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Laser gun battles, dangerous mutants, teleportation, androids, time travel. We are all familiar with these traditional science fiction motifs. But, who ever heard of mathematics in science fiction?!? Did the invading aliens in a B-movie ever threaten to Mathematics in “divide by zero” if the scientist didn’t take them to his leader? Science Fiction Alex Kasman College of Charleston

might define science fiction as fantasy ematics in science fiction. After all, the mathematical science fiction for you in fiction designed to appeal to those with famous precision of mathematics makes this brief article. However, I would like Ia knowledge and interest in science. it useful in making things convincing. (In to give you a quick “whirl-wind” tour, This definition addresses not only the fact, one might argue that a frequent stopping at some of the most famous fact, but the reason that science fiction misuse of mathematics is its inclusion in landmarks and pointing out just a few of fans like to read about pushing scientific weak scientific papers for no purpose my favorite lesser-known gems. topics just a bit farther than actual science other than to make them seem more It seems natural to start this tour with can do now. In my view, the typical read- rigorous than they really are.) Moreover, Somnium. Written in 1634, this story er of science fiction is someone whose those “crazy math nerds”—a group in about life on the moon is often referred to interest in the boundaries of scientific which I proudly count myself—would as the first science fiction story. Note that knowledge make traditional “realistic” most certainly find fictional mathematics it is not the first story about a trip to the fiction seem a little bit mundane. On the as interesting as the fictional science of moon—Lucian of Samosata wrote about other hand, these same SF readers know most science fiction. such trips more than a millenium too much science to be able to suspend As it turns out, the surprising thing is earlier—but these earlier stories were not disbelief for a traditional fantasy in which not this tremendous potential for mathe- science fiction since neither the authors there are no constraints on the author. The matics in science fiction, but the fact that nor the readers knew enough science to fact that the author of a science fiction this potential has been largely realized make them so. On the other hand, the story cannot simply turn a prince into a without many people being aware of it. author of Somnium was none other than frog—at least not without reference to The “Mathematical Fiction Homepage” the famous mathematician Johannes some bogus “scientific” explanation (see box below) is a comprehensive Kepler. Kepler’s understanding of plane- involving a matter-transducing machine listing of significant appearances of tary systems was intimately bound to his that controls the collapse of the quantum mathematics in fiction. It presently lists mathematical research, and it must be wavefunction—makes the story more 392 works of mathematical fiction—189 precisely for the reasons of interest and believable to the reader. of which are categorized as science believability that he included pages of Then, those of us who know and love fiction—and is growing all the time. mathematical computation with the story. mathematics can immediately see a Obviously, I could not hope to Equations and mathematical notation tremendous potential for the use of math- describe or analyze all 189 works of have been thrown into more recent

The Mathematical Fiction Homepage is maintained online at http://math.cofc.edu/kasman/MATHFICT/default.html and seeks to be a comprehensive listing of all significant appearances of mathematics in fiction.The site lists movies, books, stories, TV shows and plays. Wherever possible, a link exists to works that are available for free on the inter- net or to sites at which the works can be purchased. Everyone interested in mathematics is invited to use this site to find works of fiction that they can learn from and enjoy. Moreover, anyone who knows any works of mathematical fiction not already listed is urged to contact the site administrators; the list continues to grow thanks to your help!

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science fiction stories for similar Asimov. Although written in 1957 when hallmarks of science fiction, but reasons, though often by people who do ‘calculators’ were heavy mechanical explores the discovery of a proof that not really understand their significance. devices based on complex gear systems, mathematics is inconsistent—something A medical doctor named Miles Breuer Asimov speculates about a future that that of course has not yet been found, wrote several mathematical fiction might seem familiar to people in the but whose existence cannot be ruled out stories about travel to higher dimension- early 21st century, when all students according to Gödel’s incompleteness al spaces. His 1929 The Captured Cross- carry their own pocket-sized electronic theorem. As far as I am concerned, the Section begins with some vaguely rea- calculators. The story concerns a future most moving and clever bit of fictional sonable mathematical notation that was society in which all arithmetic is done mathematics is the conclusion of Carl probably learned in a Linear Algebra by computers and people have Sagan’s novel Contact. Although mathe- course but which is seriously misapplied completely forgotten how—and even matics makes brief appearances in the and misinterpreted as having something that—it is possible for them to do it film, the mathematical conclusion of the to do with the theory of relativity. More themselves. The hero of the story is a book was completely dropped from the reasonable computations of special military officer who rediscovers for movie. In the book, the main character relativity, even if very elementary ones, himself the ability to multiply numbers finds a hidden message not sent by radio appear in Poul Anderson’s Tau Zero “by hand,” a technique that is consid- waves through space, but hidden in the (1970). ered to be a new secret weapon by his very fabric of mathematics itself. Perhaps the most famous example of superiors. Whether such a “message” exists and mathematical science fiction is Edwin It is possible to encounter, and perhaps what it would mean if it did raises very Abbot Abbott’s 1884 masterpiece, even to learn, some sophisticated and real interesting philosophical questions. Flatland. This book about a world of 2- mathematical results by reading science This idea from Contact that we might dimensional polygonal creatures and fiction stories. For instance, Martin be able to communicate with aliens their brief encounters with people from Gardner’s 1946 story The No-Sided Pro- through the universal language of mathe- our 3-dimensional world was as much fessor provides a fun introduction to matics is a very common one in science about Victorian society as it was an topology, just as Stanislaw Lem’s The fiction. I could list many stories, books attempt to teach its readers about the Extraordinary Hotel (1968) introduces and movies in which mathematics is used mathematical concept of “dimension.” some of the interesting properties of in this way. However, it might be more However, its legacy has been its ability countably infinite sets. More ambitiously, interesting for me to mention two “excep- to convey an understanding of this Greg Egan wrote a story called The tions to the rule.” Mathematician David abstract concept in a way that readers Infinite Assassin (1991) whose con- Ruelle wrote an unusual bit of science find both entertaining and enlightening. clusion can only be appreciated by fiction, published only in a book of seri- Perhaps this is why so many mathemat- mathematically sophisticated readers ous essays by famous mathematicians, ical science fiction stories focus on the who understand what it means for the called Conversations on Mathematics concept of dimension. Its direct progeny, Cantor Set to be uncountable in cardinal- with a Visitor from Outer Space (1998) such as Ian Stewart’s recent Flatterland ity and yet have measure zero! which argues that our mathematics is and Rudy Rucker’s hilariously bizarre On the other hand, it is perhaps more essentially human in nature, and that short story Message Found in a Copy of interesting to read some fictional aliens could be expected to have a differ- Flatland, are just a small sampling of mathematics—instances in which the ent mathematics entirely. By concidence, the many science fiction stories that mathematical results themselves come in the same year Ted Chiang wrote a story really address the mathematical aspects from the imagination of the author rather that seems to demonstrate an example of of dimension. After Flatland, Robert than from actual mathematical research. this. Chiang’s Story of Your Life (1998) Heinlein’s 1940 short story And He Built This must be very difficult to do, but involves the difficulty in communicating a Crooked House—about a house built some stunningly clever and well done with an alien species whose mathematics, in the shape of a tesseract—is probably instances of this motif do exist in litera- unlike our algebra-based mathematics, is the most famous, but Mark Clifton’s ture. Greg Egan’s novel Distress (1995) based on variational formulations such as Star, Bright is also notable for its inclu- takes place at a realistically described Fermat’s Principle of Optics. sion of a topological twist—the Möbius mathematical physics conference where Being a mathematician myself, I Band. researchers are on the verge of complet- appreciate these stories that build upon Another rather well-known science ing a long sought after “Theory of and utilize real mathematical knowl- fiction story with mathematical content Everything.” Ted Chiang’s 1991 short edge. However, another side of mathe- is The Feeling of Power by Isaac story Division by Zero contains no other matical science fiction that I have grown

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to enjoy does not require the reader to his story Herbrand’s Conjecture and the change to suit the needs and desires of really understand any mathematics at White Sox Scandal by saying humans. This aspect of mathematics is all, but instead to appreciate it as a sort The idea is that the mathematical well-utilized in the classic science fic- The Cold Equations of poetry. Consider Kathryn Cramer’s logician Jacques Herbrand who tion story (1954) by Forbidden Knowledge, eerie short story actually did die in a mountaineer- Tom Godwin and in Yevgeny Zamy- We which is built up around an actual quote ing accident the same time that his atin’s 1924 dystopian novel . from a mathematics book by Irving article was coming out on matters Such a “cold” view of mathematics, Kaplansky that seems to have a sort of related to the Gödel consistency that it is frigid and uninteresting, is dadaist beauty to it separate from any theorem, was murdered by aliens widely held in America today. One of mathematical significance: who needed to hush him up, the most exiting possible roles for mathematics in fiction is to change this We shall have no need to assign because they were using certain viewpoint. In fact, Rudy Rucker’s meaning to Ext itself; we shall lacunae in the metamathematical tongue-in-cheek short story A New speak only of its vanishing. structures of mind or of space- Golden Age time to sneak through in order to considers the conse- A more humorous approach to mathe- devour humanity. A psychic quences of a machine that could do just matical poetry is taken by Norman named Izzy Molson gets wise to that; it can allow non-mathematicians Kagan in his clever story The Mathe- them and fights them. In the to see and appreciate mathematics as nauts, which develops the notion that course of this fight, the value of mathematicians do. My own apprecia- one can travel through space merely by p changes, wreaking havoc on tion for mathematics was probably thinking about abstract mathematics. It earthly geometry. influenced by reading Stanislaw Lem’s contains passages such as 1961 science fiction novel Return from Another aspect of mathematics not the Stars. This book has several pages I was looking over Ephraim directly related to its content is also of mathematical dialogue that lies Cohen’s latest paper, Nymphoma- important to its use in fiction, and that is niac Nested Complexes with Ross- somewhere between the genres of the stereotype of mathematicians. On the “fictional mathematics” and “mathe- ian Irrelevancies (old Ice Cream one hand, there is the somewhat positive matics as poetry” described above. But, Cohen loves sexy titles), when the stereotype—one that I cannot say I mind most moving to me was a description trouble started. We’d abstract- terribly much—which says that mathe- by an astronaut of why he became ed...were ready for the first tests. I maticians are very intelligent. Thus, it is interested in mathematics during an made the Dold invariant and easy for a writer to get across the fact incredibly long journey: shoved off through one of the pas- that one of the characters is supposed to sages that linked the isomorpho- be very smart...they simply make him or I only came to understand this mechanism and the lab. her a mathematician. One example there. Because mathematics stands Finally, I could not discuss the use of might be the mathematician character above everything. The works of mathematics as poetry in science fiction played by Jeff Goldblum in the film ver- Abel and Kronecker are as good without mentioning Eliot Fintushel, sion of Jurassic Park. On the other today as they were four hundred whose bizarre stories are regularly hand, mathematicians also have a repu- years ago, and it will always be so. published in monthly science fiction tation—completely undeserved in my New roads arise, but the old ones magazines to wide acclaim despite their experience—of being cold and heartless. lead on. They do not become over- frequent (and bizarre) references to Many mathematical villians in books grown. There...there you have eter- obscure mathematical topics such as the and stories display this supposedly evil nity. Only mathematics does not Banach-Tarski paradox. In an e-mail side of mathematics. It must be admit- fear it. Up there, I understood how message to me, Fintushel summarized ted, however, that mathematics does not final it is. And strong. ■

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