Stanislas |last2=Dehaene-Lambertz |first2=Ghislaine |last3=Cohen |first3=Laurent}}</ref> was probably that of [[number]]s: 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.
| |
| | |
| [[File:maya.svg|thumb|[[Mayan numerals]]]]
| |
| Evidenced by [[tally sticks|tallies]] found on bone, in addition to recognizing how to [[counting|count]] physical objects, [[prehistoric]] peoples may have also recognized how to count abstract quantities, like time – days, [[season]]s, years.<ref>See, for example, Raymond L. Wilder, ''Evolution of Mathematical Concepts; an Elementary Study'', ''passim''</ref>
| |
| | |
| More complex mathematics did not appear until around 3000 BC, when the [[Babylonia]]ns and Egyptians began using arithmetic, algebra and geometry for [[taxation]] and other financial calculations, for building and construction, and for [[astronomy]].<ref>Kline 1990, Chapter 1.</ref> The earliest uses of mathematics were in [[trade|trading]], [[land measurement]], [[painting]] and [[weaving]] patterns and the recording of time.
| |
| | |
| In [[Babylonian mathematics]] [[elementary arithmetic]] ([[addition]], [[subtraction]], [[multiplication]] and [[division (mathematics)|division]]) first appears in the archaeological record. [[Numeracy]] pre-dated [[writing]] and [[numeral system]]s have been many and diverse, with the first known written numerals created by [[Ancient Egypt|Egyptians]] in [[Middle Kingdom of Egypt|Middle Kingdom]] texts such as the [[Rhind Mathematical Papyrus]].{{Citation needed|date=August 2009}}
| |
| | |
| Between 600 and 300 BC the [[Ancient Greeks]] began a systematic study of mathematics in its own right with [[Greek mathematics]].<ref>"''[http://books.google.com/books?id=drnY3Vjix3kC&pg=PA1&dq&hl=en#v=onepage&q=&f=false A History of Greek Mathematics: From Thales to Euclid]''". Thomas Little Heath (1981). ISBN 0-486-24073-8</ref>
| |
| | |
| 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 [[theorem]]s and their [[mathematical proof|proofs]]."<ref>Sevryuk</ref>
| |
| | |
| ===Etymology===
| |
| The word ''mathematics'' comes from the [[ancient Greek language|Greek]] μάθημα (''máthēma''), which, in the ancient Greek language, means "that which is learnt",<ref>{{cite web |title=mathematic |publisher=[[Online Etymology Dictionary]] |url=http://www.etymonline.com/index.php?term=mathematic&allowed_in_frame=0}}</ref> "what one gets to know", hence also "study" and "science", and in modern Greek just "lesson". The word ''máthēma'' is derived from μανθάνω (''manthano''), while the modern Greek equivalent is μαθαίνω (''mathaino''), both of which mean "to learn". In Greece, the word for "mathematics" came to have the narrower and more technical meaning "mathematical study" even in Classical times.<ref>Both senses can be found in Plato. {{LSJ|maqhmatiko/s|μαθηματική|ref}}</ref> Its adjective is {{lang|grc|μαθηματικός}} (''mathēmatikós''), meaning "related to learning" or "studious", which likewise further came to mean "mathematical". In particular, {{lang|grc|μαθηματικὴ τέχνη}} (''mathēmatikḗ tékhnē''), {{lang-la|ars mathematica}}, meant "the mathematical art".
| |
| | |
| 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: a particularly notorious one is [[Saint Augustine]]'s warning that Christians should beware of ''mathematici'' meaning astrologers, which is sometimes mistranslated as a condemnation of mathematicians.{{citation needed|date=October 2013}}
| |
| | |
| The apparent plural form in English, like the French plural form {{lang|fr|''les mathématiques''}} (and the less commonly used singular derivative {{lang|fr|''la mathématique''}}), goes back to the Latin neuter plural {{lang|la|''mathematica''}} ([[Cicero]]), based on the Greek plural {{lang|el|τα μαθηματικά}} (''ta mathēmatiká''), used by [[Aristotle]] (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 [[physics]] and [[metaphysics]], which were inherited from the Greek.<ref>''[[The Oxford Dictionary of English Etymology]]'', ''[[Oxford English Dictionary]]'', ''sub'' "mathematics", "mathematic", "mathematics"</ref> In English, the noun ''mathematics'' takes singular verb forms. It is often shortened to ''maths'' or, in English-speaking North America, ''math''.<ref name=maths>[http://oed.com/view/Entry/114982 "maths, ''n.''"] and [http://oed.com/view/Entry/114962 "math, ''n.3''"]. ''Oxford English Dictionary,'' on-line version (2012).</ref>
| |
| | |
| ==Definitions of mathematics==
| |
| {{main|Definitions of mathematics}}
| |
| | |
| [[Aristotle]] defined mathematics as "the science of quantity", and this definition prevailed until the 18th century.<ref name=Franklin>James Franklin, "Aristotelian Realism" in ''Philosophy of Mathematics", ed. A.D. Irvine, [http://books.google.com/books?id=mbn35b2ghgkC&pg=PA104#v=onepage&q&f=false p. 104]. Elsevier (2009).</ref> Starting in the 19th century, when the study of mathematics increased in rigor and began to address abstract topics such as [[group theory]] and [[projective geometry]], which have no clear-cut relation to quantity and measurement, mathematicians and philosophers began to propose a variety of new definitions.<ref name="Cajori">{{cite book |title=A History of Mathematics |publisher=American Mathematical Society (1991 reprint) |author=Cajori, Florian |authorlink=Florian Cajori |year=1893 |pages=[http://books.google.com/books?id=mGJRjIC9fZgC&pg=PA285 285–6] |isbn=0-8218-2102-4}}</ref> 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.<ref name="Mura">{{cite journal |title=Images of Mathematics Held by University Teachers of Mathematical Sciences |author=Mura, Robert |journal=Educational Studies in Mathematics |date=Dec 1993 |volume=25 |issue=4 |pages=[http://www.jstor.org/stable/10.2307/3482762 375–385] |ref=harv}}</ref> There is not even consensus on whether mathematics is an art or a science.<ref name="Runge">{{cite book |title=[[Iris Runge]]: A Life at the Crossroads of Mathematics, Science, and Industry |publisher=Springer |author=Tobies, Renate and Helmut Neunzert |year=2012 |pages=[http://books.google.com/books?id=EDm0eQqFUQ4C&pg=PA9 9] |isbn=3-0348-0229-3 |quote=It is first necessary to ask what is meant by ''mathematics'' in general. Illustrious scholars have debated this matter until they were blue in the face, and yet no consensus has been reached about whether mathematics is a natural science, a branch of the humanities, or an art form.}}</ref> A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable.<ref name=Mura/> Some just say, "Mathematics is what mathematicians do."<ref name=Mura/>
| |
| | |
| Three leading types of definition of mathematics are called [[logicist]], [[intuitionist]], and [[Formalism (mathematics)|formalist]], each reflecting a different philosophical school of thought.<ref name=Snapper>{{Cite journal |doi=10.2307/2689412 |title=The Three Crises in Mathematics: Logicism, Intuitionism, and Formalism |journal=Mathematics Magazine |date=September 1979 |first=Ernst |last=Snapper |volume=52 |issue=4 |pages=207–16 |id= |jstor=2689412 |postscript=. |ref=harv}}</ref> All have severe problems, none has widespread acceptance, and no reconciliation seems possible.<ref name=Snapper/>
| |
| | |
| An early definition of mathematics in terms of logic was [[Benjamin Peirce]]'s "the science that draws necessary conclusions" (1870).<ref name="Peirce">{{cite book |title=Linear Associative Algebra |author=Peirce, Benjamin |authorlink=Benjamin Peirce |year=1882 |page=1 |url=http://books.google.com/books?id=De0GAAAAYAAJ&pg=PA1#v=onepage&q&f=false}}</ref> In the ''[[Principia Mathematica]]'', [[Bertrand Russell]] and [[Alfred North Whitehead]] advanced the philosophical program known as [[logicism]], and attempted to prove that all mathematical concepts, statements, and principles can be defined and proven entirely in terms of [[symbolic logic]]. A logicist definition of mathematics is Russell's "All Mathematics is Symbolic Logic" (1903).<ref name=Russell>Bertrand Russell, ''The Principles of Mathematics,'' [http://books.google.com/books?id=kj0a_aV2mxIC&pg=PA5#v=onepage&q&f=false p. 5]. University Press, Cambridge (1903)</ref>
| |
| | |
| [[Intuitionist]] 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."<ref name=Snapper/> 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 proven to exist even though they cannot be constructed, intuitionism allows only mathematical objects that one can actually construct.
| |
| | |
| [[Formalism (mathematics)|Formalist]] definitions identify mathematics with its symbols and the rules for operating on them. [[Haskell Curry]] defined mathematics simply as "the science of formal systems".<ref name="Curry">{{cite book |title=Outlines of a Formalist Philosophy of Mathematics |publisher=Elsevier |author=Curry, Haskell |authorlink=Haskell Curry |year=1951 |pages=[http://books.google.com/books?id=tZHrBQgp1bkC 56] |isbn=0-444-53368-0}}</ref> A [[formal system]] 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.
| |
| | |
| ==Inspiration, pure and applied mathematics, and aesthetics==
| |
| {{main|Mathematical beauty}}
| |
| | |
| {{multiple image
| |
| | footer = [[Isaac Newton]] (left) and [[Gottfried Wilhelm Leibniz]] (right), developers of infinitesimal calculus
| |
| | width1 = 184
| |
| | image1 = GodfreyKneller-IsaacNewton-1689.jpg
| |
| | alt1 = Isaac Newton
| |
| | width2 = 200
| |
| | image2 = Gottfried Wilhelm von Leibniz.jpg
| |
| | alt2 = Gottfried Wilhelm von Leibniz
| |
| }}
| |
| | |
| Mathematics arises from many different kinds of problems. At first these were found in [[commerce]], [[land measurement]], [[architecture]] and later [[astronomy]]; today, 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 interaction|fundamental forces of nature]], continues to inspire new mathematics.<ref>{{cite book |title=The Feynman Integral and Feynman's Operational Calculus |author=Johnson, Gerald W.; Lapidus, Michel L. |publisher=[[Oxford University Press]] |year=2002 |isbn=0-8218-2413-9}}</ref> 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 [[pure mathematics]] and [[applied mathematics]]. However pure mathematics topics often turn out to have applications, e.g. [[number theory]] in [[cryptography]]. This remarkable fact that even the "purest" mathematics often turns out to have practical applications is what [[Eugene Wigner]] has called "[[The Unreasonable Effectiveness of Mathematics in the Natural Sciences|the unreasonable effectiveness of mathematics]]".<ref>{{cite journal |last=Wigner |first=Eugene |year=1960 |title=The Unreasonable Effectiveness of Mathematics in the Natural Sciences |url=http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html |journal=[[Communications on Pure and Applied Mathematics]] |volume=13 |issue=1 |pages=1–14 |doi=10.1002/cpa.3160130102 |ref=harv}}</ref> 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.<ref>{{cite web |url=http://www.ams.org/mathscinet/msc/pdfs/classification2010.pdf |title=Mathematics Subject Classification 2010 |format=PDF |date= |accessdate=2010-11-09}}</ref> 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 (mathematics)|proof]], such as [[Euclid]]'s proof that there are infinitely many [[prime number]]s, 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. He identified criteria such as significance, unexpectedness, inevitability, and economy as factors that contribute to a mathematical aesthetic.<ref>{{cite book |title=A Mathematician's Apology |author=Hardy, G.H. |publisher=Cambridge University Press |year=1940 |isbn=0-521-42706-1}}</ref> Mathematicians often strive to find proofs that are particularly elegant, proofs from "The Book" of God according to [[Paul Erdős]].<ref>{{cite book |title=Proof and Other Dilemmas: Mathematics and Philosophy |author=Gold, Bonnie; Simons, Rogers A. |publisher=MAA |year=2008}}</ref><ref>{{cite book |title=Proofs from ''The Book'' |last1=Aigner |first1=Martin |last2=Ziegler |first2=Günter M. |authorlink2=Günter M. Ziegler |publisher=Springer |year=2001 |isbn=3-540-40460-0}}</ref> The popularity of [[recreational mathematics]] is another sign of the pleasure many find in solving mathematical questions.
| |
| | |
| ==Notation, language, and rigor==
| |
| {{main|Mathematical notation}}
| |
| | |
| [[File:Leonhard Euler 2.jpg|upright|thumb|[[Leonhard Euler]], who created and popularized much of the mathematical notation used today]]
| |
| Most of the mathematical notation in use today was not invented until the 16th century.<ref>[http://jeff560.tripod.com/mathsym.html Earliest Uses of Various Mathematical Symbols] (Contains many further references).</ref> Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery.<ref>Kline, p. 140, on [[Diophantus]]; p. 261, on [[Franciscus Vieta|Vieta]].</ref> [[Leonhard Euler|Euler]] (1707–1783) was responsible for many of the notations in use today. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. It is extremely compressed: a few symbols contain a great deal of information. Like [[musical notation]], modern mathematical notation has a strict syntax (which to a limited extent varies from author to author and from discipline to discipline) and encodes information that would be difficult to write in any other way.
| |
| | |
| Mathematical [[language]] can be difficult to understand for beginners. Words such as ''or'' and ''only'' have more precise meanings than in everyday speech. Moreover, words such as ''[[open set|open]]'' and ''[[field (mathematics)|field]]'' have been given specialized mathematical meanings. Technical terms such as ''[[homeomorphism]]'' and ''[[Integral|integrable]]'' have precise meanings in 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 [[rigor]]. Mathematicians want their theorems to follow from axioms by means of systematic reasoning. This is to avoid mistaken "[[theorem]]s", based on fallible intuitions, of which many instances have occurred in the history of the subject.<ref>See ''[[false proof]]'' for simple examples of what can go wrong in a formal proof.</ref> 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. Misunderstanding the rigor is a cause for some of the common misconceptions of mathematics<!--This part needs improvement |>>-->. Today, mathematicians continue to argue among themselves about [[computer-assisted proof]]s. Since large computations are hard to verify, such proofs may not be sufficiently rigorous.<ref>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).</ref>
| |
| | |
| [[Axiom]]s 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 [[independence (mathematical logic)|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.<ref>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."</ref>
| |
| | |
| ==Fields of mathematics==
| |
| {{see also|Areas of mathematics|Glossary of areas of mathematics}}
| |
| | |
| [[File:Abacus 6.png|right|thumb|An [[abacus]], a simple calculating tool used since ancient times]]
| |
| Mathematics can, broadly speaking, be subdivided into the study of quantity, structure, space, and change (i.e. [[arithmetic]], [[algebra]], [[geometry]], and [[mathematical analysis|analysis]]). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to [[mathematical logic|logic]], to [[set theory]] ([[foundations of mathematics|foundations]]), to the empirical mathematics of the various sciences ([[applied mathematics]]), and more recently to the rigorous study of [[uncertainty]].
| |
| | |
| ===Foundations and philosophy===
| |
| In order to clarify the [[foundations of mathematics]], the fields of [[mathematical logic]] and [[set theory]] were developed. Mathematical logic includes the mathematical study of [[logic]] and the applications of formal logic to other areas of mathematics; set theory is the branch of mathematics that studies [[set (mathematics)|sets]] or collections of objects. [[Category theory]], which deals in an abstract way with [[mathematical structure]]s and relationships between them, is still in development. The phrase "crisis of foundations" describes the search for a rigorous foundation for mathematics that took place from approximately 1900 to 1930.<ref>Luke Howard Hodgkin & Luke Hodgkin, ''A History of Mathematics'', Oxford University Press, 2005.</ref> 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 theory|controversy over Cantor's set theory]] and the [[Brouwer–Hilbert controversy]].
| |
| | |
| Mathematical logic is concerned with setting mathematics within a rigorous [[axiom]]atic 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 [[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''). 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 [[recursion theory]], [[model theory]], and [[proof theory]], and is closely linked to [[theoretical computer science|theoretical]] [[computer science]],{{Citation needed|date=March 2011}} as well as to [[category theory]].
| |
| | |
| [[Theoretical computer science]] includes [[computability theory (computation)|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 [[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 the rapid advancement of computer hardware. A famous problem is the "[[P = NP problem|{{nowrap|'''P''' {{=}} '''NP'''?}}]]" problem, one of the [[Millennium Prize Problems]].<ref>[http://www.claymath.org/millennium/P_vs_NP/ Clay Mathematics Institute], P=NP, claymath.org</ref> 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 [[data compression|compression]] and [[Entropy (information theory)|entropy]].
| |
| | |
| :{|style="border:1px solid #ddd; text-align:center; margin:auto" cellspacing="15"
| |
| |<math>p \Rightarrow q \,</math>|| [[File:Venn A intersect B.svg|128px]] || [[File:Commutative diagram for morphism.svg|96px]] || [[File:DFAexample.svg|96px]]
| |
| |-
| |
| |[[Mathematical logic]] || [[Set theory]] || [[Category theory]] || [[Theory of computation]]
| |
| |}
| |
| | |
| ===Pure mathematics===
| |
| ====Quantity====
| |
| <!--This section is linked from [[List of basic mathematics topics]]-->
| |
| The study of quantity starts with [[number]]s, first the familiar [[natural number]]s and [[integer]]s ("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]]. The [[twin prime]] 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 [[subset]] of the [[rational number]]s ("[[Fraction (mathematics)|fractions]]"). These, in turn, are contained within the [[real number]]s, which are used to represent [[Continuous function|continuous]] quantities. Real numbers are generalized to [[complex number]]s. These are the first steps of a hierarchy of numbers that goes on to include [[quaternion]]s and [[octonion]]s. Consideration of the natural numbers also leads to the [[transfinite number]]s, which formalize the concept of "[[infinity]]". Another area of study is size, which leads to the [[cardinal number]]s and then to another conception of infinity: the [[aleph number]]s, which allow meaningful comparison of the size of infinitely large sets.
| |
| | |
| :{|style="border:1px solid #ddd; text-align:center; margin:auto" cellspacing="20"
| |
| |<math>1, 2, 3,\ldots\!</math> || <math>\ldots,-2, -1, 0, 1, 2\,\ldots\!</math> || <math> -2, \frac{2}{3}, 1.21\,\!</math> || <math>-e, \sqrt{2}, 3, \pi\,\!</math> || <math>2, i, -2+3i, 2e^{i\frac{4\pi}{3}}\,\!</math>
| |
| |-
| |
| |[[Natural number]]s|| [[Integer]]s || [[Rational number]]s || [[Real number]]s || [[Complex number]]s
| |
| |}
| |
| | |
| ====Structure====
| |
| <!--This section is linked from [[List of basic mathematics topics]]-->
| |
| Many mathematical objects, such as [[set (mathematics)|sets]] of numbers and [[function (mathematics)|functions]], exhibit internal structure as a consequence of [[operation (mathematics)|operations]] or [[relation (mathematics)|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 [[number theory]] studies properties of the set of [[integer]]s that can be expressed in terms of [[arithmetic]] operations. Moreover, it frequently happens that different such structured sets (or [[mathematical structure|structures]]) exhibit similar properties, which makes it possible, by a further step of [[abstraction]], to state [[axiom]]s for a class of structures, and then study at once the whole class of structures satisfying these axioms. Thus one can study [[group (mathematics)|groups]], [[ring (mathematics)|rings]], [[field (mathematics)|fields]] and other abstract systems; together such studies (for structures defined by algebraic operations) constitute the domain of [[abstract algebra]]. 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 [[Galois theory]], which involves field theory and group theory. Another example of an algebraic theory is [[linear algebra]], which is the general study of [[vector space]]s, whose elements called [[vector (geometric)|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 [[geometry]] and [[algebra]] have very strong interactions in modern mathematics. [[Combinatorics]] studies ways of enumerating the number of objects that fit a given structure.
| |
| | |
| :{|style="border:1px solid #ddd; text-align:center; margin:auto" cellspacing="15"
| |
| |<math>\begin{matrix} (1,2,3) & (1,3,2) \\ (2,1,3) & (2,3,1) \\ (3,1,2) & (3,2,1) \end{matrix}</math> || [[File:Elliptic curve simple.svg|96px]] || [[File:Rubik's cube.svg|96px]] || [[File:Group diagdram D6.svg|96px]] || [[File:Lattice of the divisibility of 60.svg|96px]] || [[File:Braid-modular-group-cover.svg|96px]]
| |
| |-
| |
| |[[Combinatorics]] || [[Number theory]] || [[Group theory]] || [[Graph theory]] || [[Order theory]] || [[Algebra]]
| |
| |}
| |
| | |
| ====Space====
| |
| <!--This section is linked from [[List of basic mathematics topics]]-->
| |
| The study of space originates with [[geometry]] – in particular, [[Euclidean geometry]]. [[Trigonometry]] is the branch of mathematics that deals with relationships between the sides and the angles of triangles and with the trigonometric functions; it combines space and numbers, and encompasses the well-known [[Pythagorean 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]]. [[Convex geometry|Convex]] and [[discrete geometry]] were developed to solve problems in [[geometry of numbers|number theory]] and [[functional analysis]] but now are pursued with an eye on applications in [[convex optimization|optimization]] and [[computational geometry|computer science]]. Within differential geometry are the concepts of [[fiber bundles]] and calculus on [[manifold]]s, in particular, [[vector calculus|vector]] and [[tensor calculus]]. 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 group]]s 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; 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]], [[homotopy theory]], and [[Morse theory]]. Topology also includes the now solved [[Poincaré conjecture]], and the still unsolved areas of the [[Hodge conjecture]]. Other results in geometry and topology, including the [[four color theorem]] and [[Kepler conjecture]], have been proved only with the help of computers.
| |
| | |
| :{|style="border:1px solid #ddd; text-align:center; margin:auto" cellspacing="15"
| |
| |[[File:Illustration to Euclid's proof of the Pythagorean theorem.svg|96px]] || [[File:Sinusvåg 400px.png|96px]] || [[File:Hyperbolic triangle.svg|96px]] || [[File:Torus.png|96px]] || [[File:Mandel zoom 07 satellite.jpg|96px]] || [[File:Measure illustration.png|70px]]
| |
| |-
| |
| |[[Geometry]] || [[Trigonometry]] || [[Differential geometry]] || [[Topology]] || [[Fractal|Fractal geometry]] || [[Measure theory]]
| |
| |}
| |
| | |
| ====Change====
| |
| <!--This section is linked from [[List of basic mathematics topics]]-->
| |
| Understanding and describing change is a common theme in the [[natural science]]s, and [[calculus]] was developed as a powerful tool to investigate it. [[Function (mathematics)|Functions]] arise here, as a central concept describing a changing quantity. The rigorous study of [[real number]]s and functions of a real variable is known as [[real analysis]], with [[complex analysis]] the equivalent field for the [[complex number]]s. [[Functional analysis]] focuses attention on (typically infinite-dimensional) [[Space#Mathematics|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 equation]]s. Many phenomena in nature can be described by [[dynamical system]]s; [[chaos theory]] makes precise the ways in which many of these systems exhibit unpredictable yet still [[deterministic system (mathematics)|deterministic]] behavior.
| |
| | |
| {|style="border:1px solid #ddd; text-align:center; margin:auto" cellspacing="20"
| |
| |[[File:Integral as region under curve.svg|96px]] || [[File:Vector field.svg|96px]] || [[File:Airflow-Obstructed-Duct.png|96px]] || [[File:Limitcycle.svg|96px]] || [[File:Lorenz attractor.svg|96px]] || [[File:Conformal grid after Möbius transformation.svg|96px]]
| |
| |-
| |
| |[[Calculus]] || [[Vector calculus]]|| [[Differential equation]]s || [[Dynamical system]]s || [[Chaos theory]] || [[Complex analysis]]
| |
| |}
| |
| | |
| ===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 [[pure mathematics]].
| |
| | |
| ====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 [[random sampling]] and with randomized [[design of experiments|experiments]];<ref>[[C.R. Rao|Rao, C.R.]] (1997) ''Statistics and Truth: Putting Chance to Work'', World Scientific. ISBN 981-02-3111-3</ref> 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 study|observational studies]], statisticians "make sense of the data" using the art of [[statistical model|modelling]] and the theory of [[statistical inference|inference]] – with [[model selection]] and [[estimation theory|estimation]]; the estimated models and consequential [[Scientific method#Predictions from the hypothesis|predictions]] should be [[statistical hypothesis testing|tested]] on [[Scientific method#Evaluation and improvement|new data]].<ref>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.</ref>
| |
| | |
| [[Statistical theory]] studies [[statistical decision theory|decision problem]]s such as minimizing the [[risk]] ([[expected loss]]) of a statistical action, such as using a [[statistical method|procedure]] in, for example, [[parameter estimation]], [[hypothesis testing]], and [[selection algorithm|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 [[cost]], under specific constraints: For example, designing a survey often involves minimizing the cost of estimating a population mean with a given level of confidence.<ref name="RaoOpt">{{cite book |editor1-last=Arthanari |editor1-first=T.S. |editor2-last=Dodge |editor2-first=Yadolah |editor2-link=Yadolah Dodge |last=Rao |first=C.R. |authorlink=C.R. Rao |chapter=Foreword |title=Mathematical programming in statistics |series=Wiley Series in Probability and Mathematical Statistics |publisher=Wiley |location=New York |year=1981 |pages=vii–viii |isbn=0-471-08073-X |mr=607328 |ref=harv}}</ref> Because of its use of [[mathematical optimization|optimization]], the mathematical theory of statistics shares concerns with other [[decision science]]s, such as [[operations research]], [[control theory]], and [[mathematical economics]].<ref name="Whittle">{{harvtxt|Whittle|1994|pp=10–11 and 14–18}}: {{cite book |first=Peter |last=Whittle |authorlink=Peter Whittle |chapter=Almost home |editor-link=Frank Kelly (mathematician) |editor-first=F.P. |editor-last=Kelly |year=1994 |title=Probability, statistics and optimisation: A Tribute to Peter Whittle |location=Chichester |publisher=John Wiley |isbn=0-471-94829-2 |pages=1–28 |ref=harv |url=http://www.statslab.cam.ac.uk/History/2history.html#6._1966--72:_The_Churchill_Chair |edition=previously "A realised path: The Cambridge Statistical Laboratory upto 1993 (revised 2002)"}}</ref>
| |
| | |
| ====Computational mathematics====
| |
| [[Computational mathematics]] proposes and studies methods for solving [[mathematical problem]]s that are typically too large for human numerical capacity. [[Numerical analysis]] studies methods for problems in [[analysis (mathematics)|analysis]] using [[functional analysis]] and [[approximation theory]]; numerical analysis includes the study of [[approximation]] and [[discretization]] broadly with special concern for [[rounding error]]s. Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially [[algorithm]]ic [[numerical linear algebra|matrix]] and [[graph theory]]. Other areas of computational mathematics include [[computer algebra]] and [[symbolic computation]].
| |
| | |
| {|style="border:1px solid #ddd; text-align:center; margin:0 auto" cellspacing="20"
| |
| |-
| |
| |[[File:Gravitation space source.png|96px]] || [[File:BernoullisLawDerivationDiagram.svg|96px]] || [[File:Composite trapezoidal rule illustration small.svg|96px]] || [[File:Maximum boxed.png|96px]] || [[File:Two red dice 01.svg|96px]] || [[File:Oldfaithful3.png|96px]] || [[File:Caesar3.svg|96px]]
| |
| |-
| |
| |[[Mathematical physics]] || [[Fluid dynamics]] || [[Numerical analysis]] || [[Mathematical optimization|Optimization]] || [[Probability theory]] || [[Statistics]] || [[Cryptography]]
| |
| |-
| |
| |[[File:Market Data Index NYA on 20050726 202628 UTC.png|96px]] || [[File:Arbitrary-gametree-solved.svg|96px]] || [[File:Signal transduction pathways.svg|96px]] || [[File:CH4-structure.svg|96px]] || [[File:GDP PPP Per Capita IMF 2008.svg|96px]] || [[File:Simple feedback control loop2.svg|96px]]
| |
| |-
| |
| |[[Mathematical finance]] || [[Game theory]] || [[Mathematical biology]] || [[Mathematical chemistry]]|| [[Mathematical economics]] || [[Control theory]]
| |
| |}
| |
| | |
| ==Mathematics as profession==
| |
| Arguably the most prestigious award in mathematics is the [[Fields Medal]],<ref>"''The Fields Medal is now indisputably the best known and most influential award in mathematics.''" Monastyrsky</ref><ref>Riehm</ref> established in 1936 and now awarded every four years. 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 [[Abel Prize]], was introduced in 2003. The [[Chern Medal]] 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 [[open problem]]s, 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. A solution to each of these problems carries a $1 million reward, and only one (the [[Riemann hypothesis]]) is duplicated in Hilbert's problems.
| |
| <!--<<<
| |
| ==Common misconceptions==
| |
| {{Unreferenced section|date=August 2009}}
| |
| {{Original research|date=August 2009}}
| |
| Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of [[unsolved problems in mathematics|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:
| |
| *misunderstanding of the implications of [[mathematical rigor]];
| |
| *attempts to circumvent the usual criteria for publication of [[mathematical paper]]s in a [[learned journal]] after [[peer review]], often in the belief that the journal is biased against the author;
| |
| *lack of familiarity with, and therefore underestimation of, the existing literature.
| |
| | |
| Like [[astronomy]], mathematics owes much to [[List of amateur mathematicians|amateur contributors]] such as [[Pierre de Fermat|Fermat]] and [[Marin Mersenne|Mersenne]].-->
| |
| | |
| ==Mathematics as science==
| |
| [[File:Carl Friedrich Gauss.jpg|upright|thumb|left|[[Carl Friedrich Gauss]], known as the "prince of mathematicians"<ref>{{cite book
| |
| |last=Zeidler
| |
| |first=Eberhard
| |
| |title=Oxford User's Guide to Mathematics
| |
| |location=Oxford, UK
| |
| |publisher=Oxford University Press
| |
| |year=2004
| |
| |isbn=0-19-850763-1
| |
| |page=1188}}</ref>]]
| |
| | |
| [[Carl Friedrich Gauss|Gauss]] referred to mathematics as "the Queen of the Sciences".<ref name="Waltershausen"/> In the original Latin ''Regina Scientiarum'', as well as in [[German language|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 ''[[natural science]]'' follows the rise of [[Baconian method|Baconian science]], which contrasted "natural science" to [[scholasticism]], the [[Organon|Aristotelean method]] of inquiring from [[first principles]]. The role of empirical experimentation and observation is negligible in mathematics, compared to natural sciences such as [[experimental psychology|psychology]], [[biology]], or [[physics]]. [[Albert Einstein]] 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."<ref name=certain>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]]''.</ref> More recently, [[Marcus du Sautoy]] has called mathematics "the Queen of Science ... the main driving force behind scientific discovery".<ref>[[Marcus du Sautoy]], ''[http://www.bbc.co.uk/programmes/b00stcgv A Brief History of Mathematics: 10. Nicolas Bourbaki]'', [[BBC Radio 4]], 1 October 2010.</ref>
| |
| | |
| Many philosophers believe that mathematics is not experimentally [[falsifiability|falsifiable]], and thus not a science according to the definition of [[Karl Popper]].<ref>{{cite book |title=Out of Their Minds: The Lives and Discoveries of 15 Great Computer Scientists |author=Shasha, Dennis Elliot; Lazere, Cathy A. |publisher=Springer |year=1998 |page=228}}</ref> However, in the 1930s [[Gödel's incompleteness theorems]] convinced many mathematicians{{Who|date=January 2011}} that mathematics cannot be reduced to logic alone, and Karl Popper concluded that "most mathematical theories are, like those of [[physics]] and [[biology]], [[hypothesis|hypothetico]]-[[deductive]]: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently."<ref>Popper 1995, p. 56</ref> 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. The theoretical physicist [[J.M. Ziman]] proposed that science is ''public knowledge'', and thus includes mathematics.<ref>Ziman</ref> Mathematics shares much in common with many fields in the physical sciences, notably the [[deductive reasoning|exploration of the logical consequences]] of assumptions. [[Intuition (knowledge)|Intuition]] and [[experiment]]ation also play a role in the formulation of [[conjecture]]s 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 mathematicians{{Who|date=August 2009}} 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{{Who|date=August 2009}} 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 [[university|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]].{{Citation needed|date=August 2009}}
| |
| | |
| ==See also==
| |
| {{main|Lists of mathematics topics}}
| |
| *[[Mathematics and art]]
| |
| *[[Mathematics education]]
| |
| | |
| {{Portal bar|Mathematics}}
| |
| | |
| ==Notes==
| |
| {{Reflist|30em
| |
| |refs=
| |
| <ref name=future>[[Lynn Steen|Steen, L.A.]] (April 29, 1988). ''The Science of Patterns'' [[Science (journal)|Science]], 240: 611–616. And summarized at [http://www.ascd.org/publications/curriculum-handbook/409/chapters/The-Future-of-Mathematics-Education.aspx Association for Supervision and Curriculum Development], www.ascd.org.</ref>
| |
| <ref name=devlin>[[Keith Devlin|Devlin, Keith]], ''Mathematics: The Science of Patterns: The Search for Order in Life, Mind and the Universe'' (Scientific American Paperback Library) 1996, ISBN 978-0-7167-5047-5</ref>}}
| |
| | |
| ==References==
| |
| {{refbegin|30em}}
| |
| *[[Richard Courant|Courant, Richard]] and [[Herbert Robbins|H. Robbins]], ''What Is Mathematics? : An Elementary Approach to Ideas and Methods'', Oxford University Press, USA; 2 edition (July 18, 1996). ISBN 0-19-510519-2.
| |
| *{{cite book
| |
| |last=Einstein
| |
| |first=Albert
| |
| |authorlink=Albert Einstein
| |
| |title=Sidelights on Relativity: I. Ether and relativity. II. Geometry and experience (translated by G.B. Jeffery, D.Sc., and W. Perrett, Ph.D).
| |
| |publisher=E.P. Dutton & Co., New York
| |
| |url=http://searchworks.stanford.edu/view/1216826
| |
| |year=1923}}
| |
| *[[Marcus du Sautoy|du Sautoy, Marcus]], ''[http://www.bbc.co.uk/podcasts/series/maths A Brief History of Mathematics]'', [[BBC Radio 4]] (2010).
| |
| *Eves, Howard, ''An Introduction to the History of Mathematics'', Sixth Edition, Saunders, 1990, ISBN 0-03-029558-0.
| |
| *[[Morris Kline|Kline, Morris]], ''Mathematical Thought from Ancient to Modern Times'', Oxford University Press, USA; Paperback edition (March 1, 1990). ISBN 0-19-506135-7.
| |
| *{{Cite journal |url=http://www.fields.utoronto.ca/aboutus/FieldsMedal_Monastyrsky.pdf |year=2001 |title=Some Trends in Modern Mathematics and the Fields Medal |author=Monastyrsky, Michael |publisher=Canadian Mathematical Society |accessdate=2006-07-28 |format=PDF |ref=harv}}
| |
| *[[Oxford English Dictionary]], second edition, ed. John Simpson and Edmund Weiner, [[Clarendon Press]], 1989, ISBN 0-19-861186-2.
| |
| *''[[The Oxford Dictionary of English Etymology]]'', 1983 reprint. ISBN 0-19-861112-9.
| |
| *Pappas, Theoni, ''The Joy Of Mathematics'', Wide World Publishing; Revised edition (June 1989). ISBN 0-933174-65-9.
| |
| *{{Cite journal |pages=97–229 |title=Linear associative algebra |first=Benjamin |last=Peirce |authorlink=Benjamin Peirce |editor-link=Charles Sanders Peirce |editor-first=Charles Sanders |editor-last=Peirce |edition=Corrected, expanded, and annotated revision with an 1875 paper by B. Peirce and annotations by his son, C.S. Peirce, of the 1872 lithograph |journal=American Journal of Mathematics |volume=4 |publisher=Johns Hopkins University |year=1881 |url=http://books.google.com/?id=De0GAAAAYAAJ&pg=PA1&dq=Peirce+Benjamin+Linear+Associative+Algebra+&q= |id=Corrected, expanded, and annotated revision with an 1875 paper by B. Peirce and annotations by his son, C. S. Peirce, of the 1872 lithograph ed. ''Google'' [http://books.google.com/books?id=LQgPAAAAIAAJ&pg=PA221 Eprint] and as an extract, D. Van Nostrand, 1882, ''Google'' [http://books.google.com/books?id=De0GAAAAYAAJ&printsec=frontcover Eprint] |issue=1–4 |doi=10.2307/2369153 |ref=harv}}.
| |
| *Peterson, Ivars, ''Mathematical Tourist, New and Updated Snapshots of Modern Mathematics'', Owl Books, 2001, ISBN 0-8050-7159-8.
| |
| *{{Cite book |first=Karl R. |last=Popper |authorlink=Karl Popper |title=In Search of a Better World: Lectures and Essays from Thirty Years |chapter=On knowledge |publisher=Routledge |year=1995 |isbn=0-415-13548-6}}
| |
| *{{cite journal
| |
| |last=Riehm
| |
| |first=Carl
| |
| |title=The Early History of the Fields Medal
| |
| |journal=Notices of the AMS
| |
| |volume=49
| |
| |issue=7
| |
| |pages=778–782
| |
| |publisher=AMS
| |
| |date=August 2002
| |
| |url=http://www.ams.org/notices/200207/comm-riehm.pdf
| |
| |format=PDF
| |
| |ref=harv}}
| |
| *{{cite journal |last=Sevryuk |first=Mikhail B. |date=January 2006 |title=Book Reviews |journal=[[Bulletin of the American Mathematical Society]] |volume=43 |issue=1 |pages=101–109 |url=http://www.ams.org/bull/2006-43-01/S0273-0979-05-01069-4/S0273-0979-05-01069-4.pdf |format=PDF |accessdate=2006-06-24 |doi=10.1090/S0273-0979-05-01069-4 |ref=harv}}
| |
| *{{cite book |last=Waltershausen |first=Wolfgang Sartorius von |authorlink=Wolfgang Sartorius von Waltershausen |title=Gauss zum Gedächtniss |year=1856, repr. 1965 |publisher=Sändig Reprint Verlag H. R. Wohlwend |isbn=3-253-01702-8 |url=http://www.amazon.de/Gauss-Ged%e4chtnis-Wolfgang-Sartorius-Waltershausen/dp/3253017028 |asin=B0000BN5SQ}}
| |
| {{refend}}
| |
| | |
| ==Further reading==
| |
| {{refbegin|30em}}
| |
| *Benson, Donald C., ''The Moment of Proof: Mathematical Epiphanies'', [[Oxford University Press]], USA; New Ed edition (December 14, 2000). ISBN 0-19-513919-4.
| |
| *[[Carl B. Boyer|Boyer, Carl B.]], ''A History of Mathematics'', Wiley; 2nd edition, revised by Uta C. Merzbach, (March 6, 1991). ISBN 0-471-54397-7.—A concise history of mathematics from the Concept of Number to contemporary Mathematics.
| |
| *[[Philip J. Davis|Davis, Philip J.]] and [[Reuben Hersh|Hersh, Reuben]], ''[[The Mathematical Experience]]''. Mariner Books; Reprint edition (January 14, 1999). ISBN 0-395-92968-7.
| |
| *[[Jan Gullberg|Gullberg, Jan]], ''Mathematics – From the Birth of Numbers''. [[W. W. Norton & Company]]; 1st edition (October 1997). ISBN 0-393-04002-X.
| |
| *[[Hazewinkel, Michiel]] (ed.), ''[[Encyclopaedia of Mathematics]]''. [[Kluwer Academic Publishers]] 2000. – A translated and expanded version of a Soviet mathematics encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM, and [http://www.encyclopediaofmath.org/ online].
| |
| *Jourdain, Philip E. B., ''The Nature of Mathematics'', in ''The World of Mathematics'', James R. Newman, editor, [[Dover Publications]], 2003, ISBN 0-486-43268-8.
| |
| *Maier, Annaliese, ''At the Threshold of Exact Science: Selected Writings of Annaliese Maier on Late Medieval Natural Philosophy'', edited by Steven Sargent, Philadelphia: University of Pennsylvania Press, 1982.
| |
| {{refend}}
| |
| | |
| ==External links==
| |
| {{Sister project links|Mathematics}}
| |
| {{Library resources box
| |
| |by=no
| |
| |onlinebooks=no
| |
| |others=no
| |
| |about=yes
| |
| |label=Mathematics}}
| |
| {{WVS}}
| |
| {{refbegin}}
| |
| *{{In Our Time|Mathematics|p00545hk|Mathematics}}
| |
| *[http://freebookcentre.net/SpecialCat/Free-Mathematics-Books-Download.html Free Mathematics books] Free Mathematics books collection.
| |
| *[[Encyclopaedia of Mathematics]] online encyclopaedia from [http://www.encyclopediaofmath.org/ Springer], Graduate-level reference work with over 8,000 entries, illuminating nearly 50,000 notions in mathematics.
| |
| *[http://hyperphysics.phy-astr.gsu.edu/Hbase/hmat.html HyperMath site at Georgia State University]
| |
| *[http://www.freescience.info/mathematics.php FreeScience Library] The mathematics section of FreeScience library
| |
| *Rusin, Dave: ''[http://www.math-atlas.org/ The Mathematical Atlas]''. A guided tour through the various branches of modern mathematics. (Can also be found at [http://www.math.niu.edu/~rusin/known-math/index/index.html NIU.edu].)
| |
| *Polyanin, Andrei: ''[http://eqworld.ipmnet.ru/ EqWorld: The World of Mathematical Equations]''. An online resource focusing on algebraic, ordinary differential, partial differential ([[mathematical physics]]), integral, and other mathematical equations.
| |
| *Cain, George: [http://www.math.gatech.edu/~cain/textbooks/onlinebooks.html Online Mathematics Textbooks] available free online.
| |
| *[http://www.tricki.org/ Tricki], Wiki-style site that is intended to develop into a large store of useful mathematical problem-solving techniques.
| |
| *[http://math.chapman.edu/cgi-bin/structures?HomePage Mathematical Structures], list information about classes of mathematical structures.
| |
| *[http://www-history.mcs.st-and.ac.uk/~history/ Mathematician Biographies]. The [[MacTutor History of Mathematics archive]] Extensive history and quotes from all famous mathematicians.
| |
| *''[http://metamath.org/ Metamath]''. A site and a language, that formalize mathematics from its foundations.
| |
| *[http://www.nrich.maths.org/public/index.php Nrich], a prize-winning site for students from age five from [[University of Cambridge|Cambridge University]]
| |
| *[http://garden.irmacs.sfu.ca/ Open Problem Garden], a [[wiki]] of open problems in mathematics
| |
| *''[http://planetmath.org/ Planet Math]''. An online mathematics encyclopedia under construction, focusing on modern mathematics. Uses the [[CC BY SA|Attribution-ShareAlike]] license, allowing article exchange with Wikipedia. Uses [[TeX]] markup.
| |
| *[http://www-math.mit.edu/daimp Some mathematics applets, at MIT]
| |
| *Weisstein, Eric et al.: ''[http://www.mathworld.com/ MathWorld: World of Mathematics]''. An online encyclopedia of mathematics.
| |
| *Patrick Jones' [https://www.youtube.com/user/patrickJMT Video Tutorials] on Mathematics
| |
| *[http://en.citizendium.org/wiki/Theory_(mathematics) Citizendium: Theory (mathematics)].
| |
| *[[Marcus du Sautoy|du Sautoy, Marcus]], ''[http://www.bbc.co.uk/podcasts/series/maths A Brief History of Mathematics]'', [[BBC Radio 4]] (2010).
| |
| *[http://mathoverflow.net/ MathOverflow] A Q&A site for research-level mathematics
| |
| {{refend}}
| |
| | |
| {{Mathematics-footer}}
| |
| | |
| [[Category:Mathematics| ]]
| |
| [[Category:Formal sciences]]
| |
| | |
| {{Link GA|de}}
| |
| {{Link GA|zh}}
| |
| {{Link FA|ia}}
| |
| {{Link FA|ka}}
| |
| {{Link FA|la}}
| |
| {{Link FA|vo}}
| |