The analysis of quantitative relationships. The discipline is nowadays divided into arithmetic, algebra, and geometry, the first exploring the calculative relationships between specified numbers, the second relationships including unspecified numbers, and the third the analysis of spatial relationships. Counting and calculation presumably evolved in prehistory in close association with agricultural endeavour, for the purposes of assessing wealth, negotiating trade, and determining the calendar by astronomical means. The use of measurements in construction must also have provided a powerful incentive for the sophistication of arithmetic and geometry.
The gradual complication of technology and the increasing importance of money required ever-greater facility and ingenuity in calculation as Classical culture developed, but the philosophical importance of mathematics was secured when the followers of Pythagoras asserted that numerical relationships held the key to the interpretation and understanding of the universe, attributing a mystical significance to them whose echoes still linger. The abstract ruminations of the post-Pythagoreans and the practical calculations of the Egyptians and the Alexandrian school laid the foundations for what are now known as pure and applied mathematics. Although the former is fundamental to philosophy and the scientific worldview it fostered, the latter can comfortably exist in a near-static culture, as it continued to do for more than a thousand years after the practically inclined Romans abandoned the more abstruse components of Greek philosophy. Roman numerals apparently served well enough for the utilitarian purposes of the Romans' expert surveyors, but pure mathematics required the ''place-value'' system of modern numerical notation and the development of a zero symbol to make further progress.
The arithmetical wonders that inspired Pythagoras quickly gave birth to the occult methodology of numerology, which attributed special significance to various numbers and employed numerical resolutions of names in divination. The early literary extrapolation of numerology was widespread; one reference in the Biblical book of Revelation to the significant ''number of the beast'' (666) is echoed in much subsequent literary imagery, frequently cited in twentieth- century horror fiction. The supposed unluckiness of the number thirteen and the magical qualities of such numbers as five and seven also proved remarkably tenacious. Five is awarded special status in Thomas Browne's lyrical and mystical essay The Garden of Cyrus (1658), advertised as a celebration of ''the Quincunciall'', while seven exercised sufficient authority over Isaac Newton—an enthusiastic numerologist in the context of his Biblical studies—to make him insist that there are seven colours in the visual spectrum, although orange and indigo are surely fringe effects rather than distinct colour bands. The unsettling potential of arithmetic is touched upon in a legend to the effect that the Pythagorean who first discovered irrational numbers (numbers not expressible as fractional ratios between other numbers— pi and the square root of 2 are the most familiar examples) was put to death by his peers lest the secret get out. The eventual addition to the arithmetic canon of imaginary numbers, extrapolated from the square root of minus one, and transfinite numbers, extrapolated from the concept of infinity, amplified this sense of strangeness considerably and lent further encouragement to the notion of unusual mathematical concepts as quasi-magical arcana.
The development of mathematics was significantly extended in the Islamic world from the ninth century onwards, although it might have made much faster progress had Arab merchants adopted the Indian numerical system they first encountered in the sixth century. When Arabic numerals were imported into Europe at the end of the twelfth century, their facilitation of economic and theoretical calculations helped to provide the foundations of the European Renaissance. The earliest elaborations of mathematical lore produced in thirteenth-century Europe included the identification of the Fibonacci sequence by the thusnicknamed Leonardo of Pisa in Liber Abaci (1202).
Algebra took its name from a treatise written by the Arabic mathematician al-Khwarizmi, although its popularisation in Europe had to await its elaboration in Gerolamo Cardano's Artis magnae sive de regulis algebraicis (1545). European mathematics made rapid progress in the seventeenth century in close association with the advancement of science. John Napier published his discovery of logarithms in 1614, a year before John Kepler devised new methods of geometrical analysis that were further developed by Bonaventura Cavalieri in 1635. Rene´ Descartes published his method of analytical geometry in 1637, assisting the conceptualisation of infinite space and opening up the possibility of considering time as a dimension. His work was carried forward by Pierre de Fermat, who also helped Blaise Pascal found probability theory in 1654. Gottfried Leibniz and Isaac Newton developed what is now known as calculus in the 1670s, fervently disputing the priority of their discoveries in the late 1680s.
In the eighteenth century, the evolution of mathematics became a determining feature of the Age of Enlightenment, symbolised in France by the work of encyclopaedist Jean Le Rond d'Alembert and the triumph of Joseph-Louis Lagrange's Me´canique analytique (1788), which provided a comprehensive analysis of the mathematical foundations of mechanics. Pierre-Simon Laplace extended this work into the field of celestial mechanics in Me´canique Ce´leste (1799– 1825), and developed several new analytical instruments, whose catalogue was further extended by Jean-Baptiste Fourier. Applied mathematics made a huge contribution to the development of human science, not merely in the assembly and correlation of social statistics—following precedents set by John Gaunt in the seventeenth century—but in such philosophical considerations as Jeremy Bentham's utilitarian calculations of the greatest good for the greatest number in Introduction to the Principles of Morals and Legislation (1789). These various threads of thought came together in Robert Malthus' Essay on Population (1798) and their combination continued to complicate mathematical schools of economics.
The literary reflection of this pattern of progress and application was inhibited by fundamental differences of style and manner between mathematical and verbal descriptions and conclusions. Although the spaces of exotic geometry were eventually opened up to literary exploitation, the potential deployments of arithmetic and algebra remained far more limited. The occult aspects of numerology remained far more evident in eighteenth-century fiction than pure or applied mathematics. By the end of the century, mathematics seemed to lay people to be so far removed from the realms of common sense as to require a special kind of mind to engage with it. Such significant mathematical puzzles as ''Fermat's last theorem'' took on a quasi-iconic status as emblems of extreme difficulty, reflected in such stories as Jerry Oltion's ''Fermat's Lost Theorem'' (1994).
Early nineteenth-century crusades for popular education and attempts to popularise science paid as much attention to elementary numeracy as to fundamental literacy, but the deterrent effect of mathematical expressions was soon realised and such representations were largely reserved to the pages of academic publications. The aesthetic appreciation of mathematics became an esoteric achievement, although it did find occasional literary expression, as in George Boole's ''Sonnet to the Number Three'' (ca. 1850). The intrusions of mathematics into the fiction of such mathematicians as Charles Dodgson (Lewis Carroll) and C. H. *Hinton were often calculatedly surreal, founding a tradition of uninhibited bizarrerie carried forward into the twentieth century in such mathematicians' fantasies as David Eugene Smith's Every Man a Millionaire: A Balloon Trip in the Mathematical Stratosphere of Social Relations (1937), J. L. Synge's Kandelman's Krim (1957), Philip J. Davis' Thomas Gray, Philosopher Cat (1988) and Thomas Gray in Copenhagen (1995), Clifford A. Pickover's Chaos in Wonderland (1995), and Eliot Fintushel's ''Milo and Sylvia'' (2000). Homer Nearing's The Sinister Researches of C. P. Ransom (1954), whose protagonist is head of the mathematics department of Uh-Uh University, is similar in spirit, as is Paul Di Filippo's ''Math Takes a Holiday'' (2001).
Heroic efforts were made in the twentieth century to make mathematics accessible and interesting to lay readers, one of the leading figures in the crusade being Eric Temple Bell—author of The Queen of the Sciences (1931), The Handmaiden of the Sciences (1937), and Men of Mathematics (1937)—who also wrote science fiction as John Taine. Lancelot Hogben's Mathematics for the Million (1937) sold very well, but its sales reflected the good intentions of its readers more than any triumph of educational achievement. The deterrent effect of mathematics continued to increase in the latter half of the century, to the point at which Stephen Hawking was advised before writing A Brief History of Time (1988) that every equation would halve his sales. Martin Gardner made elaborate use of a strategy that attempted to popularise mathematics by means of puzzles, in his columns in Scientific American and Isaac Asimov's Science Fiction Magazine, although the principal effect of his endeavours was to help secure a cult following for mathematical puzzles. Although generally esoteric, the hobby in question proved capable of spawning a popular fad in the early twenty-first century Sudoku craze. Several notable showcase anthologies of mathematical fiction were produced in this cause, including Clifton Fadiman's Fantasia Mathematica (1958) and The Mathematical Magpie (1962) and Rudy Rucker's Mathenauts (1987); Gardner's contributions included Science Fiction Puzzle Tales (1981), Puzzles from Other Worlds (1984), and The No-Sided Professor (1987). One simple arithmetical phenomenon that is widely featured in story form is the remarkable increase obtained by geometric series, featured in a traditional anecdote in which a wealthy potentate agrees to reward a petitioner by placing one grain of rice on the first of a chessboard's sixty-four squares, two on the second, four on the third, and so on, not realising that although 64 is not a huge number, 263 most definitely is. Ingenious exactions of this kind feature in a number of Faustian fantasies, including James Dalton's The Gentleman in Black (1831), whose hero unwisely commits himself to doubling the number of sins he commits every year in return for unlimited wealth, and Alexandre Dumas' Le meneur de loups (1857; trans. as TheWolf-Leader). The power of compound interest to magnify sums of money over long periods of time is used as a plot lever in Euge`ne Sue's Le Juif errant (1845; trans. As The Wandering Jew), Edmond About's L'homme a` l'oreille casse´e (1861; trans. as The Man with the Broken Ear), and H. G. Wells' When the Sleeper Wakes (1899) and is extravagantly foregrounded in Harry Stephen Keeler's ''John Jones' Dollar'' (1915). The formula was reused in Charles Eric Maine's The Man Who Owned the World (1961), but the folly of neglecting the similarly geometrical erosions of inflation is pointed out in Frederik Pohl's The Age of the Pussyfoot (1968). Ideas are spread in a similar fashion in Edward Everett Hale's ''Ten Times One Is Ten'' (1871), which anticipates the theory of ''chain letters'' and ''pyramid selling'', as featured in numerous twentieth-century get-rich-quick schemes and such literary works as W. Laird Clowes' The Great Peril, and How it Was Averted (1893) and Katherine MacLean's ''The Snowball Effect'' (1952). Early pulp science fiction provided a showcase for neo-Pythagorean celebrations of the cosmic significance of mathematics. New computer-generated equations alter reality in Nathan Schachner's ''The Living Equation'' (1934) and ''The Orb of Probability'' (1935), while the universe proves to be reducible to manipulable mathematical statements in John Russell Fearn's ''Mathematica'' and ''Mathematica Plus'' (both 1936). Although the fashionability of numerology as a divinatory means was far outstripped in the second half of the twentieth century by astrology and cartomancy, the mysticism of numbers remained more robust in literary imagery. Mathematical species of magic were sometimes dressed in apologetic disguise, as in L. Ron Hubbard's ''The Dangerous Dimension'' (1938), and sometimes featured explicitly, as in L. Sprague *de Camp and Fletcher Pratt's Harold Shea series (launched 1940). Both strategies were extensively echoed; John Rankine's ''Six Cubed Plus One'' (1966) and Stephen G. Spruill's The Janus Equation (1979) are examples of the former strategy, Geoffrey A. Landis' ''Elemental'' (1984) the latter. Jamil Nasir's The Higher Space (1996) offers a hybrid account of thaumatomathematics.
The extrapolation of similar modes of thinking to take aboard irrational, imaginary, and transfinite numbers is often treated in a whimsical Carrollian manner, as in such stories as James Blish's ''FYI'' (1953), J. W. Swanson's ''Godel Numbers'' (1969), Charles Mobbs' ''Art Thou Mathematics'' (1978), Rudy Rucker's White Light (1980), Ted Chiang's ''Division by Zero'' (1991), and John Barrow's play Infinities (2002), although Carl Sagan's Contact (1985) offers a more earnest account of pi as an encoded message and Stephen Baxter's ''The Logic Pool'' (1994) offers a reverent view of ''metamathematics''. Late twentieth-century advances in mathematical theory that gave rise to similar literary spin-off included Rene´ Thom's ''catastrophe theory'', as set out in Stabilite´ structurelle et morphoge`nese (1972; trans. As Structural Stability and Morphogenesis) and *chaos theory.
The widespread use of statistical analysis in various kinds of social and economic research in the twentieth century encouraged the production of numerous works in which such research bears strange fruit. William Tenn's ''Null-P'' (1951) puts forward the ironic proposition that the perfectly average man would be the perfect democratic representative. Several statistical cycles peak simultaneously in Robert A. Heinlein's ''The Year of the Jackpot'' (1952). James Blish's ''Statistician's Day'' (1970) imagines statistics being converted from a form of measurement into a means of social design to which reality is then adjusted. The relentless search for statistical anomalies in parapsychological research is mirrored in many psi stories, including Raymond F. Jones' ''The Non-Statistical Man'' (1956). Ominous breaches of the ''law of averages'' are also featured in such whimsical stories as Robert M. Coates' ''The Law'' (1974) and the opening scene of Tom Stoppard's Rosencrantz and Guildenstern Are Dead (1968). Stories featuring mathematicians at work are understandably rare, given the nature of their labour, although there are numerous elliptical accounts of mathematical geniuses such as Robert A. Heinlein's ''Misfit'' (1939) and J. T. Lambery Jr.'s ''Young Beaker'' (1973). William Orr's ''Euclid Alone'' (1975) is a notable exception.