Mathematics in science fiction

Summary: Mathematics plays many roles in science fiction, sometimes as content or characters, other times bringing elements to life on the screen.

Like mathematics, writing science fiction is a craft grounded in deduction and extrapolation. The writer begins with certain axioms: the world as we know it and the world as we believe it could be. He introduces certain new variables: a thinking robot, an alien invasion, human clones. Explicitly or implicitly, the story explores the consequences, the corollaries of these new things according to the implications of those initial axioms. Stories that do not do this are considered fantasy or sometimes science-fantasy or soft science fiction, if they otherwise contain the set-dressing of the science fiction genre.

The setting is often the future, an alternative history, or some alternate reality, and may include use of time travel. Alien characters frequently interact with humans in science fiction. Many books, movies, comics, graphic novels, computer games, and Internet applications use science fiction themes, sometimes as a context in which to explore deeper philosophical questions. Mathematics and science play a variety of roles within the science fiction genre. Sometimes, mathematics and science are written into the story to give validity and believability to the futuristic setting or to the technology. Mathematics is also used to bring fantastic science fiction elements to life on screen, such as in the groundbreaking Star Wars franchise or the 2010 film Avatar. At other times, characters in science fiction works are mathematicians or scientists who act as the primary heroes or villains, or who explain scientific elements to the audience. The inclusion of mathematically talented characters in science fiction is sometimes done to exploit commonly held stereotypes about mathematicians for narrative purposes, such as genius or aloofness. In other works, mathematics becomes the explicit subject of the story, and the mathematics of science fiction in both written and visual media has been explored in college courses and mathematics research. Mathematicians or individuals that have mathematical training often create science fiction, and science fiction may inform mathematical research. The widely noted “Big Three” authors of twentieth-century science fiction—Arthur C. Clark, Robert A. Heinlein, and Isaac Asimov—all had mathematical training or mathematically based science backgrounds and made nonfiction contributions to areas such as satellites, rocketry, robotics, and ethics.

Early History of Science Fiction

Because of the varying definitions of science fiction, it is difficult to determine exactly what might be the first science fiction story. The Mesopotamian epic poem The Epic of Gilgamesh, which is among the oldest surviving works of literature, is cited by some scholars as containing elements of science fiction. Some researchers note that the Bible, when examined as a work of literature, has stories that could be classified as science fiction, such as the ascension of the prophet Elijah to heaven in a fiery chariot. In the second century, the Greek satirist Lucien of Samosata wrote about interplanetary travel and alien life forms in his True Histories (or True Tales). English lawyer and philosopher Thomas More’s 1516 work Utopia described a perfect society, which became a common theme among later science-fiction writers. Some scholars argue that such early works cannot be claimed as the first science fiction because neither the audience nor the authors likely knew enough about the underlying science. Correspondingly, they might claim that the origin of science fiction coincided with the post-medieval scientific revolution and discoveries in science and mathematics made by people such as Isaac Newton and Galileo Galilei. Mathematician and astronomer Johannes Kepler wrote a story in 1634 called Somnium, which imagined that a student of astronomer Tycho Brahe had been transported to the moon and described how Earth might look when viewed from that location. It contained mathematical computations and is considered by some to be a scientific treatise, while others cite it as the first science fiction, including both Asimov and astronomer and author Carl Sagan. Author Brian Aldiss asserts that science fiction derives many of its structure and conventions from the Gothic horror genre, which suggests that Mary Shelley’s 1818 novel Frankenstein is “the first seminal work to which the label SF can be logically attached.” This labeling is perhaps because of its introduction of science fiction themes like a mad scientist, the potential misuse of technology, and the presence of an non-human being as a main character.

The Foundations of Twentieth-Century Science Fiction

Jules Verne and Herbert George (H. G.) Wells are often jointly known as the “fathers of science fiction” for their creative influence on the development of twentieth-century science fiction. Jules Verne consistently incorporated the newest technological discoveries and experiments of his lifetime into his work. Many of his most popular novels, like A Journey to the Center of the Earth (1864), From the Earth to the Moon (1865), Twenty Thousand Leagues Under the Sea (1869), and Around the World in Eighty Days (1873), have been widely translated into other languages and adapted into plays, movies, television shows, and cartoons. Some scholars have called Verne’s books visionary and even prophetic for describing mathematical and scientific phenomena such as weightlessness and heavier-than-air flight before they were well-known or understood. His attention to realistic scientific principles and detailed descriptions of problems and solutions would later challenge many real-life mathematicians, scientists, and engineers. Physicist and engineer Hermann Oberth and scientist Konstantin Tsiolkovsky, who are known as the “fathers of rocketry and astronautics” along with physicist Robert H. Goddard, reported being inspired by Verne’s books.

Like Verne’s work, the novels of H. G. Wells have been widely adapted into various other media, and the 1895 novel The Time Machine, in particular, is cited as inspiring many other works of fiction. The invention of the now commonly used term “time machine” is attributed to Wells, as is the notion of time being the fourth dimension. In the 1897 novel The Invisible Man, a scientist named Griffin makes himself invisible by changing the refractive index of his body so that it neither absorbs or reflects light. Some of Wells’s books were considered to be exceptionally bold and compelling. His 1898 novel The War of the Worlds is well-grounded in mathematical and scientific theories from the time it was written, like mathematician Pierre-Simon Laplace’s formulation of the nebular hypothesis. It shared a vision of space travel common to late nineteenth-century novels, including Verne’s From the Earth to the Moon. Large cylinders were fired from cannons on the Mars surface to transport the aliens to Earth. Later mathematical models and calculations necessary to send people into Earth’s orbit and to the moon, as well as to guide probes to Mars and the far reaches of the solar system, demonstrated that the parabolic trajectories were often quite complex and that the forces required to propel a cylinder from Mars to the Earth would likely be lethal to passengers. Wells was also a science teacher and political activist who recognized and asserted the importance of quantitative knowledge, noting: “Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write.”

Mathematical Science Fiction

While mathematics is widely used to help build or validate the setting or technology of a science fiction story, such as in the works of Verne and Wells, in some cases it is a central component of the plot. There are many mathematical science fiction novels that have been written about a variety of themes. The 1946 short story No-Sided Professor, written by mathematician and author Martin Gardner, discusses the Möbius strip, a one-sided figure named for mathematician August Möbius. It addresses the possibility of a zero-sided figure and other concepts in topology. Occam’s Razor, by author David Duncan in 1956, posits the notion of discontinuous time, which can be bridged by minimal surfaces in certain topologies. Asimov’s 1957 novel The Feeling of Power addresses scientific computing in a futuristic society in which people have lost the ability to perform basic arithmetic calculations. The rediscovery of hand-multiplication therefore becomes a new “secret weapon” for the society’s military. Author Stanislaw Lem discusses countably infinite sets in his 1968 novel The Extraordinary Hotel, while author and mathematician Greg Egan’s 1991 work The Infinite Assassin includes a discussion of the Cantor set, named for mathematician Georg Cantor, an important concept in topology and some other mathematical fields. Other mathematical science fiction urges appreciation of mathematics as if it is a form of poetry. Examples include author Kathryn Cramer’s 1987 work Forbidden Knowledge, author Normal Kagan’s 1964 work The Mathenauts, and multiple stories by author Eliot Fintushel. Mathematician and author Vernor Vinge often addressed the mathematical themes of superhuman artificial intelligence and a predicted technological singularity: a point in time where the exponential growth of technology results in essentially instantaneous change. These themes are also found Clark’s 2001: A Space Odyssey and its sequels. The term “technological singularity” is credited to mathematician Irving Good and is also linked to Moore’s law, named for Intel co-founder Gordon Moore, which mathematically models the trend in the evolution of computer processor speeds.

Several science fiction novels challenge the foundations of mathematics itself or the commonly proposed notion of mathematics as a universal language. Author Ted Chiang’s Division by Zero, a 1991 short story, discusses the discovery of a proof that mathematics is inconsistent, which may be possible according to Gödel’s Incompleteness Theorems, named for mathematician Kurt Gödel. Chiang’s later 1998 work Story of Your Life involves humans trying to communicate with aliens whose mathematics is based on variational formulations rather than algebra. In the same year, mathematician and author David Reulle’s Conversations on Mathematics With a Visitor from Outer Space, which was published in a collection of nonfiction mathematical essays, argued that mathematics on Earth is essentially human in nature, so humans should not expect aliens to share human’s unique mathematical language. Sagan’s 1985 novel Contact alternatively suggested that humans and aliens may communicate via mathematics, but rather than the typical mode of receiving radio waves containing messages from space, communications are instead embedded within the very framework of mathematics itself.

Since Wells introduced the notion of the fourth dimension in The Time Machine, dimensionality in many forms has been a widely used theme in science fiction, including mathematical science fiction. In the 1940 novel And He Built a Crooked House by Heinlein, a mathematical architect designs a house that is constructed as an “inverted double cross” representation of an unfolded tesseract net in three-dimensional space. Following an earthquake, the structure spontaneously shifts and folds itself into an actual tesseract, whose four-dimensional properties are explored and described by characters. The satirical Edwin Abbot novel Flatland: A Romance of Many Dimensions, which was written largely as a social commentary on Victorian norms and mores, may also be considered science fiction because it depicts an alternate two-dimensional world inhabited by polygonal creatures, which is visited by three-dimensional creatures in a manner that resembles twentieth- and twenty-first-century depictions of human-alien interactions. More than a century after its initial publication, Flatland remains popular in the mathematical community because of its entertaining and enlightening discussions of what some people consider to be an abstract mathematical concept, and it was once described by Asimov as, “The best introduction one can find into the manner of perceiving dimensions.”

Other authors have used the novel as inspiration. Mathematician and author Ian Stewart’s 2001 work Flatterland: Like Flatland, Only More So, explores several mathematical topics such as Feynman diagrams, named for physicist Richard Feynman, superstring theory, quantum mechanics, fractal geometry, and the recurring science fiction theme of time travel. He includes mathematical jokes and puns such as a one-sided cow named Moobius to make concepts relatable to a broader audience. Stewart also co-authored the semi-fictional Science of Discworld series, which compares mathematically and scientifically the natural laws of sperical planets or “roundworlds” like Earth to the created or imagined physical laws of the flat, disc-shaped setting of author Terry Pratchet’sDiscworld novels. Some other works that are commonly cited as extensions of ideas found in Flatland include mathematician and author Dionys Burger’s 1953 novel Sphereland: A Fantasy About Curved Spaces and an Expanding Universe and two works by mathematician and author Rudy Rucker: the 1983 short story Message Found in a Copy of Flatland and the 2002 novel Spaceland.

Mathematics is a living discipline that is constantly evolving, and mathematical science fiction sometimes underscores this point. Gardner’s 1952 story The Island of Five Colors is the sequel to the No-Sided Professor. The characters in the story attempt to solve the Four Color theorem, which was unproven at the time. It illustrates the inherent time dependence of some elements of science fiction, since imagined creations and the mathematics on which they are based frequently become reality later. Gardner stated: “the true four-color theorem, unproved when I wrote my story, has since been established by computer programs, though not very elegantly. As science fiction, the tale is now as dated as a story about Martians or about the twilight zone of Mercury.” At the same time, others argue that the themes of such novels are still useful and relevant when considering the nature of mathematics and that these stories do not automatically lose value as entertainment or inspiration simply because the mathematical or scientific frameworks become somewhat out of date.

Mathematicians as Science Fiction Authors

Mathematicians and mathematically trained individuals such as Martin Gardner, Isaac Asimov, Greg Egan, Ian Stewart, and Vernor Vinge often contribute to both science fiction writing and mathematical or scientific research, and several mathematicians have won the Hugo Award, the premier prize in science fiction and fantasy literature. Perhaps one of the most well-known of these is Rudy Rucker, who is considered among the founding fathers of the science fiction subgenre of cyberpunk, a style that draws inspiration from Gothic horror like Frankenstein, film noir, punk, computer science, and cybernetics, a discipline whose twentieth-century development is attributed to mathematician Norbert Weiner. Rucker credits his mathematical background for influencing not only the content of what he writes but also the way in which he writes: “I think of the writing process itself as a fractal. I have the big arc of the plot, the short-story-like chapters, the scenes within the chapter, the actions that make up the scenes, and nicely formed sentences to describe the actions, the carefully chosen words in the sentence. And hidden beneath each word is another fractal, the entire language with all my ramifying mental associations.” He also notes that both mathematics and science fiction writing can be thought of as ways of exploring the consequences of imposed constraints or assumptions, and that science fiction writing provides a structure for carrying out interesting “thought experiments” about concepts such as alternative mathematical structures to explain the nature of reality.

Visual Media

Science fiction has long been translated to other forms of media, and mathematics plays a dual role as a subject and a technique for bringing both realistic and fantastic images to life. Further, mathematicians are often involved as writers or consultants. Stanley Kubrick’s 2001: A Space Odyssey is one of the most well-known science fiction films. Mathematician Irving Good consulted on the film, as did novelist Clark, and it is praised for its scientific realism and pioneering special effects. The Star Wars franchise, launched in 1977, now includes books, comics, movies, video games, and Web media. It was groundbreaking in its use of mathematically based special effects techniques, including extensive stop motion animation and then later computer animation for backgrounds, props, costumes, and even entire characters. Effects that were once limited to big-budget films have now made their way onto television. The Star Trek franchise is notable not just for its visual imagery but also for references to real-world mathematical concepts including π and Fermat’s last theorem, named for mathematician Pierre de Fermat. Other examples of shows that contain frequent real-world mathematical include SyFy’s Eureka and the animated series Futurama. Producer and writer Ken Keeler has a mathematics Ph.D. from Harvard. Along with other mathematically trained writers, he co-creates many of the mathematical references found on Futurama, and once notably constructed a new mathematical proof to validate an episode’s plot twist.

Bibliography

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Chartier, Timothy, and Dan Goldman. “Mathematical Movie Magic.” Math Horizons 11 (April 2004).

Gouvêa, Fernando. “As Others See Us: Four Science Fictional Mathematicians.” Math Horizons 11 (April 2004).

Kasman, Alex. “Mathematical Fiction.” http://kasmana.people.cofc.edu/MATHFICT/.

———, “Mathematics in Science Fiction.” Math Horizons 11 (April 2004) http://kasmana.people.cofc.edu/MATHFICT/sf-mathhoriz.pdf.

Raham, Richard. Teaching Science Fact With Science Fiction. Santa Barbara, CA: Libraries Unlimited, 2004.

Shubin, Tatiana. “A Conversation With Rudy Rucker.” Math Horizons 11(April 2004).