I Grattan-Guinness. New Dictionary of the History of Ideas. Editor: Maryanne Cline Horowitz. Volume 4. Detroit: Charles Scribner’s Sons, 2005.
This article comprises a compact survey of the development of mathematics from ancient times until the early twentieth century. The treatment is broadly chronological, and most of it is concerned with Europe.
It seems unavoidable that mathematical thinking played a role in human theorizing from the start of the race, and in various ways. Arithmetic (as the later branch of mathematics became known) would have been one of them, motivated initially by forming integers in connection with counting. But other branches surely include geometry, linked to the appreciation of line, surface, and space; trigonometry, inspired by awareness of angles; mechanics, related to the motion of bodies large and small and the (in)stability of structures; part-whole theory, from consideration of collections of things; and probability, coming from judging and guessing about situations. In all cases the thinking would have started out as very intuitive, gradually becoming more explicit and less particular.
Some of the associated contexts would have been provided by study of the environment (such as days) and the heavens (such as new and full moons), which was a major concern of ancient cultures in all parts of the world; in those times mathematics and astronomy were linked closely. For example, the oldest recognized artifact is a bone from Africa, thought to be about thirty-seven thousand years old, upon which phases of the moon seem to have been recorded.
Among the various ancient cultures, the Babylonians have left the earliest extant evidence of their mathematical practice. They counted with tokens from the eighth millennium; and from the late fourth millennium they expressed numbers and properties of arithmetic in a numeral system to base 10 and handled fractions in expansions of powers of 1/60. Many surviving artifacts seem to relate to education—for example, exercises requiring calculations of unknown quantities, which correspond to the solution of equations but are not to be so identified. They also developed geometry, largely for terrestrial purposes. The Egyptians pursued similar studies, even also finding a formula (not the same one) for the volume of the rectangular base of a pyramid of given sides. They also took up the interesting mathematical problem of representing a fraction as the sum of reciprocals.
A major mathematical question for these cultures concerned the relationship between circles and spheres and rectilinear objects such as lines and cubes. They involve the quantity that we symbolize by, and ancient evidence survives of methods of approximating to its value. But it is not clear that these cultures knew that the same quantity occurs in all the relationships.
On Greek Mathematics
The refinement of mathematics was effected especially by the ancient Greeks, who flourished for about a millennium from the sixth century B.C.E. Pythagoras and his clan are credited with many things, starting with their later compatriots: the eternality of integers; the connection between ratios of integers and musical intervals; the theorem relating the sides of a right-angled triangle; and so on. Their contemporary Thales (c. 625-c. 547 B.C.E.) is said to have launched trigonometry with his appreciation of the angle. However, nothing survives directly from either man.
A much luckier figure concerning survival is Euclid (fl. c. 300 B.C.E.), especially with his Elements. While no explanatory preface survives, it appears that most of the mathematics presented was his rendition of predecessors’ work, but that (some of) the systematic organization that won him so many later admirers might be his. He stated explicitly the axioms and assumptions that he noticed; one of them, the parallel axiom, lacked the intuitive clarity of the others, and so was to receive much attention in later cultures.
The Elements comprised thirteen Books: Books 7-9 dealt with arithmetic, and the others presented basic plane (Books 1-6) and solid (Books 11-13) geometry of rectilinear and circular figures. The extraordinary Book 10 explored properties of ratios of smaller to longer lines, akin to a theory of irrational numbers but again not to be so identified. A notable feature is that Euclid confined the role of arithmetic within geometry to multiples of lines (say, “twice this line is … “), to a role in stating ratios, and to using reciprocals (such as 1/5); he was not concerned with lengths—that is, lines measured arithmetically. Thus, he said nothing about the value of, for it relates to measurement.
The Greeks were aware of the limitations of straight line and circle. In particular, they found many properties and applications of the “conic sections”: parabola, hyperbola, and ellipse. Hippocrates of Chios (fl. c. 600 B.C.E.) is credited with three “classical problems” (a later name) that his compatriots (rightly) suspected could not be solved by ruler and compass alone: (1) construct a square equal in area to a given circle; (2) divide any angle into three equal parts; and (3) construct a cube twice the volume of a given one. The solutions that they did find enlarged their repertoire of curves.
Among later Greeks, Archimedes (c. 287-212 B.C.E.) stands out for the range and depth of his work. His work on circular and spherical geometry shows that he knew all four roles for; but he also wrote extensively on mechanics, including floating bodies (the “eureka!” tale) and balancing the lever, and focusing parabolic mirrors. Other figures developed astronomy, partly as applied trigonometry, both planar and spherical; in particular, Ptolemy (late second century) “compiled” much knowledge in his Almgest, dealing with both the orbits and the distances of the heavenly bodies from the central and stationary Earth.
Mathematics developed well from antiquity also in the Far East, with distinct traditions in India, China, Japan, Korea, and Vietnam. Arithmetic, geometry, and mechanics were again prominent; special features include a powerful Chinese method equivalent to solving a system of linear equations, a pretty theory of touching circles in Japanese “temple geometry,” and pioneering work on number theory by the Indians. They also introduced the place-value system of numerals to base 10, of which we use a descendant that developed after several changes in adopted symbols.
This system of numerals was mediated into Europe by mathematicians working in medieval Islamic civilization, often though not always writing in Arabic. They became the dominant culture in mathematics from the ninth century and continued strongly until the fourteenth. They assimilated much Greek mathematics; indeed, they are our only source for some of it.
The first major author was al-Khwarizmi (fl. c. 800-847), who laid the foundations of algebra, especially the solution of equations. He and his followers launched the theory using words rather than special symbols to mark unknowns and operations. Other interests in geometry included attempts to prove Euclid’s parallel axiom and applications to optics and trigonometry; an important case of the latter was determining the qibla (that is, the direction of Mecca) for any time and place at times of Muslim prayer. Their massive contributions to astronomy included theory and manufacture of astrolabes.
The Wakening Europe from the Twelfth Century
From the decline of the Roman Empire (including Greece) Euclid was quiescent mathematically, though the Carolingian kingdom inspired some work, at least in education. The revival dates from around the late twelfth century, when universities also began to be formed. The major source for mathematics was Latin translations of Greek and Arabic writings (and re-editions of Roman writers, especially Boethius). In addition, the Italian Leonardo Fibonacci (c. 1170-c. 1240) produced a lengthy Liber Abbaci in 1202 that reported in Latin many parts of Arabic arithmetic and algebra (including the Indian numerals); his book was influential, though perhaps less than is commonly thought. The Italian peninsula was then the most powerful region of Europe, and much commercial and “research” mathematics was produced there; the German states and the British Isles also came to boast some eminent figures. In addition, a somewhat distinct Hebrew tradition arose—for example, in probability theory.
A competition developed between two different methods of reckoning. The tradition was to represent numbers by placing pebbles (in Latin, calculi) in determined positions on a flat surface (in Latin, abacus, with one b), and to add and subtract by moving the pebbles according to given rules. However, with the new numerals came a rival procedure of calculating on paper, which gradually supervened; for, as well as also allowing multiplication and division, the practitioner could show and check his working, an important facility unavailable to movers of pebbles.
Mathematics rapidly profited from the invention of printing in the late fifteenth century; not only were there printed Euclids, but also many reckoning books. Trigonometry became a major branch in the fifteenth and sixteenth centuries, not only for astronomy but also, as European imperialism developed, for cartography, and the needs of navigation and astronomy made the spherical branch more significant than the planar. Geometry was applied also to art, with careful studies of perspective; Piero della Francesca (c. 1420-1492) and Albrecht Dürer (1471-1528) were known not only as great artists but also as significant mathematicians.
Numerical calculation benefited greatly from the development of logarithms in the early seventeenth century by John Napier (1550-1617) and others, for then multiplication and division could be reduced to addition and subtraction. Logarithms superseded a clumsier method called “prosthaphairesis” that used certain trigonometrical formulas.
In algebra the use of special symbols gradually increased, until in his Géométrie (1637), René Descartes (1596-1650) introduced (more or less) the notations that we still use, and also analytic geometry. His compatriot Pierre de Fermat (1601-1665) also worked in these areas and contributed some theorems and conjectures to number theory. In addition, he corresponded with Blaise Pascal (1623-1662) on games of chance, thereby promoting parts of probability theory.
In mechanics a notable school at Merton College, Oxford, had formed in the twelfth century to study various kinds of terrestrial and celestial motion. The main event in celestial mechanics was Nicolaus Copernicus’s (1473-1543) De revolutionibus (1453; On the revolutions), where rest was transferred from the Earth to the sun (though otherwise the dynamics of circular and epicyclical motions was not greatly altered). In the early seventeenth century the next stages lay especially with Johannes Kepler’s (1571-1630) abandonment of circular orbits for the planets and Galileo Galilei’s (1564-1642) analysis of (locally) horizontal and vertical motions of bodies.
The Epoch of Newton and Leibniz
By the mid-seventeenth century, science had become professionalized enough for some national societies to be instituted, especially the Royal Society of London and the Paris Académie des Sciences. At that time two major mathematicians emerged: Isaac Newton (1642-1727) in Cambridge and Gottfried Wilhelm von Leibniz (1646-1716) in Hanover. Each man invented a version of the differential and integral calculus, Newton first in creation but Leibniz first in print. The use here of Leibniz’s adjectives recognizes the superior development of his version. During the early 1700s Newton became so furious (or envious?) that he promoted a charge of plagiarism against Leibniz, complete with impartial committee at the Royal Society. It was a disaster for Britain: Newton’s followers stuck with their master’s theory of “fluxions” and “fluents,” while the Continentals developed “differentials” and “integrals,” with greater success. The accusation was also mathematically stupid, for conceptually the two calculi were quite different: Newton’s was based upon (abstract) time and unclearly grounded upon the notion of limit, while Leibniz’s used infinitesimal increments on variables, explicitly avoiding limits. So even if Leibniz had known of Newton’s theory (of which the committee found no impartial evidence), he rethought it entirely.
Leibniz’s initial guard was largely Swiss: brothers Jakob (1654-1705) and Johann Bernoulli (1667-1748) from the 1680s, then from the 1720s Johann’s son Daniel (1700-1782) and their compatriot Leonhard Euler (1707-1783), who was to be the greatest of the lot. During the eighteenth century they and other mathematicians (especially in Paris) expanded calculus into a vast territory of ordinary and then partial differential equations and studied many related series and functions. The Newtonians kept up quite well until Colin Maclaurin (1698-1746) in the 1740s, but then faded badly.
The main motivation for this vast development came from applications, especially to mechanics. Here Newton and Leibniz differed again. In his Principia mathematica (1687) Newton announced the laws that came to carry his name: (1) a body stays in equilibrium or in uniform motion unless disturbed by a force; (2) the ratio of the magnitude of the force and the mass of the body determines its acceleration; and (3) to any force of action there is one of reaction, equal in measure and opposite in sense. In addition, for both celestial and terrestrial mechanics, which he novelly united, the force between two objects lies along the straight line joining them, and varies as the inverse square of its length.
With these principles Newton could cover a good range of mechanical phenomena. His prediction that the Earth was flattened at the poles, corroborated by an expedition undertaken in the 1740s, was a notable success. He also had a splendid idea about why the planets did not exactly follow the elliptical orbits around the sun that the inverse square law suggested: they were “perturbed” from them by interacting with each other. The study of perturbations became a prime topic in the eighteenth century, with Euler’s work being particularly significant. Euler also showed that law 2 could be applied to any direction in a mechanical situation, thus greatly increasing its utility. He and others made important contributions to the mechanics of continuous media, especially fluid mechanics and elasticity theory, where Newton had been somewhat sketchy.
Mathematics in the Eighteenth Century: The Place of Lagrange
However, Newton’s theory was not alone in mechanics. Leibniz and others developed an alternative approach, partly inspired by Descartes, in which the “living forces” (roughly, kinetic energy) of bodies were related to their positions. Gradually this became a theory of living forces converted into “work” (a later term), specified as (force x traversed distance). Engineers became keen on it for its utility in their concerns, especially when impact between bodies was involved; from the 1780s Lazare Carnot (1753-1823) urged it as a general approach for mechanics.
Carnot thereby challenged Newton’s theory, but his main target was a recent new tradition partly launched by Jean d’Alembert (1718-1783) in midcentury and developed further by Joseph-Louis Lagrange (1736-1813). Suspicious of the notion of force, d’Alembert had proposed that it be defined by Newton’s law 2, which he replaced by one stating how systems of bodies moved when disturbed from equilibrium. At that time Euler and others proposed a “principle of least action,” which asserted that the action (a mechanical notion defined by an integral) of a mechanical system took its optimal value when equilibrium was achieved. Lagrange elaborated upon these principles to create Méchanique analitique (1788), in which he challenged the other two traditions; in particular, dynamics was reduced mathematically to statics. For him a large advantage of his principles was that they were formulated exclusively in algebraic terms; as he proclaimed in the preface of his book, there were no diagrams, and no need for them. A main achievement was a superb though inconclusive attempt to prove that the system of planets was stable; predecessors such as Newton and Euler had left that matter to God.
Lagrange formulated mechanics this way in order to make it (more) rigorous. Similarly, he algebraized the calculus by assuming that any mathematical function could be expressed in an infinite power series (the so-called Taylor series), and that the basic notions of derivative (his word) and integral could be determined solely by algebraic manipulations. He also greatly expanded the calculus of variations, a key notion in the principle of least action.
As in mechanics, Lagrange’s calculus challenged the earlier ones, Newton’s and Leibniz’s, and as there, reaction was cautious. A good example for both contexts was Pierre-Simon Laplace (1749-1827), a major figure from 1770. While strongly influenced by Lagrange, he did not confine himself to the constraints of Lagrange’s book when writing his own four-volume Traité de mécanique céleste (1799-1805; Treatise on celestial mechanics). His exposition of celestial and planetary mechanics used many differential equations, series, and functions.
The French Revolution and a New Professionalization
Laplace published his large book in a new professional and economic situation for science. After the Revolution of 1789 in France, higher education and its institutions there were reformed, with a special emphasis upon engineering. In particular, a new school was created, the École Polytechnique (1794), with leading figures as professors (such as Lagrange) and as examiners (Laplace), and with enrollment of students determined by talent, not birth. A new class of scientists and engineers emerged, with mathematics taught, learned, researched, and published on a scale hitherto unknown.
Of this mass of work only a few main cases can be summarized here. Joseph Fourier (1768-1830) is noteworthy for his mathematical analysis of heat diffusion, both the differential equation to represent it (the first important such equation found outside mechanics) and solutions by certain infinite series and by integrals that both now bear his name. From the 1820s they attracted much attention, not only for their use in heat theory but especially for the “pure” task of establishing conditions for their truth. New techniques for rigor had just become available, mainly from Augustin-Louis Cauchy (1789-1857), graduate of the École Polytechnique and now professor there. He taught a fourth approach to the calculus (and also function and series), based like Newton’s upon limits but now fortified by a careful theory of them; although rather unintuitive, its mathematical merits gradually led worldwide to its preference over the other three approaches.
Ironically, Cauchy’s own analysis of Fourier series failed, but a beautiful treatment following his approach came in 1829 from J. P. G. Dirichlet (1805-1859)—a French-sounding name of a young German who had studied with the masters in Paris. Dirichlet also exemplifies a novelty of that time: other countries producing major mathematicians. Another contemporary example lies in elliptic functions, which Carl Jacobi (1804-1851) and the young Norwegian Niels Henrik Abel (1802-1829) invented independently following much pioneering work on the inverse function by A. M. Legendre (1752-1833).
Jacobi and Abel drew upon a further major contribution to mathematics made by Cauchy when, by analogy with the calculus, he developed a theory of functions of the complex variable x + √ − 1y (x and y real), complete with an integral. His progress was fitful, from the 1810s to the 1840s; after that, however, his theory became recognized as a major branch of mathematics, with later steps taken especially by the Germans.
Between 1810 and 1830 the French initiated other parts of mathematical physics in addition to Fourier on heat: Siméon-Denis Poisson (1781-1840) on magnetism and electrostatics; André-Marie Ampère (1775-1836) on electrodynamics; and Augustin Jean Fresnel (1788-1827) on optics with his wave theory. Mathematics played major roles: many analogies were taken from mechanics, which itself developed massively, with Carnot’s energy approach elaborated by engineers such as Gaspard-Gustave Coriolis (1792-1843), and continuum mechanics extended, especially by Cauchy.
Geometry was also taught and studied widely. Gaspard Monge (1746-1818) sought to develop “descriptive geometry” into a geniune branch of mathematics and gave it prominence in the first curriculum of the École Polytechnique; however, this useful theory of engineering drawing could not carry such importance, and Laplace had its teaching reduced. But former student Jean Victor Poncelet (1788-1867) was partly inspired by it to develop “the projective properties of figures” (Traité des propries projectives de figures, 1822), where he studied characteristics independent of measure, such as the order of points on a line.
The main mathematician outside France at this time was C. F. Gauss (1777-1855), director of the Göttingen University Observatory. Arguably he was the greatest of all, with major work published in number theory, celestial mechanics, and aspects of analysis and probability theory. But he was not socially active, and he left many key insights in his manuscripts (for example, on elliptic functions).
Other major contributors outside France include George Green (1793-1841), who, in An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism (1828), produced a wonderful theorem in potential (his word) theory that related the state of affairs inside an extended body to that on its surface. But he published his book very obscurely, and it became well-known only on the reprint during the 1850s initiated by William Thomson (later Lord Kelvin), who was making notable contributions of his own to the theory.
By the 1840s Britain and the Italian and German states were producing quality mathematicians to complement and even rival the French, and new posts were available in universities and engineering colleges everywhere. Among the Germans, two figures stand out.
From around 1860 Karl Weierstrass (1815-1897) gave lecture courses on many aspects of real-and complex-variable analysis and parts of mechanics at Berlin University, attended by students from many countries who then went home and taught likewise. Meanwhile at Göttingen, Bernhard Riemann (1826-1866) rethought complex-variable analysis and revolutionized the understanding of both Fourier series and the foundations of geometry. Much of this work was published only after his early death in 1866, but it soon made a great impact. The work on Fourier series led Georg Cantor (1845-1918) to develop set theory from the 1870s. On geometry Riemann showed that the Euclidean was only one of many possible geometries, and that each of them could be defined independently of any embedding space. The possibility of non-Euclidian geometries, using alternatives to the parallel axiom, had been exhibited around 1830 in little-recognized work by Janos Bolyai (1802-1860) and Nicolai Lobachevsky (1793-1856) (and, in manuscript, Gauss); Riemann, however, went much further and brought us proper understanding of the plurality of geometries.
Weierstrass emulated and indeed enhanced Cauchy-style rigor, carefully formulating definitions and distinctions and presenting proofs in great detail. By contrast, Riemann worked intuitively, offering wonderful but often proof-free insights grounded upon some “geometric fantasy,” as Weierstrass described it. A good example is their revisions of Cauchy’s complex-variable analysis: Weierstrass relied solely on power series expansions of the functions, whereas Riemann invented surfaces now named after him that were slit in many remarkable ways. Among many consequences of the latter, the German Felix Klein (1849-1925) and the Frenchman Henri Poincaré (1854-1912) in the early 1880s found beautiful properties of functions defined on these surfaces, which they related to group theory as part of the rise of abstract algebras.
Another example of the gap between Riemann and Weierstrass is provided by potential theory. Riemann used a principle employed by his mentor Dirichlet (and also envisaged by Green) to solve problems in potential theory, but in 1870 Weierstrass exposed its fallibility by a counterexample, and so methods became far more complicated.
Better news for potential theory had come at midcentury with the “energetics” physics of Thomson, Hermann von Helmholtz (1821-1894), and others. The work expression of engineering mechanics was extended into the admission of potentials, which now covered all physical factors (such as heat) and not just the mechanical ones that had split Carnot from Lagrange. The latter’s algebraic tradition in mechanics had been elaborated by Jacobi and by the Irishman William Rowan Hamilton (1805-1865), who also introduced his algebra of quaternions.
Among further related developments, the Scot James Clerk Maxwell (1831-1879) set out theories of electricity and magnetism (including, for him, optics) in his Treatise on Electricity and Magnetism (1873). Starting out from the electric and magnetic potentials as disturbances of the ether rather than Newton-like forces acting at a distance through it, he presented relationships between his basic notions as differential equations (expressed in quaternion form). A critical follower was the Englishman Oliver Heaviside, who also analyzed electrical networks by means of a remarkable but mysterious operator algebra. Other “Maxwellians” preferred to replace dependence upon fields with talk about “things,” such as electrons and ions; the relationship between ether and matter (J. J. Larmor, Aether and Matter, 1900) was a major issue in mathematical physics at century’s end.
The Early Twentieth Century
A new leader emerged: the German David Hilbert (1862-1943). Work on abstract algebras and the foundations of geometry led him to emphasize the importance of axiomatizing mathematical theories (including the axioms of Euclidean geometry that Euclid had not noticed) and to study their foundations metamathematically. But his mathematical knowledge was vast enough for him to propose twenty-three problems for the new century; while a personal choice, it exercised considerable influence upon the community. He presented it at the International Congress of Mathematicians, held in Paris in 1900 as the second of a series that manifested the growing sense of international collaboration in mathematics that still continues.
One of Hilbert’s problems concerned the foundations of physics, which he was to study intensively. In physics Albert Einstein (1879-1955) proposed his special theory of relativity in 1905 and a general theory ten years later; according to both, the ether was not needed. Mathematically, the general theory both deployed and advanced tensor calculus, which had developed partly out of Riemann’s interpretation of geometry.
Another main topic in physics was quantum mechanics, which drew upon partial differential equations and vector and matrix theory. One of its controversies concerned Werner Heisenberg’s principle of the uncertainty of observation: should it be interpreted statistically or not? The occurrence of this debate, which started in the mid-1920s, was helped by the increasing presence of mathematical statistics. Although probability must have had an early origin in mathematical thinking, both it and mathematical statistics had developed very slowly in the nineteenth century—in strange contrast to the mania for collecting data of all kinds. Laplace and Gauss had made important contributions in the 1810s, for example, over the method of least-squares regression, and Pafnuty Chebyshev (1821-1894) was significant from the 1860s in Saint Petersburg (thus raising the status of Russian mathematics). But only from around 1900 did theorizing in statistics develop strongly, and the main figure was Karl Pearson (1857-1936) at University College, London, and his students and followers. Largely to them we owe the definition and theory of basic notions such as standard deviation and correlation coefficient, basic theorems concerning sampling and ranking, and tests of significance.
Elsewhere, Cantor’s set theory and abstract algebras were applied to many parts of mathematics and other sciences in the new century. A major beneficiary was topology, the mathematics of location and place. A few cases had emerged in the nineteenth century, such as the “Möbius strip” with only one side and one edge, Riemann’s fantastical surfaces, and above all a remarkable classification of deformable manifolds by Poincaré; most of the main developments, however, date from the 1920s. General theories were developed of covering, connecting, orientating, and deforming manifolds and surfaces, along with many other topics. A new theory of dimensions was also proposed because Cantor had refuted the traditional understanding by mapping one-one all the points in a square onto all the points on any of its sides. German mathematicians were prominent; so were Americans in a country that had risen rapidly in mathematical importance from the 1890s.
The amount of mathematical activity has usually increased steadily or even exponentially, and the growth from the mid-twentieth century has been particularly great. For example, the German reviewing journal Zentralblatt Math published at the beginning of the twenty-first century a six-hundred-page quarto volume every two weeks, using a classification of mathematics into sixty-three numbered sections. To suggest the rate of increase, the other reviewing journal, the U.S. periodical Mathematical Reviews, published 3,800 octavo pages in 1980, 7,500 pages a decade later, and 9,800 pages in 2003. It would be impossible to summarize this mountain of work, even up to 1970; instead, some main points are noted relating to the previous sections and to the companion articles on algebras and on logic.
Not only has the amount increased; the variety of theories has also greatly expanded. All the topics and branches mentioned above continue to develop (and also many more that were not noted), and new topics emerge and fresh applications are found. For example, beginning with the 1940s mathematics became widely utilized in the life sciences and medicine and has expanded greatly in economics and other social sciences relative to previous practice.
Much of that work lies in statistics, which after its very slow arrival has developed a huge community of practitioners in its own right. Often it functions rather separately from mathematics, with its own departments in universities.
Another enormous change has been the advent of computing, again particularly since World War II and indeed much stimulated by war work as on cryptography and the calculation of parameters in large technological artifacts. Mathematics plays a role both in the design, function, and programming of computers themselves and in the formulation of many mathematical theories. An important case is in numerical mathematics, where approximations are required and efficient algorithms sought to effect them. This kind of mathematics has been practiced continuously from ancient times, especially in connection with all sorts of applications. Quite often algorithms were found to be too slow or mathematically cumbersome to be practicable; but now computer power makes many of them feasible in “number crunching” (to quote a popular oversimplification of such techniques).
A feature of many mathematical theories is linearity, in which equations or expressions of the form
(A) ax by cz … and so on finitely or even infinitely
make sense, in a very wide range of interpretations of the letters, not necessarily within an algebra itself (for example, Fourier series shows it). But a dilemma arises for many applications, for the world is not a linear place, and in recent decades nonlinear theories have gained higher status, partly again helped by computing. The much-publicized theory of fractals falls into this category.
From the Greeks onward, mathematicians have often been fascinated by major unsolved problems and by the means of solving them. In the late 1970s a proof was produced of the four-color theorem, stating that any map drawn upon a surface can be colored with four colors such that bounding regions do not share the same color. The proof was controversial, for a computer was used to check thousands of special cases, a task too large for people. Another example is “Fermat’s last theorem,” that the sum of the nth powers of two positive integers is never equal to the nth power of another integer when n 2. The name is a misnomer, in that Fermat only claimed a proof but did not reveal it; the modern version (1994) uses modern techniques far beyond his ken.
This article has focused upon the main world cultures, but every society has produced mathematics. The “fringe” developments are studied using approaches collectively known as ethnomathematics. While the cultures involved developed versions of arithmetic and geometry and also some other branches, several of them also followed their own concerns; some examples, among many, are intricate African drawings made in one unbroken line, Celtic knitting patterns, and sophisticated rows of knotted strings called quipus used in Mexico to maintain accounts.
A thread running from antiquity in all cultures, fringe or central, is recreational mathematics. Unfortunately, the variety is far too great even for summary here. Often it consists of exercises, perhaps posed for educational use, or perhaps just for fun; an early collection is attributed to Alcuin in the ninth century, for use in the Carolingian Empire. Solutions sometimes involve intuitive probability, or combinatorics to work out all options; with games such as chess and bridge, however, the analysis is much more sophisticated. Several puzzles appear in slightly variant forms in different cultures, suggesting transmission. Some are puzzles in logic or reasoning rather than mathematics as such, and it is striking that for some games the notion of decidability was recognized (that is, is there a strategy that guarantees victory?) long before it was studied metamathematically in the foundations of mainstream mathematics.
Lastly, since the early 1970s interest in the history of mathematics has increased considerably. There are now several journals in the field along with a variety of books and editions, collectively covering all main cultures and periods. One main motive for people to take up historical research was their dislike of the normal unmotivated way in which mathematics was (and is) taught and learned; thus, the links between history and mathematics education are strong. For, despite many appearances to the contrary, mathematics is a human activity.