MATHEMATICS

   The imaginations of pure mathematicians have provided sf writers with important motifs. For example, the notions taken from geometry and topology of a fourth and other DIMENSIONS (which see for a listing of relevant sf stories) have the essential qualities of strangeness and mystery, making them an enjoyable struggle for the untrained intuition to accept. A surprising number of sf writers have been mathematicians, or at least have trained in mathematics; among them have been Lewis CARROLL, Arthur C. CLARKE, Paul DAVIES, Ralph Milne FARLEY, Martin GARDNER, NormanKAGAN, Johannes KEPLER, Donald KINGSBURY, Homer NEARING, Larry NIVEN, Esther ROCHON, Rudy RUCKER, Bertrand RUSSELL, Boris STRUGATSKI, John TAINE, Vernor VINGE and David ZINDELL.In discussing the use of mathematical ideas in sf, the boundary between sf and fantasy must be drawn according to somewhat different principles from those used in the case of the natural sciences. Since many mathematical ideas derive their piquancy from the fact that they are definitely incompatible with the world we live in, a story illustrating such an idea cannot claim any credence as a record of possible events, and should perhaps be classed as a fantasy. Yet an important consideration in judging a story of this type is its fidelity to mathematical truth, in which respect it belongs not just to sf but to sf at the furthest remove from fantasy, to that subgenre comprising stories which turn on a point of established science.In the field of geometry these points are illustrated by the prototype of all stories which use the idea of space having other than three dimensions, E. A. ABBOTT's Flatland (1884 as by A Square). Written in a period when therewas great interest among mathematicians in n-dimensional geometry, this fantasy offers an indirect approach to the problems we, as three-dimensional creatures, have in understanding four-dimensional space by examining the difficulties two-dimensional beings would have in understanding three-dimensional space - an explanatory device which was to become a standard feature of sf invoking a fourth dimension. With sentient lines, triangles and polygons as its inhabitants, the book's only three-dimensional character being a visiting sphere, Flatland makes no pretence of being related to the real world. The book has been made into a short animated film, Flatland (1965), dir Eric Martin, with narration by Peter Cook. C.H. HINTON developed Abbott's speculations, adding some ofhis own, in several pieces in Scientific Romances (coll 1886) and Scientific Romances: Second Series (coll 1902), and in his sequel AnEpisode of Flatland (1907). In Bolland (1957; trans as Sphereland 1965 US) Dionys BURGER wrote another sequel designed to explain in the same way Einstein's theories about curved space. Greg BEAR's stylish story "Tangents" (1986) imagines the intrusion of higher-dimensional beings into our three-dimensional space, in a sophisticated reworking of the theme of Miles J. BREUER's "The Captured Cross-Section" (1929).Among the manystories using fourth and other dimensions, two deserve mention here for their emphasis on particular mathematical points. H.G. WELLS's "The Plattner Story" (1896) turns on the fact that a three-dimensional object,if rotated through half a turn in a fourth dimension, becomes its mirror image (in the story this happens to Gottfried Plattner, who afterwards finds his heart is on the right). The reception of this point by literary readers amusingly illustrates how, if science can lend credibility to sf, sf removes credibility from science: one critic (Allan Rodway, in Science and Modern Writing (1964)) told his readers that this was "neither scientific nor mathematical". In fact it is excellent mathematics. In "And He Built a Crooked House" (1940) Robert A. HEINLEIN describes a house ofeight cubical rooms which fit together like the eight three-dimensional "faces" of a four-dimensional cube (a tesseract). The story ostensiblytakes place in the real world, but Heinlein's main concern is not to persuade the reader that his house is physically possible but to show us something which is mathematically feasible though seemingly paradoxical. He is therefore careful to be mathematically correct in describing thestructure of his house, while emphasizing its startling features. His one slip, as it happens, offends against both requirements; the mathematical truth is even stranger than he realizes.Other writers have set stories in frankly imaginary worlds for the sake of unusual topological structures of space, but few have been so careful to define the structures as Heinlein was. It is common for the topological oddity to be revealed only at the last, as a shock ending, as in David I. MASSON's "Traveller's Rest" (1965) - though this is only one element of a subtle and complex story in whichthe structures of time and language undergo variations related to that of the structure of space - and Arthur C. CLARKE's "Wall of Darkness" (1949), which uses a similar idea. Christopher PRIEST's INVERTED WORLD (1974) features (or appears to, for the whole thing could be a trick of perception) a hyperboloid world where variations of subjective experience take place according to one's position in the world. (Several mathematical stories, including Priest's, are discussed under PERCEPTION.) Topology is also likely to be abused as a catch-all explanation for any weird happening: in "A Subway Named Mobius" (1950) by A.J. Deutsch, for example, it is supposed that a subway network has become so complex that trains mysteriously disappear and reappear, although no proper topological explanation is presented.This careless attitude to topology is comparable with the numerology (PSEUDO-SCIENCE) of such stories as "Six Cubed Plus One" (1966) by John Rankine (Douglas R. MASON), in which magicalproperties are attributed to special numbers. (A sardonic comment on cavalier attitudes to mathematics was made by L. Sprague DE CAMP and Fletcher PRATT in The Incomplete Enchanter (1942), in which a series of propositions in mathematical logic is used as a magic talisman.)Transfinite arithmetic shares with topology the appeal of the unfamiliar and the smack of paradox, and infinity has its own sensational connotations. For these reasons transfinite numbers are often called upon to establish an atmosphere of mathematical mysticism, but few authors have found it possible to do more with them. They appealed to the quirkiness of James BLISH, who in "FYI" (1953) seized on the fact that they do not andcannot count material objects and contemplated the Universe being reconstructed to accommodate them.The two other areas of mathematics which have provided material for sf stories are statistics and logic. The concepts of statistics and probability theory are easy to misunderstand, as has been demonstrated in many sf stories; also, being abstractions which can masquerade as concrete instances, they are easy to ridicule, and this can be seen in Russell Maloney's "Inflexible Logic" (1940), which shows us monkeys typing famous works of literature, William TENN's "Null-P" (1951), in which an exactly average man is discovered, and JackC. HALDEMAN's "A Very Good Year" (1984) in which the absence of death for a whole year is statistically compensated for in the next. A rather more serious point about statistics was made by Robert M. COATES in "The Law" (1974), which describes the "Law of Averages" breaking down and so promptsconsideration of why human beings in large numbers normally do behave in predictable ways.The perennial fascination of logical paradoxes was exploited by Gordon R. DICKSON in "The Monkey Wrench" (1951). This story uses the paradox of Epimenides the Cretan ("this statement is false") to deflate a computer engineer's pride in the perfection of his machine, thus giving a reassuring reminder of the insufficiency of logic. An opposite effect was achieved by Frederik POHL in a number of stories, notably "The Schematic Man" (1968), which describes a man coding himself as a computerprogramme, and so raises the question of what makes the real world more than a mathematical model. Logical paradoxes in fictional form were a speciality of Lewis Carroll, whose A Tangled Tale (1886) and The Game of Logic (1887) are devoted to them as, in part, are the Alice books. Closerto our own time, Martin GARDNER, whose mathematical-puzzle column appeared in Scientific American 1957-81 and in ISAAC ASIMOV'S SCIENCE FICTION MAGAZINE from 1977, has written many fictionalized mathematicaldiversions, such as those collected in Science Fiction Puzzle Tales (coll 1981) and Puzzles from Other Worlds (coll 1984).Mathematics whose point isnot primarily mathematical can also appear in sf; the use of an occasional mathematical formula is seen by some sf writers, as by some scientists, as conferring intellectual respectability. A rare example of a genuine mathematical argument occurs in a footnote to Fred HOYLE's The Black Cloud (1957): it is a nice calculation, and has probably added to a number ofreaders' enjoyment of the book. Hoyle also gave a mathematical explanation of an sf speculation in the preface to Fifth Planet (1963).Examples of popular exposition of mathematical ideas in sf are the explanation of the calculus of variations in David DUNCAN's Occam's Razor (1957) and that of coordinate systems and relativity in Miles J. Breuer's "The Gostak and the Doshes" (1930). Both authors proceed to tell stories which have onlytenuous connections with the mathematical ideas they have expounded.Though the mathematical genius Libby in Robert Heinlein's "Misfit" (1939) proves resourceful, mathematicians as characters in GENRE SF have often been stereotyped as absent-minded, ineffectual and unworldly; they are clearly descended from the inhabitants of Jonathan SWIFT's Laputa in Gulliver's Travels (1726; rev 1735). Sf is popular among mathematicians, however, andit is not surprising that there should have been some attempts to adjust this image. This can be seen particularly in the stories of Norman KAGAN, whose portrayals of zany, hyperactive maths students, although they sometimes appear self-congratulatory, may be rather closer to reality. Kagan's stories make witty use of many parts of mathematics; whileostensibly concerned with sf speculations - in "Four Brands of Impossible" (1964) the use of a different logic to describe the world, in "TheMathenauts" (1964) a journey into various mathematical spaces - they are really about the experience of doing mathematics. An important mathematical sf protagonist is Shevek, in Ursula K. LE GUIN's The Dispossessed (1974), whose new mathematics is the basis for building theANSIBLE, a FASTER-THAN-LIGHT communications device. A particularly interesting mathematician is the elderly protagonist of "Euclid Alone" (1975) by William F. Orr, himself a mathematician. A student successfullyproves one of Euclid's axioms to be wrong. His teacher is left with the moral quandary of whether or not to suppress the discovery, which may, ultimately, destroy the serenity of everyone in the world. Orr's story can be found in Mathenauts (anth 1987) ed Rudy RUCKER, the only anthology of sf mathematical stories since Fantasia Mathematica (anth 1958) and The Mathematical Magpie (anth 1962), both ed Clifton Fadiman.Mathematics hasentered fiction in strange ways. Some of the oddest are discussed in the terminology entry OULIPO. Certainly stories of COMMUNICATIONS can feature mathematics, through the idea of mathematics as a universal language. Some notable mathematical incursions into sf during the 1980s are the mathematical harmonies in Kim Stanley ROBINSON's The Memory of Whiteness (1985), the cosmic message concealed in the endless series of numbersfollowing pi's decimal point in Contact (1985) by Carl SAGAN, and the disquisition on the Mandelbrot set in Arthur C. Clarke's The Ghost from the Grand Banks (1990), one of the few sf stories to use the mathematics of fractals. But the most important mathematical sf writers of the past decade have been Rudy Rucker and David Zindell, both mathematicians. Rucker's stories do not merely turn on mathematical points; they are oftenset in worlds generated by mathematical ideas, whose exploration is itself an act of mathematical intellection, in which the author delights, as he does in raunchy humour. Such tales include much of his work, notably White Light, or What is Cantor's Continuum Problem? (1980) - a crazed fantasiamoving in physical (though afterlife) analogues of Hilbertian space, transfinite numbers and a lot else - The Sex Sphere (1983) and The Secret of Life (1985). Zindell's Neverness (1988) is one of the few successful books whose assumption is that mathematics is romantic. In this novel, to win an ice-race is to solve a theorem. The sequence where the protagonist can map the space windows only through mathematics - fountains and arpeggios of mathematics - is sustained and moving, and conveys with great conviction even to the nonmathematical reader what the high delight of mathematical thought must feel like.

   TSu/PN