# Mathematics in Science Fiction: Mathematics as Science Fiction.

Both classical and current fiction include mathematics as a literary device. As commonly used in science fiction, topology includes the study of spatial dimension, including the arcane Fourth Dimension, famously employed by H. G. Wells (and many writers since) for time travel In the following essay, David Fowler suggests that mathematics itself is a branch of science fiction.

Imagine visiting a large commercial bookstore in the company of an extraterrestrial being--one suitably disguised and equipped for perception and discourse through our narrow spectral bands. Imagine further that you pause before a section marked "Science Fiction" or, more likely, "Science Fiction/Fantasy." Half the most recent titles are from a popular vampire series, and the leading seller, described as "hard science," deals with alien parasites inserting themselves into human brains. Together you wander through other sections of the store, and as you leave, your companion informs you, "On my planet, the section you call Science Fiction would contain the books you call Mathematics."

Your ET then vanishes, leaving you wondering if there was a glitch in the translation software. Did your otherworldly companion fail to understand that we on Earth do include mathematical themes in our fiction, including science fiction? That error can easily be corrected. Is this visitor telling you that our mathematical knowledge is itself a large work of fiction? This is certainly a much harder question to address. Let's begin with the easier question.

Mathematics in Fiction

Mathematical references in literature are abundant. A simple use of mathematics is to show that a character has a high degree of intelligence, as when Stieg Larsson's heroine, Lisbeth Salander, in the midst of ongoing danger, solves Fermat's "Last Theorem" using only the logical tools that would have been available to Fermat in 1637. At a somewhat higher level of use, a mathematical example may be used as a plot development device. Dan Brown uses the Fibonacci numbers this way in The Da Vinci Code. Mathematics appears in more subtle fashion as the maze-solving scheme used by Brother William in Umberto Eco's The Name of the Rose. The scheme is a "depth-first search algorithm" in computer-science jargon, with an underlying mathematical justification.

Readers of the above novels will not be challenged mathematically by the above examples, whose superficial explanation is sufficient to keep a story moving. Turning to a classical example--The Brothers Karamazov--one encounters a passage that certainly demands a pause for reflection: Ivan's comparison of his theological doubt to the existence of non-Euclidean geometry. Ivan Karamazov sees his inability to grasp non-Euclidean geometry as evidence that he can't understand God. Although Euclid may no longer be the ultimate source of geometric truth, Ivan still accepts the work of the modern non-Euclidean thinkers as a standard of truth. Albert Einstein's supposed claim to have "learned more from Dostoevsky than from any scientific thinker" may be apocryphal; but in any case, Dostoevsky's reflections on time and space can be viewed as compatible, in qualitative fashion, with the framework of special relativity.

In the above examples, mathematics is used as a Gold Standard for reasoning. Characters In fiction are brilliant thinkers if they know mathematics, even if they are evil characters such as Professor Moriarty, created by Conan Doyle to bedevil Sherlock Holmes: "He is a man of good birth and excellent education, endowed by nature with a phenomenal mathematical faculty. At the age of twenty-one he wrote a treatise upon the binomial theorem, which has had a European vogue."

Rarely are mathematicians described as foolish, although Swift's Island of Laputa persists as an image of mathematicians as fools. As Alfred Whitehead pointed out, "Swift describes the mathematicians of that country as silly and useless dreamers.... On the other hand, the mathematicians of Laputa ... ruled the country and maintained their ascendancy over their subjects." Whitehead also remarked that Newton had just published his Principia and suggested that "Swift might just as well have laughed at an earthquake."

Finally, mathematicians may be inserted into fictional narratives to enhance historical plausibility. Neal Stephenson includes Alan Turing in his novel Cryptonomicon. David Leavit's recent novel The Indian Clerk, based on episodes in the life of the mathematician Srinivasa Ramanujan, includes Bertrand Russell, George Hardy, and a panoply of Cambridge University scholars.

"Topological" Science Fiction

Given that mathematical allusions may be found in works ranging from popular fiction to literary classics, consider now the fuzzy category of "science fiction." Within this domain, mathematics sometimes supplies the "science." Again, mathematics is used as a high standard for reasoning, but mathematics is also used as a device for speculation: "a fantastic event of development considered rationally," in the words of James Gunn.

An early example is the character Professor Surd, in Edward Page Mitchell's 1873 story "The Tachypomp." Professor Surd sets the following problem to a would-be suitor of Surd's daughter, Abscissa: "Discover the principle of infinite speed. I mean the law of motion which shall accomplish an infinitely great distance in an infinitely short time." The suitor seeks out his mathematics tutor for assistance, who explains a principle of relative motion in terms of a person walking through a railroad train from the back car toward the engine. The tutor proposes a machine consisting of one train engine pulling a large flat car on which rails are laid for a second engine, effectively doubling the rate of motion. He then describes a bootstrapping system of engines driven by electromagnets--a linear accelerator that could in theory take a passenger to any arbitrary speed.

One hundred seventeen years later, the mathematician Ian Malcolm, in Michael Crichton's Jurassic Park, provides an ongoing narrative in terms of chaos theory, a cutting-edge development at that time. Crichton's evident enthusiasm for mathematics is reminiscent of an earlier writer--Robert Heinlein. In his 1952 novel, The Rolling Stones, Heinlein describes mathematics with the same enthusiasm one might apply to a Chesley Bonestell Mars landscape: "They moved on [from analytical geometry] to more rarefied heights ... the complex logics of matrix algebra, frozen in beautiful arrays ... the tensor calculus that unlocks the atom ... the wild and wonderful field equations that make Man king of the universe ... the crashing, mind-splitting intuition of Forsyte's Solution that had opened the 21st century and sent mankind another step toward the stars."

A favorite mathematical subject of science-fiction writers has produced enough stories to define a subgenre: topological fiction, where "topology" refers to the study of continuous deformation of shapes in space. Robert Heinlein's "And He Built a Crooked House," published in 1940, and Martin Gardner's "The No-Sided Professor," published in 1946, are among the first in science fiction to introduce readers to the Moebius band, the Klein bottle, and the hypercube (tesseract). A 1950 example is "A Subway Named Moebius" by A. J. Deutsch, in which the tunnels of a subway system become a network of such complexity that trains vanish into a higher dimension. Mark Clifton's 1952 story "Star Bright" describes a child who travels through time and space using her superior intelligence and understanding of space-time structures. These stories are a reasonably good qualitative introduction to these surfaces, if not to the rigorous study of topology as a mathematical discipline.

Dimension and Time in Science Fiction

Dimension theory, as a special branch of topology, generates its own special niche in the science-fiction library, although in most cases no real mathematical theory is applied or even referenced. An exception is Abbot's Flatland, written in 1884, in which a three-dimensional sphere visits the inhabitants of a two-dimensional world. This narrative continues to have a compelling grip on the mathematical imagination. Rudy Rucker, mathematician and science-fiction writer, has his own higher-dimensional version of the story, called Spaceland. Ian Stewart, prolific writer of expository mathematical books, has a version called Flatterland.

The seminal work on dimension, of course, is H. G. Wells's Time Machine. As the inventor of the machine explains to his guests:
``` Space, as our mathematicians have it, is
spoken of as having three dimensions,
which one may call Length, Breadth, and
Thickness, and is always definable by reference
to three planes, each at right angles to
the others. But some philosophical people
have been asking why THREE dimensions
particularly--why not another direction at
right angles to the other three?--and have
even tried to construct a Four-Dimension
geometry. [...] "Scientific people ... know
very well that Time is only a kind of Space."
```

This sentence, remarkable in that it was published in 1895, before Einstein's 1905 paper on special relativity and Minkowski's 1908 recasting of relativity into a four-dimensional framework, seems unusually prescient--at least as a qualitative, if not a mathematical, description of relativity. Certainly the power of its imaginative insight makes Wells a candidate for the title "Father of Science Fiction." An enormous volume of writing about time travel has emerged since then, meticulously documented by Paul Nahin in his book Time Machines: Time Travel in Physics, Metaphysics, and Science Fiction.

On the other hand, the description of "dimension" as given by Wells is little more than a combination of classical analytical (Cartesian) geometry and introductory mechanical drawing. The Time Traveler uses a block of wood to illustrate dimension as consisting of length, breadth, and height. This is not to diminish Wells's insight: a mathematical formalization of dimension theory was first published by Karl Menger in 1928, and Hurewicz and Wallman's Dimension Theory was published in 1941.

Jorge Luis Borges constructed a more refined example connecting spatial dimension and time. "El Jardin de senderos que se bifurcan" was first published in 1941. His first work to be translated into English, the story appeared in Ellery Queen's Mystery Magazine in August 1948 as "The Garden of Forking Paths." The "garden" is really two constructs: a spatial maze but also a network of alternative paths in time. In his explanation, Borges seems to anticipate fractal geometry, just as Wells seems to anticipate relativity and Mitchell a particle accelerator. These works are not mathematically based, and they don't describe experimental equipment for actual tests, but they can be seen as "thought experiments" for imagining alternatives to conventional thinking. To explore this idea further, we must take a temporary break from mathematics and science fiction and consider examples of perceptual space.

A Thought Experiment

In ordinary conversation, correspondences between space and time occur whenever we make a statement such as "I live about ten minutes from here" or use some conceptual metaphoric phrase like "I don't have far to go to finish this report." The following thought experiment suggests a way one might conceptualize distance in acoustic terms.

Imagine yourself standing in what seems at first to be a dense, dry fog, not uncomfortable but preventing you from seeing even a close distance from your face. You mutter a startled comment, and you can suddenly see to the limit of your outstretched hands. You shout, and you can see that you're standing in the center of a field. The louder you shout, the further you can see, but of course there is a limit to how loudly you can cry out. As you move about, the visual field moves with you, showing you as great a space as your last utterance generated.

Eventually you find yourself back in the ordinary world, but you no longer can express distances in the old, familiar metric terms. You tell your friends that a light rainfall amounted to about a "whisper of precipitation," and that you've parked your car "about a loud shout from your office." Your physical space is the one in which you've always lived, but your perceptual space is now defined by a "sonic metric" rather than a geometric one.

Alternative Perceptual Spaces

In The Dream Seekers: Native American Visionary Traditions of the Great Plains, Lee Irwin describes the perceptual spaces of Native Americans on the Great Plains:
``` It is important to realize that the character
of both space and time in Native American
religious topology is relative and elastic. A
direction is not something to be measured
in a strict Cartesian sense as a rigidly fixed,
three-dimensional spatial grid, and time
is not a rigidly conceived, unidirectional
linear flow from past to future.
```

[ILLUSTRATION OMITTED]

Irwin uses the term "topology" to describe perceptual space in both dreaming and living worlds and suggests that, to the Plains people, these spaces share a unity:
``` The unified topology of the dreaming and
lived worlds can only be grasped if the
"checkered tablecloth of Cartesian space-time"
is pulled out from under the externally
observed world, leaving the multidimensional,
unbound, fluid contours of
a noncausal, visionary space-time to
expand or contract according to individual
experience.
```

One might wonder why, lacking "the checkered tablecloth of Cartesian space-time," the Plains Indians' decorative patterns were so carefully geometric. The frequent appearance of geometric designs, in particular triangular patterns, in Native art, can be explained as an artist's concentration on symmetry rather than a Euclidean investigation into the properties of triangles and the intersections of lines in a plane.

Perceptions of symmetric patterns are evidently built into the neural structures of humans and perhaps many other species. The survival value of quick symmetry detection is apparent. If the symmetry detection equipment is present in the brain, it could be used to generate new images as well as to perceive images already existing in the world.

The symmetry-processing mechanisms in the brain can generate new mathematics as well as new art. The mathematician Felix Klein--the same Klein for whom the topological "bottle" is named--proposed in 1892 that different geometries could be classified by families of symmetry transformations. Studies of symmetry, art, and perception seem to have an underlying mathematical as well as a neurological association.

The question addressed in the final section of this essay is this: How much of mathematics is generated by the human brain, rather than existing in the universe and discovered by the brain? As George Lakoff and Rafael Nunez put the question in their book on embodied cognition: Where Does Mathematics Come From?

Mathematics as Fiction

A common assumption in science fiction, and among at least some scientists and mathematicians, is that mathematics would be commonly understood by intelligent beings from other cosmic locations. Notation would certainly differ, but prime numbers would still be different from composite numbers, ellipses would describe the orbits of planets, equilateral triangles would be similar, etc. Most mathematicians are comfortable with this Platonic point of view. They believe they are discovering mathematics rather than inventing it. Mathematicians who are inclined toward philosophical discussion--and many are not--may claim that the objects studied by mathematicians must be real, since the universe is real and mathematics seems indispensable to a scientific explanation of the physical world. Recall how in the bookstore we communicated with the alien through a narrow spectral band. Almost all our knowledge of the universe comes to us through the mathematical analysis of data collected by machines built to extend our limits of perception. In his essay "Fictionalism, Theft, and the Story of Mathematics," Mark Balaguer proposes, among other arguments, that rather than describe a mathematical statement as "true," it could be described as "true within the story of mathematics."

[ILLUSTRATION OMITTED]

Mathematical science fiction then could be described as the intersection of two stories, one of them being a "fantastic event" and the other being some part of the "story of mathematics." One might then conceive of many "stories of mathematics" existing throughout different civilizations, on Earth and on other planets. These stories could describe possible connections between perceptual and physical spaces, resulting in a "geometrical space," or something like geometry, as it exists in any given civilization.

We can even get a sense of what a different story could look like by examining a relatively new kind of mathematical object--the cellular automaton. The physicist Stephen Wolfram, in his book A New Kind of Science, has demonstrated the emergence of complex patterns from extremely simple rules. His cellular automata can, for example, generate prime numbers--what we on earth call "prime numbers."

Although the patterns may not look like arithmetic as we know it, one can imagine an alien glancing at the diagram and remarking, through its translating equipment, "Oh, yes, I've read that one before."

WORKS CITED

Balaguer, Mark. "Fictionalism, Theft, and the Story of Mathematics." Philosophica Mathematica 3:17 (2009): 131-62.

Boi, Ludano. "Questions Regarding Husserlian Geometry and Phenomenology: A Study of the Concept of Manifold and Spatial Perception." Husserl Studies 20:3 (2004).

Gunn, James E., and Matthew Candelaria. Speculations on Speculation: Theories of Science Fiction. Lanham, Md.: Scarecrow, 2005.

Hurewicz, Witold, and Henry Wallman. Dimension Theory. Princeton, N.J.: Princeton University Press, 1941.

Irwin, Lee. The Dream Seekers: Native American Visionary Traditions of the Great Plains. Norman: University of Oklahoma Press, 1994.

Knapp, Liza. "The Fourth Dimension of the Non-Euclidean Mind: Time in Brothers Karamazov, or Why Ivan Karamazov's Devil Does not Carry a Watch." Dostoevsky Studies 8 (1987).

Lakoff, George, and Rafael Nunez. Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. New York: Basic Books, 2001.

Nahin, Paul. Time Machines: Time Travel in Physics, Metaphysics, and Science Fiction. New York: Springer, 2001.

Wolfram, Stephen. A New Kind of Science. Champaign, Ill.: Wolfram Media, 2002.