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Mathematics
Mathematics is the academic discipline, and its supporting body of knowledge, that involves the study of such concepts as quantity, structure, space and change. The mathematician Benjamin Peirce called it "the science that draws necessary conclusions".[Peirce, p.97]
Other practitioners of mathematics maintain that mathematics is the science of pattern, and that mathematicians seek out patterns whether found in numbers, space, science, computers, imaginary abstractions, or elsewhere.[Lynn Steen|Steen, L.A. (April 29, 1988). ''The Science of Patterns.'' Science (journal)|Science, 240: 611–616. and summarized at Association for Supervision and Curriculum Development.][ 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 9780716750475 ] Mathematicians explore such concepts, aiming to formulate new conjectures and establish their truth by Rigour#Mathematical rigour|rigorous deductive reasoning|deduction from appropriately chosen axioms and definitions.[Jourdain]
Through the use of abstraction (mathematics)|abstraction and logical reasoning, mathematics evolved from counting, calculation, measurement, and the systematic study of the shapes and motion (physics)|motions of physical objects. Knowledge and use of basic mathematics have always been an inherent and integral part of individual and group life. Refinements of the basic ideas are visible in mathematical texts originating in the Egyptian mathematics|ancient Egyptian, Babylonian mathematics|Mesopotamian, Indian mathematics|Indian, Chinese mathematics|Chinese, Greek mathematics|Greek and Islamic mathematics|Islamic worlds. axiom|Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Euclid's Elements|''Elements''. The development continued in fitful bursts until the Renaissance period of the 16th century, when mathematical innovations interacted with new timeline of scientific discoveries|scientific discoveries, leading to an acceleration in research that continues to the present day.[Eves]
Today, mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences such as economics and psychology. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new disciplines. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind, although practical applications for what began as pure mathematics are often discovered later.[Peterson]
Etymology
The word "mathematics" comes from the ancient Greek language|Greek μάθημα (''máthēma''), which means ''learning'', ''study'', ''science'', and additionally came to have the narrower and more technical meaning "mathematical study", even in Classical times. Its adjective is μαθηματικός (''mathēmatikós''), ''related to learning'', or ''studious'', which likewise further came to mean ''mathematical''. In particular, (''mathēmatikḗ tékhnē''), in Latin ''ars mathematica'', meant ''the mathematical art''.
The apparent plural form in English language|English, like the French language|French plural form ''les mathématiques'' (and the less commonly used singular derivative ''la mathématique''), goes back to the Latin neuter plural ''mathematica'' (Cicero), based on the Greek plural τα μαθηματικά (''ta mathēmatiká''), used by Aristotle, and meaning roughly "all things mathematical".[''The Oxford Dictionary of English Etymology'', ''Oxford English Dictionary''] In English, however, the noun ''mathematics'' takes singular verb forms. It is often shortened to ''math'' in English-speaking North America and ''maths'' elsewhere.
History
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 was probably that of numbers: the realization that two apples and two oranges (for example) have something in common was a breakthrough in human thought.
In addition to recognizing how to counting|count ''physical'' objects, Prehistory|prehistoric peoples also recognized how to count ''abstract'' quantities, like time — days, seasons, years. Elementary arithmetic (addition, subtraction, multiplication and division (mathematics)|division) naturally followed.
Further steps needed writing or some other system for recording numbers such as Tally sticks|tallies or the knotted strings called quipu used by the Inca to store numerical data. Numeral systems have been many and diverse, with the first known written numerals created by Egyptians in Middle Kingdom of Egypt|Middle Kingdom texts such as the Rhind Mathematical Papyrus. The Indus Valley civilization developed the modern decimal system, including the concept of zero.
From the beginning of recorded history, the major disciplines within mathematics arose out of the need to do calculations relating to taxation and commerce, to understand the relationships among numbers, to land measurement|measure land, and to predict astronomy|astronomical events. These needs can be roughly related to the broad subdivision of mathematics into the studies of ''quantity'', ''structure'', ''space'', and ''change''.
Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries have been made throughout history and continue to be made today. According to Mikhail B. Sevryuk, in the January 2006 issue of the Bulletin of the American Mathematical Society, "The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their mathematical proof|proofs."[Sevryuk]
Inspiration, pure and applied mathematics, and aesthetics
Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. For example, the physicist Richard Feynman invented the path integral formulation of quantum mechanics using a combination of mathematical reasoning and physical insight, and today's string theory, a still-developing scientific theory which attempts to unify the four Fundamental interaction|fundamental forces of nature, continues to inspire new mathematics.[Eugene Wigner, 1960, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," ''Communications on Pure and Applied Mathematics'' '''13'''(1): 1–14.]
As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics: most mathematicians focus their research solely on one of these areas, and sometimes the choice is made as early as their undergraduate studies. Several areas of applied mathematics have merged with related traditions outside of mathematics and become disciplines in their own right, including statistics, operations research, and computer science.
For those who are mathematically inclined, there is often a definite aesthetic aspect to much of mathematics. Many mathematicians talk about the ''elegance'' of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty in a simple and elegant proof (mathematics)|proof, such as Euclid's proof that there are infinitely many prime numbers, and in an elegant numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in ''A Mathematician's Apology'' expressed the belief that these aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics.Notation, language, and rigor
Most of the mathematical notation in use today was not invented until the 16th century.[Earliest Uses of Various Mathematical Symbols (Contains many further references)] Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery. In the 18th century, Leonhard Euler|Euler was responsible for many of the notations in use today. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. It is extremely compressed: a few symbols contain a great deal of information. Like musical notation, modern mathematical notation has a strict syntax and encodes information that would be difficult to write in any other way.
Mathematical language can also be hard for beginners. Words such as ''or'' and ''only'' have more precise meanings than in everyday speech. Additionally, words such as ''open set|open'' and ''field (mathematics)|field'' have been given specialized mathematical meanings. Mathematical jargon includes technical terms such as ''homeomorphism'' and ''integrability|integrable''. But there is a reason for special notation and technical jargon: mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as "rigor".
Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow from axioms by means of systematic reasoning. This is to avoid mistaken "theorems", based on fallible intuitions, of which many instances have occurred in the history of the subject.[See false proof for simple examples of what can go wrong in a formal proof. The Four color theorem#History|history of the Four Color Theorem contains examples of false proofs accepted by other mathematicians.] The level of rigor expected in mathematics has varied over time: the Greeks expected detailed arguments, but at the time of Isaac Newton the methods employed were less rigorous. Problems inherent in the definitions used by Newton would lead to a resurgence of careful analysis and formal proof in the 19th century. Today, mathematicians continue to argue among themselves about computer-assisted proofs. Since large computations are hard to verify, such proofs may not be sufficiently rigorous.[ 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). ]
Axioms in traditional thought were "self-evident truths", but that conception is problematic. At a formal level, an axiom is just a string of symbolic logic|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.[ 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." ]
Mathematics as science
Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences".[Waltershausen] In the original Latin ''Regina Scientiarum'', as well as in German language|German ''Königin der Wissenschaften'', the word corresponding to ''science'' means (field of) knowledge. Indeed, this is also the original meaning in English, and there is no doubt that mathematics is in this sense a science. The specialization restricting the meaning to ''natural'' science is of later date. If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. Albert Einstein has stated that ''"as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.''"[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''.]
Many philosophers believe that mathematics is not experimentally falsifiability|falsifiable, and thus not a science according to the definition of Karl Popper.[Popper 1995, p. 56] Other thinkers, notably Imre Lakatos, have applied a version of falsificationism to mathematics itself.
An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is ''public knowledge'' and thus includes mathematics.[Ziman] In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. intuition (knowledge)|Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences. Experimental mathematics continues to grow in importance within mathematics, and computation and simulation are playing an increasing role in both the sciences and mathematics, weakening the objection that mathematics does not use the scientific method. In his 2002 book ''A New Kind of Science'', Stephen Wolfram argues that computational mathematics deserves to be explored empirically as a scientific field in its own right.
The opinions of mathematicians on this matter are varied. Many mathematicians feel that to call their area a science is to downplay the importance of its aesthetic side, and its history in the traditional seven liberal arts; others feel that to ignore its connection to the sciences is to turn a blind eye to the fact that the interface between mathematics and its applications in science and engineering has driven much development in mathematics. One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematics is ''created'' (as in art) or ''discovered'' (as in science). It is common to see 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.
Mathematical awards are generally kept separate from their equivalents in science. The most prestigious award in mathematics is the Fields Medal|Fields Medal,["''The Fields Medal is now indisputably the best known and most influential award in mathematics.''" Monastyrsky][Riehm] established in 1936 and now awarded every 4 years. It is often considered, misleadingly, the equivalent of science's Nobel Prizes. The Wolf Prize in Mathematics, instituted in 1978, recognizes lifetime achievement, and another major international award, the Abel Prize, was introduced in 2003. These are awarded for a particular body of work, which may be innovation, or resolution of an outstanding problem in an established field. A famous list of 23 such open problems, called "Hilbert's problems", was compiled in 1900 by German mathematician David Hilbert. This list achieved great celebrity among mathematicians, and at least nine of the problems have now been solved. A new list of seven important problems, titled the "Millennium Prize Problems", was published in 2000. Solution of each of these problems carries a million reward, and only one (the Riemann hypothesis) is duplicated in Hilbert's problems.
Fields of mathematics
As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomy|astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e., arithmetic, algebra, geometry, and 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.
Quantity
The study of quantity starts with numbers, first the familiar natural numbers and integers ("whole numbers") and arithmetical operations on them, which are characterized in arithmetic. The deeper properties of integers are studied in number theory, whence such popular results as Fermat's Last Theorem. Number theory also holds two widely considered unsolved problems: the twin prime conjecture and Goldbach's conjecture.
As the number system is further developed, the integers are recognized as a subset of the rational numbers ("Fraction (mathematics)|fractions"). These, in turn, are contained within the real numbers, which are used to represent Continuous function|continuous quantities. Real numbers are generalized to complex numbers. These are the first steps of a hierarchy of numbers that goes on to include quarternions and octonions. Consideration of the natural numbers also leads to the transfinite numbers, which formalize the concept of counting to infinity. Another area of study is size, which leads to the cardinal numbers and then to another conception of infinity: the aleph numbers, which allow meaningful comparison of the size of infinitely large sets.
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Applied mathematics
Applied mathematics considers the use of abstract mathematical tools in solving concrete problems in the sciences, business, and other areas. An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a role. Most experiments, surveys and observational studies require the informed use of statistics. (Many statisticians, however, do not consider themselves to be mathematicians, but rather part of an allied group.) Numerical analysis investigates computational methods for efficiently solving a broad range of mathematical problems that are typically too large for human numerical capacity; it includes the study of rounding errors or other sources of error in computation.
Image:Gravitation space source.png | Mathematical physics
Image:BernoullisLawDerivationDiagram.svg | Fluid mechanics|Mathematical fluid dynamics
Image:Composite trapezoidal rule illustration small.svg | Numerical analysis
Image:Maximum boxed.png | Optimization (mathematics)|Optimization
Image:Two red dice 01.svg | Probability theory
Image:Oldfaithful3.png | Statistics
Image:Market Data Index NYA on 20050726 202628 UTC.png | Financial mathematics
Image:Arbitrary-gametree-solved.svg | Game theory
Common misconceptions
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. Mathematicians publish many thousands of papers embodying new discoveries in mathematics every month.
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 papers 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 amateur contributors such as Pierre de Fermat|Fermat, Marin Mersenne|Mersenne, and Srinivasa Ramanujan|Ramanujan. See further the List of amateur mathematicians.
Mathematics and physical reality
Mathematical concepts and theorems need not correspond to anything in the physical world. Insofar as a correspondence does exist, while mathematicians and physicists may select axioms and postulates that seem reasonable and intuitive, it is not necessary for the basic assumptions within an axiomatic system to be true in an empirical or physical sense. Thus, while many axiom systems are derived from our perceptions and experiments, they are not dependent on them.
For example, we could say that the physical concept of two apples may be accurately mathematical model|modeled by the natural number 2. On the other hand, we could also say that the natural numbers are ''not'' an accurate model because there is no standard "unit" apple and no two apples are exactly alike. The modeling idea is further complicated by the possibility of fraction (mathematics)|fractional or partial apples. So while it may be instructive to visualize the axiomatic definition of the natural numbers as collections of apples, the definition itself is not dependent upon nor derived from any actual physical entities.
Nevertheless, mathematics remains extremely useful for solving real-world problems.
See also
Notes
References
- 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; 2 edition (March 6, 1991). ISBN 0-471-54397-7. — A concise history of mathematics from the Concept of Number to contemporary Mathematics.
- Courant, R. and 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.
- 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. — A gentle introduction to the world of mathematics.
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- Eves, Howard, ''An Introduction to the History of Mathematics'', Sixth Edition, Saunders, 1990, ISBN 0-03-029558-0.
- Gullberg, Jan, ''Mathematics — From the Birth of Numbers''. W. W. Norton & Company; 1st edition (October 1997). ISBN 0-393-04002-X. — An encyclopedic overview of mathematics presented in clear, simple language.
- 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 online http://eom.springer.de/default.htm.
- Jourdain, Philip E. B., ''The Nature of Mathematics'', in ''The World of Mathematics'', James R. Newman, editor, Dover, 2003, ISBN 0-486-43268-8.
- 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.
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- 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.
- JSTOR.
- Peterson, Ivars, ''Mathematical Tourist, New and Updated Snapshots of Modern Mathematics'', Owl Books, 2001, ISBN 0-8050-7159-8.
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External links
- Online Encyclopaedia of Mathematics http://eom.springer.de from Springer. Graduate-level reference work with over 8,000 entries, illuminating nearly 50,000 notions in mathematics.
- Some mathematics applets, at MIT
- Rusin, Dave: ''The Mathematical Atlas''. A guided tour through the various branches of modern mathematics. (Can also be found here.)
- Stefanov, Alexandre: ''Textbooks in Mathematics''. A list of free online textbooks and lecture notes in mathematics.
- Weisstein, Eric et al.: ''MathWorld: World of Mathematics''. An online encyclopedia of mathematics.
- Polyanin, Andrei: ''EqWorld: The World of Mathematical Equations''. An online resource focusing on algebraic, ordinary differential, partial differential ( mathematical physics), integral, and other mathematical equations.
- ''Planet Math''. An online mathematics encyclopedia under construction, focusing on modern mathematics. Uses the GNU Free Documentation License| GFDL, allowing article exchange with Wikipedia. Uses TeX markup.
- ''Metamath''. A site and a language, that formalize mathematics from its foundations.
- Mathematician Biographies. The MacTutor History of Mathematics archive Extensive history and quotes from all famous mathematicians.
- Cain, George: Online Mathematics Textbooks available free online.
- Math & Logic: The history of formal mathematical, logical, linguistic and methodological ideas. In ''The Dictionary of the History of Ideas.''
- Nrich, a prize-winning site for students from age five from University of Cambridge| Cambridge University
- 'FreeScience Library->Mathematics ' The mathematics section of FreeScience library
- Open Problem Garden, a wiki of open problems in mathematics
- Applications of High School Algebra
- HyperMath site at Georgia State University
Category:Mathematics|Mathematics
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zh-yue:數學
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