mathematics

(măth'ə-măt'ĭks)

[From Middle English mathematik, from Old French mathematique, from Latin mathēmatica, from Greek mathēmatikē (tekhnē), mathematical (science), feminine of mathēmatikos, mathematical. See mathematical.]

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Greek tradition ascribed the invention of geometry to the Egyptians and its introduction into Greece to Thalēs and Pythagoras. The first Greek mathematician of whose work there is clear evidence is Hippocratēs (2) (late fifth century BC). See also APOLLONIUS OF PERGE, ARCHIMEDES, ARCHYTAS, ARISTARCHUS OF SAMOS, CONON (2), ERATOSTHENES, EUCLID, EUDOXUS, HIPPARCHUS (2), HYPATIA, METON, PROCLUS, PTOLEMY, THEON, and NUMBERS.

In his "Mathematical Praeface" to the Elements of Euclid of 1570, Elizabethan polymath John Dee (1527–1608) expounded on the importance and utility of mathematics to all fields of human endeavor. Field after field, he argued, from those we would find obvious (like navigation) to those we would find arcane (astrology) or outlandish (thaumaturgike), would benefit from the systematic application of mathematics. Although Dee was promoting a role for mathematics that was just taking shape during his lifetime, his vision did indeed prove prophetic. Undoubtedly, one of the most striking features of intellectual life in the early modern period is the startling expansion in the scholarly and practical domains covered by mathematics.

Mathematics and Its Critics

Prior to the sixteenth century, mathematics in the West was a well-defined and circumscribed field consisting of two main branches: arithmetic, which had obvious practical applications in commerce and banking, and Euclidean geometry, which had few practical uses apart from astrology and, occasionally, optics. While mathematics was generally admired for the certainty and universality of its claims, the world as a whole, in keeping with Aristotelian tradition, was distinctly unmathematical, being governed by qualitative rather than quantitative rules. By the eighteenth century this view had been turned on its head: not only was an ever increasing number of fields being subjected to mathematical analysis, but the world itself had come to be understood as fundamentally mathematical in nature.

These developments were by no means a foregone conclusion in the sixteenth century; if anything, they seemed highly unlikely. For mathematics, far from being universally acknowledged as central to the intellectual and technological life of the age, was at the time being challenged as never before from various quarters.

Conservative critics, defending the established order of knowledge, challenged the truth claims of mathematics as incompatible with prevailing Aristotelian standards. Prominent among them were Italian philosopher Alessandro Piccolomini (1508–1578) and the Jesuit Benito Pereira (c. 1535–1610), who challenged the explanatory value of mathematical proofs. Proper scientific explanations, they argued with perfect Aristotelian orthodoxy, were causal arguments, proceeding from the true essence of objects to their properties. Mathematics, however, had no proper subject matter at all, and it could say nothing about the essential nature of physical objects. All mathematics could do was point to logical relations between hypothetical propositions, and thus it was a fundamentally inferior type of knowledge.

Mathematics did not fare much better among the new generation of reformers, who sought to uproot the Aristotelian framework and replace it with new conceptions of knowledge. In breaking the hold of Aristotelian standards on contemporary natural philosophy, many reformers found little use for mathematics. Its rigid procedures and unchanging truths seemed an unpromising basis for a radical reform of knowledge. The study of nature, many argued, should proceed through unmediated experience and systematic trial and error. The rigorous deductive reasoning characteristic of mathematics could only lead to predetermined and unvarying results. The maverick Italian philosopher Giordano Bruno (1548–1600), for example, argued that mathematics could only describe the external appearance of phenomena, but never penetrate their hidden secrets. Similarly, in England, Francis Bacon (1561–1626) in the Novum Organum insisted that mathematics "should only give limits to natural philosophy, not generate or beget it."

Mathematics did, of course, have many prominent defenders, ranging from the Jesuit Christopher Clavius (1537–1612) to Galileo Galilei (1564–1642) and René Descartes (1596–1650), each insisting in his way on the essential role of mathematics in any meaningful scheme of knowledge. But the very range of suggestions these and other natural philosophers offered for the role of mathematics in the general scheme of knowledge makes it clear that the fundamental questions raised by the challenges to mathematics did not go away. The critiques raised the fundamental questions that would guide the development of mathematics throughout the early modern period: what is mathematics, and how is it related to the natural world? The history of mathematics in this period is the story of the various answers that were given to these questions.

The World As Mirror of Mathematics

The fundamental answer to the critiques of mathematics was given by Galileo in his Assayer of 1623, when he wrote that the universe "is written in the language of mathematics." Galileo was expressing the widely held notion among practitioners that mathematics, far from being devoid of all subject matter as claimed by its critics, had the entire natural world as its object. But while most agreed that mathematics was closely integrated with the physical world, the precise nature of their relationship remained a matter of intense dispute.

One leading approach accepted the classical view of mathematics as a rigorous deductive science of number and magnitude. The universal laws of mathematics, in this view, were the fundamental laws that governed material reality. Thus when one is investigating mathematical and geometrical relationships, one is in fact investigating the basic structure of matter.

The chief promoter of this approach was René Descartes, who viewed mathematics as a fundamental rational law laid down by God for his creation. Once God, the divine architect, had set in motion his perfectly rational universe, it would henceforth operate forever in accordance with mathematical principles. Mathematical investigations are accordingly studies of the divine plan for the natural world, and the world is the direct expression of abstract mathematical principles.

Descartes's scientific work directly reflects this fundamental understanding. In his Meditations and the Discourse on Method, Descartes insisted that by following strict rational rules one could, in principle, follow God and "create" the world step by step. Rigorous rational deduction was therefore the key to knowledge of the natural world, and Descartes proceeded to demonstrate the effectiveness of this principle in short treatises on optics and the colors of the rainbow, which were attached to the early editions of the Discourse.

Descartes's most important contribution to mathematics was also a reflection of his religious and philosophical views. The Geometry was the founding text of analytic geometry and, like Descartes's other scientific treatises, was published as an appendix to the Discourse. In essence, the new field demonstrated the fundamentally mathematical nature of the physical world. Abstract algebraic relationships (that is, y=3Dax+b) were shown to have actual physical manifestations (in this case, a straight line). In pointing out these hidden relationships Descartes was unveiling the divine mathematical laws that governed the world. Mathematics, in this view, was a perfectly rational and logical web of relationships that determined the nature of physical reality.

Mathematics As the Mirror of the World

While Descartes was honing his analytical geometry, a very different mathematical approach, based on a very different understanding of the relationship of mathematics to the world, was being developed elsewhere in Europe. The use of infinitesimals, or "indivisibles" as they were most commonly called, in calculating lengths, areas, and volumes of geometrical figures was the most dramatic and important development in seventeenth-century mathematics. Fundamentally, the procedure involved reducing geometrical objects into an infinite number of their component parts: lines were viewed as an infinite collection of points, surfaces as made up of an infinite number of lines, and solids of surfaces. The length, area, or volume of the figure as a whole would then be calculated as the infinite sum of its elementary components.

The fundamental assumptions underlying this procedure were highly questionable and seemed to fly in the face of paradoxes that had been well known since antiquity. Descartes, who was much concerned with the perfect rational structure of mathematics, rejected infinitesimals and excluded them from the bounds of mathematics. Nevertheless, the effectiveness of this approach in reaching correct and previously unknown results was undeniable, and it was embraced enthusiastically by mathematicians across Europe. Thomas Hariot (1560–1621) and John Wallis (1616–1703) in England, Galileo and his disciples Bonaventura Cavalieri (c. 1598–1647) and Evangelista Torricelli (1608–1647) in Italy, Johannes Kepler (1571–1630) in Germany, and Blaise Pascal (1623–1662) in France were but a few of the most prominent practitioners of the new methods.

The infinitesimalist mathematicians' view of the relationship between mathematics and the world was, in many ways, the reverse of Descartes's approach. Whereas Descartes assumed that pure mathematical relationships governed the structure of matter, the infinitesimalists modeled mathematics on an intuition of the physical world. Geometrical bodies could be broken down into their indivisible components because, by analogy, physical bodies could be divided in the same way. As Cavalieri, whose Geometria Indivisibilibus was the most influential book about the theory and practice of indivisibles, wrote in his introduction, "plane figures should be conceived by us in the same manner as cloths are made up of parallel threads, and solids are in fact like books, composed of parallel pages."

The infinitesimalists' approach to mathematics drew much of its inspiration from the empiricist experimental philosophy that was gaining ground throughout Europe at this time. Much as the experimentalists sought to penetrate through external appearances and bring to light the inner structure of the material world, the new mathematicians sought to uncover the "inner structure" of geometrical figures, which in their view was the true cause of all geometrical relationships. Both groups, furthermore, adopted the imagery of geographical exploration as their guiding metaphor, presenting themselves as adventurous explorers on the hazardous seas of mathematics and natural philosophy.

Like their experimentalist colleagues, the infinitesimalists made the discovery of new and correct results the true test of their success, and like them they often adopted a methodology of trial and error in searching for the correct answers. This "experimental" approach to mathematics accounts for the infinitesimalists' relative disregard for the niceties of mathematical rigor and consistency. In their view, if a method produces true results, it must be fundamentally correct, and there was no point in spending too much time on clarifying the finer logical points. The most outspoken and unapologetic proponent of this approach was probably John Wallis, who advocated applying the experimentalists' "method of induction" to mathematics, in preference to traditional rigorous mathematical deduction.

While the new infinitesimalist approaches were in wide use in the seventeenth century, they were also seriously challenged in certain influential quarters. The issues at stake were not purely mathematical in nature, but involved wide-ranging philosophical, religious, and even political considerations. For one thing, the new approaches carried the taint of atomism—the ancient view that all material objects could be reduced to indivisible particles called "atoms" (from the Greek atomos, 'uncuttable'). Indeed there was no denying that the fundamental insights of the new mathematics and even its name strongly hinted that infinitesimalist mathematics was nothing but an expansion of atomism into mathematics.

This in turn led to a deeper difficulty: the suspicion that the new mathematics was based not just on atomism, but on materialism, which is the notion that the world was composed of nothing but matter, leaving no room for a providential spiritual realm. Geometry, after all, was often taken to be the very model of pure and abstract reasoning that governs the natural world. The notion that geometry itself, far from governing physical reality, is in fact a generalization of it, seemed to turn the proper hierarchy of mind and matter on its head, and challenge those who insisted that the world was ruled by a higher intelligence.

Finally, there was the question of the certainty of knowledge. Infinitesimalist mathematics seemed to be based on nothing more than a loose analogy with the physical world, trial and error, and a willful disregard for logical paradox. If even mathematics, that paragon of certain and unchanging knowledge, turned out to be so unsound, what hope could other, less rigorous fields have of attaining true knowledge?

In an age that still considered science, philosophy, and theology to be part of a single unified worldview, these criticisms cut deep. Descartes, concerned about the rational certainty of his method, excluded infinitesimal methods from proper mathematics. Even more significant was the reaction of the Society of Jesus, the most prominent religious order in Europe and the guardian of Catholic orthodoxy. Despite having among their members some of the most important and creative mathematicians in Europe, the Jesuits banned the teaching of infinitesimals from their educational institutions.

The Calculus and Beyond

The invention of the calculus by Isaac Newton (1642–1727) and Gottfried Wilhelm Leibniz (1646–1716) in the late seventeenth century was the most important development of early modern mathematics, and it quickly transformed the landscape of the field. The calculus took as its starting point the many practical techniquesand results achievedbythe infinitesimalist mathematicians, both in the determination of surfaces and volumes of geometrical figures, and in the calculation of tangents of curves. The fundamental insight of the calculus was that these two operations, calculating tangents (differentiation) and calculating surfaces and volumes (integration), are in fact the inverse of one another.

The importance of this discovery becomes clear when curves and geometrical figures are presented not as independent geometrical figures, but as expressions of algebraic formulations in the manner of analytic geometry. When presented in this manner, differentiation no longer deals with geometrical properties of particular geometrical objects, but becomes an abstract and general relationship between algebraic expressions. For example, one can say that the parabola expressed as y=3Dx2 describes the area under the line y=3D2x, and that y=3D2x expresses the tangent of the parabola y=3Dx2 at any point. But the relationship between the two algebraic expressions is no longer dependent on their particular geometrical representation: y=3D2x is simply the differential of y=3Dx2 and y=3Dx2is the integral of y=3D2x. The inverse relationship is a fundamental relationship between abstract algebraic expressions (or functions, as they came to be called later in the eighteenth century) independent of any particular geometric representation. Both Newton and Leibniz were quick to reduce the transformations back and forth between differentials and integrals (or "fluents" and "fluxions" as Newton called them) into systematic and reliable algorithms.

In the calculus, the two competing traditions of seventeenth-century mathematics were brought together. Although it clearly grew out of the techniques developed by infinitesimalist mathematicians, the calculus was equally dependent on the algebraic formulations of analytic geometry. Furthermore, the calculus detached the infinitesimalist methods from their dependence on an intuition of physical reality. If the older approaches could be viewed as growing out of an atomistic intuition of material reality, the calculus restored the primacy of abstract logical relationship to mathematics. Particular geometric figures could be seen as examples of these abstract algebraic relations, but these relations themselves were no longer dependent on any particular physical or geometrical instances.

Mathematics in the Enlightenment

The calculus, which positioned mathematics as both an abstract system of algebraic relationships and as intimately connected to the physical world, set the tone for eighteenth-century views of the field. The most eloquent formulation of attitudes toward mathematics in the Enlightenment was given by Jean Le Rond d'Alembert (1717–1783), in his "Preliminary Discourse" to the Encyclopédie, published in 1751. Whereas seventeenth-century practitioners viewed mathematics as either a generalization of material intuitions or as a universal law governing nature, for d'Alembert mathematics was necessarily both. On the one hand, he insisted, mathematics is clearly an abstraction from nature: it is nothing but the fundamental relationships among natural objects that are arrived at when the material features such as texture and color are stripped away. On the other hand, d'Alembert argued, the laws of nature are simply elaborations of mathematical relationships, arrived at by restoring matter's physical attributes to abstract disembodied mathematics. The world, then, according to d'Alembert, is fundamentally mathematical: mathematics is derived from the physical world, while the physical world is an extension of mathematical principles.

This view of an essentially mathematical universe manifested itself in the inclusion of an evergrowing number of scholarly fields that were brought under the sway of mathematics in this period. Years before, Galileo had already introduced mathematics into the study of falling bodies and statics, and he and his followers extended his work to the field of ballistics. Cartographic work was thoroughly mathematized in the seventeenth century, and Kepler and Newton transformed the ancient science of astronomy by extending the reach of mathematics from merely describing the motions of the heavens into the realms of celestial mechanics. In optics, Descartes's ingenious application of his "method" enabled him to explain such phenomena such as the formation of the rainbow with mathematical precision.

In the eighteenth century, a new generation of mathematicians, including the Bernoullis, Leonhard Euler (1707–1783), d'Alembert, Joseph-Louis Lagrange (1736–1813), and Pierre-Simon Laplace (1749–1827), among others, added increasingly precise theories of mechanics and argued famously about proper mathematical representations of abstract concepts such as vis viva, and concrete problems like the vibrations of strings and hydromechanics. Other fields that were seemingly less malleable for quantitative analysis, like doctrines of chance, or probability, and also the "moral" sciences, known today as social sciences, were also brought under the sway of mathematics, particularly in the work of the marquis de Condorcet (1743–1794). Institutionally, the eighteenth century saw mathematics gain a quickly growing foothold in newly established engineering and military colleges.

Epilogue

Unfortunately for d'Alembert and other promoters of the mathematical universe, rigorous mathematical analysis could not be easily derived from physical reality. Inconsistencies and paradoxes seemed to crop up repeatedly when mathematics was modeled on perceptions of the physical world, as critics of infinitesimal methods and the calculus, such as George Berkeley, were quick to point out. At the same time, the physical world proved to be far more varied and surprising than could ever be derived from bare mathematical principles.

Early in the nineteenth century the interdependence of mathematics and the physical world, so eloquently presented by d'Alembert, came to an end. In their work on the foundations of the calculus, mathematicians Bernhard Bolzano (1781–1848) and Augustin-Louis Cauchy (1789–1857) reformulated mathematical analysis as rigorous and logically self-consistent, a goal that had eluded their Enlightenment predecessors. They did so, however, at a price that would have seemed too heavy for d'Alembert and his colleagues: pure mathematics, in their scheme, was finally divorced from physical reality, existing in its self-enclosed Platonic realm.

The course and development of mathematics in the early modern period had come full circle. Criticized in the sixteenth century for being irrelevant to the developing sciences, mathematicians at the time had responded by forming a closer bond than ever before between their field and the physical world. Two and a half centuries later, in an attempt to save the identity and coherence of their field, mathematicians chose to sever those same conceptual ties, and establish mathematics in its own separate and insular domain.

Bibliography

Alexander, Amir. Geometrical Landscapes: The Voyages of Discovery and the Transformation of Mathematical Practice. Stanford, 2002.

Boyer, Carl B. The History of the Calculus and its Conceptual Development. New York, 1959.

Daston, Lorraine J. Classical Probability in the Enlightenment. Princeton, 1988.

Dear, Peter. Discipline and Experience: The Mathematical Way in the Scientific Revolution. Chicago, 1995.

Hankins, Thomas L. Science and the Enlightenment. Cambridge, U.K., and New York, 1985.

—AMIR ALEXANDER

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Quotes:

"It is amusing to discover, in the twentieth century, that the quarrels between two lovers, two mathematicians, two nations, two economic systems, usually assumed insoluble in a finite period should exhibit one mechanism, the semantic mechanism of identification -- the discovery of which makes universal agreement possible, in mathematics and in life." - Alfred Korzybski

"Math is like love -- a simple idea but it can get complicated." - R. Drabek

"In studying mathematics or simply using a mathematical principle, if we get the wrong answer in sort of algebraic equation, we do not suddenly feel that there is an anti-mathematical principle that is luring us into the wrong answers." - Eric Butterworth

"Mathematics is not a book confined within a cover and bound between brazen clasps, whose contents it needs only patience to ransack; it is not a mine, whose treasures may take long to reduce into possession, but which fill only a limited number of veins and lodes; it is not a soil, whose fertility can be exhausted by the yield of successive harvests; it is not a continent or an ocean, whose area can be mapped out and its contour defined: it is limitless as that space which it finds too narrow for its aspirations; its possibilities are as infinite as the worlds which are forever crowding in and multiplying upon the astronomer's gaze." - James Joseph Sylvester

"Nobody before the Pythagorean had thought that mathematical relations held the secret of the universe. Twenty-five centuries later, Europe is still blessed and cursed with their heritage. To non-European civilizations, the idea that numbers are the key to both wisdom and power, seems never to have occurred." - Arthur Koestler

"Yet what are all such gaieties to me whose thoughts are full of indices and surds?" - Lewis Carroll

See more famous quotes about Mathematics

The study of numbers, equations, functions, and geometric shapes (see geometry) and their relationships. Some branches of mathematics are characterized by use of strict proofs based on axioms. Some of its major subdivisions are arithmetic, algebra, geometry, and calculus.

1. functioning as sing. the group of related sciences concerned with number, space, shape, and quantity, and their interrelationships.
2. functioning as sing. or pl. the use of mathematical processes or operations in calculations or the solution of problems.

• Branches and Computational Systems - mathematics: science dealing through symbolic representation with the relationship of numbers, magnitudes, and forms, including quantitative operations and the solution of quantitative problems

"Maths" and "Math" redirect here. For other uses see Mathematics (disambiguation) and Math (disambiguation).

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

Mathematics (from Greek μάθημα máthēma "knowledge, study, learning") is the study of quantity, structure, space, and change.^[2] Mathematicians seek out patterns^[3][4] and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. Since the pioneering work of Giuseppe Peano (1858-1932), David Hilbert (1862-1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. When those mathematical structures are good models of real phenomena, then mathematical reasoning often provides insight or predictions.

Through the use of abstraction and logical reasoning, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that continues to the present day.^[5]

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'.^[6] Carl Friedrich Gauss (1777–1855) referred to mathematics as "the Queen of the Sciences".^[7] Benjamin Peirce (1809-1880) called mathematics "the science that draws necessary conclusions".^[8] 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."^[9] 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".^[10]

Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, 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 and sometimes leads 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.^[11]

Contents 1 Etymology 2 History 3 Inspiration, pure and applied mathematics, and aesthetics 4 Notation, language, and rigor 5 Fields of mathematics 5.1 Foundations and philosophy 5.2 Pure mathematics 5.2.1 Quantity 5.2.2 Structure 5.2.3 Space 5.2.4 Change 5.3 Applied mathematics 5.3.1 Statistics and other decision sciences 5.3.2 Computational mathematics 6 Mathematics as profession 7 Mathematics as science 8 See also 9 Notes 10 References 11 Further reading 12 External links

Etymology

The word "mathematics" comes from the Greek μάθημα (máthēma), which means in ancient Greek what one learns, what one gets to know, hence also study and science, and in modern Greek just lesson.

The word máthēma comes from μανθάνω (manthano) in ancient Greek and from μαθαίνω (mathaino) in modern Greek, both of which mean to learn.

The word "mathematics" in Greek came to have the narrower and more technical meaning "mathematical study", even in Classical times.^[12] Its adjective is μαθηματικός (mathēmatikós), meaning related to learning or studious, which likewise further came to mean mathematical. In particular, μαθηματικὴ τέχνη (mathēmatikḗ tékhnē), Latin: 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.

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 (384-322BC), 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.^[13] In English, the noun mathematics takes singular verb forms. It is often shortened to maths or, in English-speaking North America, math.

History

Main article: History of mathematics

Greek mathematician Pythagoras (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 abstractions, or alternatively an expansion of subject matter. The first abstraction, which is shared by many animals,^[14] was probably that of numbers: 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.

In addition to recognizing how to count physical objects, prehistoric peoples also recognized how to count abstract quantities, like time – days, seasons, years.^[15] Elementary arithmetic (addition, subtraction, multiplication and division) naturally followed.

Since numeracy pre-dated writing, further steps were needed for recording numbers such as tallies or the knotted strings called quipu used by the Inca to store numerical data.^{[citation needed]} 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.^{[citation needed]}

Mayan numerals

The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns and the recording of time. More complex mathematics did not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra and geometry for taxation and other financial calculations, for building and construction, and for astronomy.^[16] The systematic study of mathematics in its own right began with the Ancient Greeks between 600 and 300 BC.^[17]

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 theorems and their proofs."^[18]

Inspiration, pure and applied mathematics, and aesthetics

Main article: Mathematical beauty

Sir Isaac Newton (1643-1727), an inventor of infinitesimal calculus.

Mathematics arises from many different kinds of problems. At first these were found in commerce, land measurement, architecture 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 forces of nature, continues to inspire new mathematics.^[19] 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. 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".^[20] 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.^[21] 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. He identified criteria such as significance, unexpectedness, inevitability, and economy as factors that contribute to a mathematical aesthetic.^[22] Mathematicians often strive to find proofs that are particularly elegant, proofs from "The Book" of God according to Paul Erdős.^[23][24] The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions.

Notation, language, and rigor

Main article: Mathematical notation

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.^[25] Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery.^[26] 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 and field have been given specialized mathematical meanings. Technical terms such as homeomorphism and 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 "theorems", based on fallible intuitions, of which many instances have occurred in the history of the subject.^[27] 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. 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.^[28]

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.^[29]

Fields of mathematics

An abacus, a simple calculating tool used since ancient times.

See also: Areas of mathematics

See also: Glossary of areas of mathematics

Mathematics can, broadly speaking, be subdivided 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.

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 sets or collections of objects. Category theory, which deals in an abstract way with mathematical structures 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.^[30] 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 set theory and the Brouwer-Hilbert controversy.

Mathematical logic is concerned with setting mathematics within a rigorous axiomatic framework, and studying the implications of such a framework. As such, it is home to Gödel's incompleteness theorems which (informally) imply 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). 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^{[citation needed]}, as well as to Category Theory.

Theoretical computer science includes 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 rapid advance of computer hardware. A famous problem is the "P=NP?" problem, one of the Millennium Prize Problems.^[31] 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 compression and entropy.

Mathematical logic	Set theory	Category theory	Theory of computation

Pure mathematics

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, 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 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 "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 as a consequence of operations or 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 integers that can be expressed in terms of arithmetic operations. Moreover, it frequently happens that different such structured sets (or structures) exhibit similar properties, which makes it possible, by a further step of abstraction, to state axioms for a class of structures, and then study at once the whole class of structures satisfying these axioms. Thus one can study groups, rings, 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 spaces, whose elements called 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.

Combinatorics	Number theory	Group theory	Graph theory	Order theory

Space

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 and discrete geometry was developed to solve problems in number theory and functional analysis but now is pursued with an eye on applications in optimization and computer science. Within differential geometry are the concepts of fiber bundles and calculus on manifolds, in particular, 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 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; 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. Other results in geometry and topology, including the four color theorem and Kepler conjecture, have been proved only with the help of computers.

Geometry	Trigonometry	Differential geometry	Topology	Fractal geometry	Measure theory

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 functions of a real variable is known as real analysis, with complex analysis the equivalent field for the complex numbers. 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	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 experiments;^[32] 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 studies, statisticians "make sense of the data" using the art of modelling and the theory of inference – with model selection and estimation; the estimated models and consequential predictions should be tested on new data.^[33]

Statistical theory studies decision problems such as minimizing the risk (expected loss) of a statistical action, such as using a procedure in, for example, parameter estimation, hypothesis testing, and 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.^[34] Because of its use of optimization, the mathematical theory of statistics shares concerns with other decision sciences, such as operations research, control theory, and mathematical economics.^[35]

Computational mathematics

Computational mathematics proposes and studies methods for solving mathematical problems that are typically too large for human numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory; numerical analysis includes the study of approximation and discretization broadly with special concern for rounding errors. Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic matrix and graph theory. Other areas of computational mathematics include computer algebra and symbolic computation.

Mathematical physics	Fluid dynamics	Numerical analysis	Optimization	Probability theory	Statistics	Cryptography
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,^[36][37] established in 1936 and now awarded every 4 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 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.

Mathematics as science

Carl Friedrich Gauss, known as the "prince of mathematicians".^[38]

Gauss referred to mathematics as "the Queen of the Sciences".^[39] In the original Latin Regina Scientiarum, as well as in 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. Of course, mathematics is in this sense a field of knowledge. The specialization restricting the meaning of "science" to natural science follows the rise of Baconian science, which contrasted "natural science" to scholasticism, the Aristotelean method of inquiring from first principles. Of course, the role of empirical experimentation and observation is negligible in mathematics, compared to natural sciences such as 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."^[10] More recently, Marcus du Sautoy has called mathematics 'the Queen of Science...the main driving force behind scientific discovery'.^[40]

Many philosophers believe that mathematics is not experimentally falsifiable, and thus not a science according to the definition of Karl Popper.^[41] However, in the 1930s Gödel's incompleteness theorems convinced many mathematicians^[who?] that mathematics cannot be reduced to logic alone, 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."^[42] 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.^[43] 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.^{[citation needed]}

The opinions of mathematicians on this matter are varied. Many mathematicians^[who?] 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?] 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.^{[citation needed]}

See also

Mathematics portal

	Book: Mathematics
Wikipedia books are collections of articles that can be downloaded or ordered in print.

Main article: Lists of mathematics topics

• Definitions of mathematics
• Mathematics and art
• Mathematics education

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. ^ "Department of Mathematics and Computer Science: Adelphi University". Academics.adelphi.edu. http://academics.adelphi.edu/artsci/math/. Retrieved 2011-11-04. 
3. ^ Steen, L.A. (April 29, 1988). The Science of Patterns Science, 240: 611–616. And summarized at Association for Supervision and Curriculum Development, www.ascd.org.
4. ^ 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
5. ^ Eves
6. ^ Marcus du Sautoy, A Brief History of Mathematics: 1. Newton and Leibniz, BBC Radio 4, 27/09/2010.
7. ^ Waltershausen
8. ^ Peirce, p. 97.
9. ^ 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).
10. ^ ^{a b} 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.
11. ^ Peterson
12. ^ Both senses can be found in Plato. Liddell and Scott, s.voceμαθηματικός
13. ^ The Oxford Dictionary of English Etymology, Oxford English Dictionary, sub "mathematics", "mathematic", "mathematics"
14. ^ S. Dehaene; G. Dehaene-Lambertz; L. Cohen (Aug 1998). "Abstract representations of numbers in the animal and human brain". Trends in Neuroscience 21 (8): 355–361. doi:10.1016/S0166-2236(98)01263-6. PMID 9720604. 
15. ^ See, for example, Raymond L. Wilder, Evolution of Mathematical Concepts; an Elementary Study, passim
16. ^ Kline 1990, Chapter 1.
17. ^ "A History of Greek Mathematics: From Thales to Euclid". Thomas Little Heath (1981). ISBN 0-486-24073-8
18. ^ Sevryuk
19. ^ Johnson, Gerald W.; Lapidus, Michel L. (2002). The Feynman Integral and Feynman's Operational Calculus. Oxford University Press. ISBN 0821824139. 
20. ^ Wigner, Eugene (1960). "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". Communications on Pure and Applied Mathematics 13 (1): 1–14. http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html. 
21. ^ "Mathematics Subject Classification 2010" (PDF). http://www.ams.org/mathscinet/msc/pdfs/classification2010.pdf. Retrieved 2010-11-09. 
22. ^ Hardy, G. H. (1940). A Mathematician's Apology. Cambridge University Press. ISBN 0521427061. 
23. ^ Gold, Bonnie; Simons, Rogers A. (2008). Proof and Other Dilemmas: Mathematics and Philosophy. MAA. 
24. ^ Aigner, Martin; Ziegler, Günter M. (2001). Proofs from The Book. Springer. ISBN 3540404600. 
25. ^ Earliest Uses of Various Mathematical Symbols (Contains many further references).
26. ^ Kline, p. 140, on Diophantus; p.261, on Vieta.
27. ^ See false proof for simple examples of what can go wrong in a formal proof.
28. ^ 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).
29. ^ 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."
30. ^ Luke Howard Hodgkin & Luke Hodgkin, A History of Mathematics, Oxford University Press, 2005.
31. ^ Clay Mathematics Institute, P=NP, claymath.org
32. ^ Rao, C.R. (1997) Statistics and Truth: Putting Chance to Work, World Scientific. ISBN 9810231113
33. ^ 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.
34. ^ Rao, C. R. (1981). "Foreword". In Arthanari, T. S.; Dodge, Yadolah. Wiley Series in Probability and Mathematical Statistics. Wiley. pp. vii–viii. ISBN 0-471-08073-X. MR607328. 
35. ^ Whittle (1994, pp. 10–11 and 14–18): Whittle, Peter (1994). "Almost home". In Kelly, F. P.. Probability, statistics and optimisation: A Tribute to Peter Whittle (previously "A realised path: The Cambridge Statistical Laboratory upto 1993 (revised 2002)" ed.). Chichester: John Wiley. pp. 1–28. ISBN 0471948292. http://www.statslab.cam.ac.uk/History/2history.html#6._1966--72:_The_Churchill_Chair. 
36. ^ "The Fields Medal is now indisputably the best known and most influential award in mathematics." Monastyrsky
37. ^ Riehm
38. ^ Zeidler, Eberhard (2004). Oxford User's Guide to Mathematics. Oxford, UK: Oxford University Press. p. 1188. ISBN 0198507631. 
39. ^ Waltershausen
40. ^ Marcus du Sautoy, A Brief History of Mathematics: 10. Nicolas Bourbaki, BBC Radio 4, 01/10/2010.
41. ^ Shasha, Dennis Elliot; Lazere, Cathy A. (1998). Out of Their Minds: The Lives and Discoveries of 15 Great Computer Scientists. Springer. p. 228. 
42. ^ Popper 1995, p. 56
43. ^ Ziman

References

• Courant, Richard 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.
• Einstein, Albert (1923). Sidelights on Relativity (Geometry and Experience). P. Dutton., Co. 
• du Sautoy, Marcus, 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.
• 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) (PDF). Some Trends in Modern Mathematics and the Fields Medal. Canadian Mathematical Society. http://www.fields.utoronto.ca/aboutus/FieldsMedal_Monastyrsky.pdf. Retrieved 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 (1881). Peirce, Charles Sanders. ed. "Linear associative algebra". American Journal of Mathematics (Johns Hopkins University) 4 (1–4): 97–229. doi:10.2307/2369153. 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 Eprint and as an extract, D. Van Nostrand, 1882, Google Eprint. http://books.google.com/?id=De0GAAAAYAAJ&pg=PA1&dq=Peirce+Benjamin+Linear+Associative+Algebra+&q=. .
• Peterson, Ivars, Mathematical Tourist, New and Updated Snapshots of Modern Mathematics, Owl Books, 2001, ISBN 0-8050-7159-8.
• 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" (PDF). Notices of the AMS (AMS) 49 (7): 778–782. http://www.ams.org/notices/200207/comm-riehm.pdf. 
• Sevryuk, Mikhail B. (January 2006). "Book Reviews" (PDF). Bulletin of the American Mathematical Society 43 (1): 101–109. doi:10.1090/S0273-0979-05-01069-4. http://www.ams.org/bull/2006-43-01/S0273-0979-05-01069-4/S0273-0979-05-01069-4.pdf. Retrieved 2006-06-24. 
• Waltershausen, Wolfgang Sartorius von (1856, repr. 1965). Gauss zum Gedächtniss. Sändig Reprint Verlag H. R. Wohlwend. ISBN 3-253-01702-8. ISSN B0000BN5SQ ASIN: B0000BN5SQ. http://www.amazon.de/Gauss-Ged%e4chtnis-Wolfgang-Sartorius-Waltershausen/dp/3253017028. 

Further reading

• 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.
• Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Mariner Books; Reprint edition (January 14, 1999). ISBN 0-395-92968-7.
• 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 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.

External links

Find more about Mathematics on Wikipedia's sister projects:
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At Wikiversity you can learn more and teach others about Mathematics at:

The School of Mathematics

• Mathematics on In Our Time at the BBC. (listen now)
• Free Mathematics books Free Mathematics books collection.
• Encyclopaedia of Mathematics online encyclopaedia from Springer, Graduate-level reference work with over 8,000 entries, illuminating nearly 50,000 notions in mathematics.
• HyperMath site at Georgia State University
• FreeScience Library The mathematics section of FreeScience library
• Rusin, Dave: The Mathematical Atlas. A guided tour through the various branches of modern mathematics. (Can also be found at NIU.edu.)
• 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.
• Cain, George: Online Mathematics Textbooks available free online.
• Tricki, Wiki-style site that is intended to develop into a large store of useful mathematical problem-solving techniques.
• Mathematical Structures, list information about classes of mathematical structures.
• Mathematician Biographies. The MacTutor History of Mathematics archive Extensive history and quotes from all famous mathematicians.
• Metamath. A site and a language, that formalize mathematics from its foundations.
• Nrich, a prize-winning site for students from age five from Cambridge University
• Open Problem Garden, a wiki of open problems in mathematics
• Planet Math. An online mathematics encyclopedia under construction, focusing on modern mathematics. Uses the Attribution-ShareAlike license, allowing article exchange with Wikipedia. Uses TeX markup.
• Some mathematics applets, at MIT
• Weisstein, Eric et al.: MathWorld: World of Mathematics. An online encyclopedia of mathematics.
• Patrick Jones' Video Tutorials on Mathematics
• Citizendium: Theory (mathematics).
• du Sautoy, Marcus, A Brief History of Mathematics, BBC Radio 4 (2010).
• MathOverflow A Q&A site for research level mathematics

v d e Areas of mathematics Areas Arithmetic · Algebra (elementary – linear – multilinear – abstract) · Geometry (Discrete geometry – Algebraic geometry – Differential geometry) · Calculus/Analysis · Set theory · Logic · Category theory · Number theory · Combinatorics · Graph theory · Topology · Lie theory · Differential equations/Dynamical systems · Mathematical physics · Numerical analysis · Computation · Information theory · Probability · Statistics · Optimization · Control theory · Game theory Divisions Pure mathematics · Applied mathematics · Discrete mathematics · Computational mathematics Category  · Mathematics portal  · Outline · Lists

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Common misspelling(s) of mathematics

• mathamatics

Dansk (Danish)
n. - matematik
n. pl. - matematik, matematikkundskaber, matematiske udregninger

Nederlands (Dutch)
wiskunde

Français (French)
n. - mathématiques, calculs
n. pl. - aspects mathématiques de (qch)

Deutsch (German)
n. pl. - Mathematik
n. - Mathematik

Ελληνική (Greek)
n. pl. - μαθηματικά, μαθηματική επιστήμη

Italiano (Italian)
matematica

Português (Portuguese)
n. pl. - matemática (f)

Русский (Russian)
математика

Español (Spanish)
n. - matemáticas
n. pl. - matemáticas, cálculos

Svenska (Swedish)
n. pl. - matematik(kunskaper)

中文（简体）(Chinese (Simplified))
数学

中文（繁體）(Chinese (Traditional))
n. pl. - 數學
n. - 數學

한국어 (Korean)
n. pl. - 수학
n. - 계산

日本語 (Japanese)
n. - 数学

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‏(الجمع) الرياضيات‏

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n. - ‮מתמטיקה‬

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