# mathematics

## Information about mathematics

Euclid, Greek mathematician, 3rd century BC, as imagined by Raphael in this detail from The School of Athens.[1]

Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".[2] Other practitioners of mathematics[3][4] maintain that mathematics is the science of pattern, that mathematicians seek out patterns whether found in numbers, space, science, computers, imaginary abstractions, or elsewhere. Mathematicians explore such concepts, aiming to formulate new conjectures and establish their truth by rigorous deduction from appropriately chosen axioms and definitions.[5]

Through the use of abstraction and logical reasoning, mathematics evolved from counting, calculation, measurement, and the systematic study of the shapes and 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 ancient Egypt, Mesopotamia, ancient India, ancient China, and ancient Greece. Rigorous arguments appear in Euclid's Elements. The development continued in fitful bursts until the Renaissance period of the 16th century, when mathematical innovations interacted with new scientific discoveries, leading to an acceleration in research that continues to the present day.[6]

Today, mathematics is used throughout the world in many fields, including natural science, engineering, medicine, and the social sciences such as economics. Applied mathematics, the application of mathematics to such 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 applications for what began as pure mathematics are often discovered later.[7]

## Etymology

The word "mathematics" (Greek: μαθηματικά or mathēmatiká) comes from the 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, μαθηματικὴ τέχνη (mathēmatikḗ tékhnē), in Latin ars mathematica, meant the mathematical art.

The apparent plural form in English, like the French plural form les mathématiques (and the less commonly used singular derivative la mathématique), goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural τα μαθηματικά (ta mathēmatiká), used by Aristotle, and meaning roughly "all things mathematical".[8] In English, however, mathematics is a singular noun, often shortened to math in English-speaking North America and maths elsewhere.

## History

A quipu, a counting device used by the Inca.

The evolution of mathematics might be seen as an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges have something in common was a breakthrough in human thought. In addition to recognizing how to count physical objects, prehistoric peoples also recognized how to count abstract quantities, like timedays, seasons, years. Arithmetic (addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to knowledge of geometry.

Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called quipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse, with the first known written numerals created by Egyptians in Middle Kingdom texts such as the Rhind Mathematical Papyrus.

From the beginnings 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 measure land, and to predict 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 proofs."[9]

## Inspiration, pure and applied mathematics, and aesthetics

Sir Isaac Newton (1643-1727), an inventor of infinitesimal calculus.
Main article: Mathematical beauty
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. Newton was one of the infinitesimal calculus inventors, although nearly all of the notation used in infinitesimal calculus was contributed by Leibniz with the exception of a dot above a variable to signify differentiation with respect to time. Feynman invented the Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. Some mathematics is only relevant 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. The 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."

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. 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, 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. Mathematicians often strive to find proofs of theorems that are particularly elegant, a quest Paul Erdős often referred to as finding proofs from "The Book" in which God had written down his favorite proofs. The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions.

## Notation, language, and rigor

In modern notation, simple expressions can describe complex concepts. This image shows the graph of cos(y arccos sin|x| + x arcsin cos|y|).
Main article: Mathematical notation

Most of the mathematical notation in use today was not invented until the 16th century.[10] Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery. In the 18th century, 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 also is hard for beginners. Words such as or and only have more precise meanings than in everyday speech. Also confusing to beginners, words such as open and field have been given specialized mathematical meanings. Mathematical jargon includes technical terms such as homeomorphism and 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.[11] 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.[12] Axioms 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 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.[13]

## Mathematics as science

Carl Friedrich Gauss, himself known as the "prince of mathematicians", referred to mathematics as "the Queen of the Sciences".
Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences".[14] In the original Latin Regina Scientiarum, as well as in 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."[15]

Many philosophers believe that mathematics is not experimentally falsifiable, and thus not a science according to the definition of Karl Popper. However, in the 1930s important work in mathematical logic showed that mathematics cannot be reduced to logic, and Karl Popper concluded that "most mathematical theories are, like those of physics and biology, hypothetico-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently."[16] 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.[17] 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 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. While some in applied mathematics feel that they are scientists, those in pure mathematics often feel that they are working in an area more akin to logic and that they are, hence, fundamentally philosophers. 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 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,[18][19] 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 1979, 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 \$1 million reward, and only one (the Riemann hypothesis) is duplicated in Hilbert's problems.

## Fields of mathematics

An abacus, a simple calculating tool used since ancient times
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 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 analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (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 ("fractions"). These, in turn, are contained within the real numbers, which are used to represent 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.
 Natural numbers Integers Rational numbers Real numbers Complex numbers

### Structure

Many mathematical objects, such as sets of numbers and functions, exhibit internal structure. The structural properties of these objects are investigated in the study of groups, rings, fields and other abstract systems, which are themselves such objects. This is the field of abstract algebra. An important concept here is that of vectors, generalized to vector spaces, and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics: quantity, structure, and space. Vector calculus expands the field into a fourth fundamental area, that of change.
 Number theory Abstract algebra Group theory Order theory

### Space

The study of space originates with geometry - in particular, Euclidean geometry. Trigonometry 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. Within differential geometry are the concepts of fiber bundles and calculus on manifolds. 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 groups 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, and includes the long-standing Poincaré conjecture and the controversial four color theorem, whose only proof, by computer, has never been verified by a human.
 Geometry Trigonometry Differential geometry Topology Fractal geometry

### Change

Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a powerful tool to investigate it. Functions arise here, as a central concept describing a changing quantity. The rigorous study of real numbers and real-valued functions is known as real analysis, with complex analysis the equivalent field for the complex numbers. The Riemann hypothesis, one of the most fundamental open questions in mathematics, is drawn from complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many problems lead naturally to relationships between a quantity and its rate of change, and these are studied as differential equations. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior.
 Calculus Vector calculus Differential equations Dynamical systems Chaos theory

### Foundations and philosophy

In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed, as well as category theory which is still in development.

Mathematical logic is concerned with setting mathematics on a rigid axiomatic framework, and studying the results of such a framework. As such, it is home to Gödel's second incompleteness theorem, perhaps the most widely celebrated result in logic, which (informally) implies that any 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). Gödel showed how to construct, whatever the given collection of number-theoretical axioms, a formal statement in the logic that is a true number-theoretical fact, but which does not follow from those axioms. Therefore no formal system is a true 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.
 Mathematical logic Set theory Category theory

### Discrete mathematics

Discrete mathematics is the common name for the fields of mathematics most generally useful in theoretical computer science. This includes computability theory, computational complexity theory, and information theory. Computability theory examines the limitations of various theoretical models of the computer, including the most powerful 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 rapid advance of computer hardware. Finally, information theory is concerned with the amount of data that can be stored on a given medium, and hence concepts such as compression and entropy.

As a relatively new field, discrete mathematics has a number of fundamental open problems. The most famous of these is the "P=NP?" problem, one of the Millennium Prize Problems.[20]
 Combinatorics Theory of computation Cryptography Graph theory

### 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.

 Mathematical physics Mathematical fluid dynamics Numerical analysis Optimization Probability Statistics Financial mathematics Game theory

## Common misconceptions

Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. Mathematicians publish many thousands of papers embodying new discoveries in mathematics every month.

Mathematics is not numerology, nor is it 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.
The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne.

### 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 most systems of axioms 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 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 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. This fact led Eugene Wigner to write an essay, The Unreasonable Effectiveness of Mathematics in the Natural Sciences.

## Notes

1. ^ 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).
2. ^ Peirce, p.97
3. ^ Steen, L.A. (April 29, 1988). The Science of Patterns. Science, 240: 611–616. and summarized at Association for Supervision and Curriculum Development.
4. ^ Devlin, Keith, Mathematics: The Science of Patterns: The Search for Order in Life, Mind and the Universe (Scientific American Paperback Library) 1996, ISBN 9780716750475
5. ^ Jourdain
6. ^ Eves
7. ^ Peterson
8. ^ The Oxford Dictionary of English Etymology, Oxford English Dictionary
9. ^ Sevryuk
10. ^ Earliest Uses of Various Mathematical Symbols (Contains many further references)
11. ^ See false proof for simple examples of what can go wrong in a formal proof. The history of the Four Color Theorem contains examples of false proofs accepted by other mathematicians.
12. ^ 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).
13. ^ 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."
14. ^ Waltershausen
15. ^ 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.
16. ^ Popper 1995, p. 56
17. ^ Ziman
18. ^ "The Fields Medal is now indisputably the best known and most influential award in mathematics." Monastyrsky
19. ^ Riehm
20. ^ Clay Mathematics Institute P=NP

## 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.
• 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.
• Davis, Philip J. and 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.
• Einstein, Albert (1923). "Sidelights on Relativity (Geometry and Experience)".
• 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.
• Kline, Morris, Mathematical Thought from Ancient to Modern Times, Oxford University Press, USA; Paperback edition (March 1, 1990). ISBN 0-19-506135-7.
• Monastyrsky, Michael (2001). "Some Trends in Modern Mathematics and the Fields Medal". Canadian Mathematical Society. Retrieved on 2006-07-28.
• 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.
• Peirce, Benjamin. "Linear Associative Algebra". American Journal of Mathematics (Vol. 4, No. 1/4. (1881).  JSTOR.
• Peterson, Ivars, Mathematical Tourist, New and Updated Snapshots of Modern Mathematics, Owl Books, 2001, ISBN 0-8050-7159-8.
• Paulos, John Allen (1996). A Mathematician Reads the Newspaper. Anchor. ISBN 0-385-48254-X.
• Popper, Karl R. (1995). "On knowledge", In Search of a Better World: Lectures and Essays from Thirty Years. Routledge. ISBN 0-415-13548-6.
• Riehm, Carl (August 2002). "The Early History of the Fields Medal". Notices of the AMS 49 (7): 778-782.
• Sevryuk, Mikhail B. (January 2006). "Book Reviews" (PDF). Bulletin of the American Mathematical Society 43 (1): 101-109. Retrieved on 2006-06-24.
• Waltershausen, Wolfgang Sartorius von (1856, repr. 1965). Gauss zum Gedächtniss. Sändig Reprint Verlag H. R. Wohlwend. ISBN 3-253-01702-8.
• Ziman, J.M., F.R.S. (1968). "Public Knowledge:An essay concerning the social dimension of science".

## External links

At Wikiversity you can learn more and teach others about Mathematics at:

Mathematics may also refer to
• Mathematics (producer), a hip-hop producer.
• Mathematics (album), an album by the band The Servant.
• Mathematics (song), a song by Mos Def
• Mathematics Magazine, a publication of the Mathematical Association of America.

..... Click the link for more information.
Math is used as a colloquial shortening of mathematics.

Math may also refer to:
• Matha, a Hindu or Jain religious order
• Math fab Mathonwy, a king in Welsh mythology:

..... Click the link for more information.
Quantity is a kind of property which exists as magnitude or multitude. It is among the basic classes of things along with quality, substance, change, and relation. Quantity was first introduced as quantum, an entity having quantity.
..... Click the link for more information.
Structure is a fundamental and sometimes intangible notion covering the recognition, observation, nature, and stability of patterns and relationships of entities. From a child's verbal description of a snowflake, to the detailed scientific analysis of the properties of magnetic
..... Click the link for more information.
The term SPACE (capitalized) can refer to:
• , a Canadian science-fiction channel
• The Society for Promotion of Alternative Computing and Employment
• DSPACE, a term in computational complexity theory

..... Click the link for more information.
Change denotes the transition that occurs between one state to another.

## Physical

Throughout history, change has been defined by varying points of view. In ancient Greek philosophy, while Herachical position may have an influence on the perception of change.
..... Click the link for more information.
Benjamin Peirce (pronounced purse), April 4, 1809 – October 6, 1880) was an American mathematician who taught at Harvard University for forty years. He made contributions to celestial mechanics, number theory, algebra, and the philosophy of mathematics.
..... Click the link for more information.
mathematician is a person whose primary area of study and research is the field of mathematics.

## Problems in mathematics

Some people incorrectly believe that mathematics has been fully understood, but the publication of new discoveries in mathematics continues at an immense
..... Click the link for more information.
In mathematics, a conjecture is a mathematical statement which appears likely to be true, but has not been formally proven to be true under the rules of mathematical logic.
..... Click the link for more information.
Deductive reasoning, according to many dictionaries[1][2][3][4], is the type of reasoning that proceeds from general principles or premises to derive particular information.
..... Click the link for more information.
axiom is a sentence or proposition that is not proved or demonstrated and is considered as self-evident or as an initial necessary consensus for a theory building or acceptation.
..... Click the link for more information.
A definition is a statement of the meaning of a term, word or phrase. The term to be defined is known as the definiendum (Latin: that which is to be defined).
..... Click the link for more information.
Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalising it so that it has wider applications.
..... Click the link for more information.
Logic (from Classical Greek λόγος logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration.
..... Click the link for more information.
Reasoning is the mental (cognitive) process of looking for reasons for beliefs, conclusions, actions or feelings.[1] Humans have the ability to engage in reasoning about their own reasoning using introspection.
..... Click the link for more information.
Counting is the mathematical action of repeatedly adding (or subtracting) one, usually to find out how many objects there are or to set aside a desired number of objects (starting with one for the first object and proceeding with an injective function from the remaining objects to
..... Click the link for more information.
A calculation is a deliberate process for transforming one or more inputs into one or more results.

The term is used in a variety of senses, from the very definite arithmetical calculation using an algorithm to the vague heuristics of calculating a strategy in a competition
..... Click the link for more information.
Measurement is the estimation of the magnitude of some attribute of an object, such as its length or weight, relative to a unit of measuremnt. Measurement usually involves using a measuring instrument, such as a ruler or scale, which is calibrated to compare the object to some
..... Click the link for more information.
Shape (OE. sceap Eng. created thing), refers to the external two-dimensional outline, appearance or configuration of some thing — in contrast to the matter or content or substance of which it is composed.
..... Click the link for more information.
cite any references or sources. Please help improve this article by citing reliable sources.
* It is in need of attention from an expert on the subject. may be able to help recruit one.
* It needs to be expanded.
..... Click the link for more information.
Editing of this page by unregistered or newly registered users is currently disabled due to vandalism.
If you are prevented from editing this page, and you wish to make a change, please discuss changes on the talk page, request unprotection, log in, or .
..... Click the link for more information.
Mesopotamia was a cradle of civilization geographically located between the Tigris and Euphrates rivers, largely corresponding to modern-day Iraq. Sumer in southern Mesopotamia is commonly regarded as the world's earliest civilization.
..... Click the link for more information.
History of India begins with the Indus Valley Civilization, which flourished in the north-western part of the Indian subcontinent from 3300 to 1700 BCE. This Bronze Age civilization was followed by the Iron Age Vedic period, which witnessed the rise of major kingdoms known as the
..... Click the link for more information.
The history of China is told in traditional historical records that refer as far back as the Three Sovereigns and Five Emperors about 5,000 years ago, supplemented by archaeological records dating to the 16th century BC. China is one of the world's oldest continuous civilizations.
..... Click the link for more information.
The term ancient Greece refers to the periods of Greek history in Classical Antiquity, lasting ca. 750 BC[1] (the archaic period) to 146 BC (the Roman conquest). It is generally considered to be the seminal culture which provided the foundation of Western Civilization.
..... Click the link for more information.
Euclid

Born fl. 300 BC

Residence Alexandria, Egypt
Nationality Greek
Field Mathematics
Known for Euclid's Elements Euclid (Greek:
..... Click the link for more information.
Euclid's Elements (Greek: Στοιχεῖα) is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria circa 300 BC.
..... Click the link for more information.
Renaissance (French for "rebirth"; Italian: Rinascimento; Spanish: Renacimiento), was a cultural movement that spanned roughly the 14th through the 17th century, beginning in Italy in the late Middle Ages and later spreading to the rest of Europe.
..... Click the link for more information.
As a means of recording the passage of time, the 16th century was that century which lasted from 1501 through 1600.

See also: 16th century in literature

## Events

### 1500s

• 1500s: Mississippian culture disappears.

..... Click the link for more information.
Science (from the Latin scientia, 'knowledge'), in the broadest sense, refers to any systematic knowledge or practice.[1] Examples of the broader use included political science and computer science, which are not incorrectly named, but rather named according to
..... Click the link for more information.

This article is copied from an article on Wikipedia.org - the free encyclopedia created and edited by online user community. The text was not checked or edited by anyone on our staff. Although the vast majority of the wikipedia encyclopedia articles provide accurate and timely information please do not assume the accuracy of any particular article. This article is distributed under the terms of GNU Free Documentation License.