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, A K Dewdney, Ralph Milne Farley, Martin Gardner, Norman Kagan, Johannes Kepler, Donald Kingsbury, Homer Nearing, Larry Niven, Esther Rochon, Rudy Rucker, Bertrand Russell, Ian Stewart, 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 there was 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 in Flatland 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), directed by Eric Martin, with narration by Peter Cook. C H Hinton developed Abbott's speculations, adding some of his own, in several pieces in Scientific Romances (coll 1886) and Scientific Romances: Second Series (coll 1902), and in his sequel An Episode of Flatland (1907). In Bolland (1957; trans as Sphereland 1965) Dionys Burger wrote another sequel designed to explain in the same way Einstein's theories about curved space; there are further Sequels by Other Hands. Greg Bear's stylish story "Tangents" (January 1986 Omni) 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" (February 1929 Amazing).
Among the many stories using fourth and other dimensions, two deserve mention here for their emphasis on particular mathematical points. H G Wells's "The Plattner Story" (April 1896 New Review) 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 ) told his readers that this was "neither scientific nor mathematical". In fact it is excellent mathematics. In "– And He Built a Crooked House" (February 1941 Astounding) Robert A Heinlein describes a house of eight cubical rooms which fit together like the eight three-dimensional "faces" of a four-dimensional cube (a tesseract). The story ostensibly takes 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 the structure 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 realized.
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" (September 1965 New Worlds) – though this is only one element of a subtle and complex story in which the structures of time and language undergo variations related to that of the structure of space (see Relativity) – and Arthur C Clarke's "The Wall of Darkness" (July 1949 Super Science Stories), 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 Möbius" (December 1950 Astounding) 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 (see Pseudoscience) of such stories as "Six Cubed Plus One" (in New Writings in SF 7, anth 1966, ed John Carnell) by John Rankine (Douglas R Mason), in which magical properties 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 [May, August 1940 Unknown; coll of linked stories 1941], in which a series of propositions in mathematical logic is used as a magic incantation of transport to worlds of fiction and Mythology.)
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 for many years few authors found it possible to do more with them. They appealed to the quirkiness of James Blish, who in "FYI" (in Star Science Fiction Stories 2, anth 1953, ed Frederik Pohl) seized on the fact that they do not and cannot count material objects and contemplated the Universe being reconstructed to accommodate them. Rudy Rucker is a mathematician with a firm grasp of the issues: his White Light, or What is Cantor's Continuum Problem? (1980) reifies the appalling tiers of mathematical infinities as an impossible mountain which the protagonist is required to climb, passing such relatively familiar benchmarks as aleph-null (the infinity of the integers or counting numbers) and C (the infinity of real numbers, which can have up to aleph-null digits after the decimal point) but reaching a setback at epsilon-zero (the point, in effect, at which all previous mathematical schemas for generating higher infinities cease to work). Here Rucker throws the reader in at the deep end of infinite set theory and other mathematical complexities; the background is more clearly explained in his nonfiction Infinity and the Mind: The Science and Philosophy of the Infinite (1982) and, in passing, in Douglas Hofstadter's Gödel, Escher, Bach: An Eternal Golden Braid (1979).
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" (February 1940 The New Yorker), which shows us monkeys producing famous works of literature by random manipulation of typewriters (a traditional Thought Experiment), William Tenn's "Null-P" (January 1951 Worlds Beyond), in which an exactly average man is discovered and hailed as the ideal, and Jack C Haldeman's "A Very Good Year" (December 1984 Analog) 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" (29 November 1974 The New Yorker), which describes the "Law of Averages" breaking down and so prompts consideration of why human beings in large numbers normally do behave in predictable ways. Robert A Heinlein's "The Year of the Jackpot" (March 1952 Galaxy) extrapolates the phenomenon of statistical "clumping" to a scenario in which numerous unrelated Disasters (predictable by study of cycles and trends) afflict Earth during the titular period. Perhaps the most famous if not the most plausible sf extrapolation of statistical theory is Isaac Asimov's predictive science of Psychohistory (which see) in the Foundation sequence. Tom Stoppard's Rosencrantz and Guildenstern are Dead (first performed 1966; 1967 chap) opens with a disconcerting breakdown of probability as 89 successive coin-tosses come up heads.
The perennial fascination of logical Paradoxes was exploited by Gordon R Dickson in "The Monkey Wrench" (August 1951 Astounding). This story uses the paradox of Epimenides the Cretan ("All Cretans are liars": in effect, "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 which was to become a notorious sf Cliché. An opposite effect was achieved by Frederik Pohl in a number of stories, notably "The Schematic Man" (January 1969 Playboy), which describes a man coding himself as a computer program (see Upload), 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 puzzle tales in A Tangled Tale (coll 1886) and rulebook for The Game of Logic (1887) make some play with them, as – more familiarly – do the Alice books. Closer to our own time, Martin Gardner, whose mathematical-puzzle column appeared in Scientific American 1957-1981 and in Asimov's Science Fiction from 1977, has written many fictionalized mathematical diversions, such as those collected in Science Fiction Puzzle Tales (coll 1981) and Puzzles from Other Worlds (coll 1984). Those who have continued the immensely influential Gardner tradition in Scientific American include A K Dewdney, Douglas Hofstadter and Ian Stewart.
Mathematics whose point is not 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 of readers' enjoyment of the book. Hoyle also gave a mathematical explanation of an sf speculation in the preface to Fifth Planet (1963). Further 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" (March 1930 Amazing). Both authors proceed to tell stories which have only tenuous connections with the mathematical ideas they have expounded. Sticking more closely to the subject, Leslie Charteris's action hero the Saint uncharacteristically takes up pencil and paper to lay bare the deceptive odds of various probability-based "sucker bets" in the non-sf "The Percentage Player" (August/September 1959 Saint Magazine).
Though the mathematical genius Libby in Robert Heinlein's "Misfit" (November 1939 Astounding) 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, and it 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; while ostensibly concerned with sf speculations – in "Four Brands of Impossible" (September 1964 F&SF) the use of a different logic to describe the world, in "The Mathenauts" (July 1964 If) a physical journey into various mathematical spaces – they are really about the experience of doing mathematics. One important mathematical sf protagonist is Shevek, in Ursula K Le Guin's The Dispossessed (1974), whose new mathematics is the basis for building the Ansible, a Faster-than-Light Communications device.
Another particularly interesting mathematician is the elderly protagonist of "Euclid Alone" (in Orbit 16, anth 1975, ed Damon Knight) by William F Orr, himself a mathematician. Here a student successfully proves 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) edited by Rudy Rucker, the only Anthology of sf mathematical stories since the classic Fantasia Mathematica (anth 1958) and The Mathematical Magpie (anth 1962), both edited by Clifton Fadiman. Similar in theme is Ted Chiang's "Division by Zero" (in Full Spectrum 3, anth 1991, ed Lou Aronica, Amy Stout and Betsy Mitchell), in which the central character is horrified to discover a proof that mathematics is inconsistent with itself, and therefore meaningless. A further seeming inconsistency in mathematics is explored by the eponymous super-Computer of Greg Egan's "Luminous" (September 1995 Asimov's), where an anomalous zone of number theory proves to be the interface to a Parallel World based on different mathematics; this crossover becomes more than theoretically disturbing in the sequel "Dark Integers" (October/November 2007 Asimov's).
Mathematics has entered fiction in strange ways. Some of the oddest are discussed in the entry for the Oulipo group. 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 numbers following pi's decimal point in Contact (1985) by Carl Sagan – this was Satirically anticipated in John T Sladek's The Müller-Fokker Effect (1970), featuring the Paranoid theory that Commies have insidiously encoded their anti-US master plan into the digits of pi – and the sidebar disquisition on the Mandelbrot set in Arthur C Clarke's The Ghost from the Grand Banks (1990), at that time one of the few sf stories to use the mathematics of fractals (later to become a Cliché). Greg Egan's "Unstable Orbits in the Space of Lies" (July 1992 Interzone) features a mapping of migrant people's movements on to the kind of complex fractal path known as a strange attractor.
But the most important mathematical sf writers since the late twentieth century have been Rudy Rucker and David Zindell, both mathematicians. Rucker's stories do not merely turn on mathematical points; they are often set 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 comprise much of his early work, notably White Light, or What is Cantor's Continuum Problem? (1980) – a crazed fantasia moving in physical (though afterlife) analogues of Hilbertian space, transfinite numbers and much 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/DRL]
see also: Basilisks; Information Theory.
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