by Madeline Barnicle
What happens when science fiction investigates the possibilities of “alternate mathematics”? Find out in the essay below, where Madeline Barnicle discusses the role of mathematics in SF across a few notable examples, from Cixin Liu’s “The Circle” to “Division by Zero” by Ted Chiang. You can also read read Barnicle’s short story “Barreira do Inferno” in our [January/February issue, on sale now!]
The genre of alternate history posits a “turning point”—what if this battle had a different outcome? what if this historical figure hadn’t died when they did?—and then imagines the repercussions days or years down the line. A “what if” scenario isn’t, in itself, a story—that still requires characters, plot, themes, and all the other elements—but it can provide the framework for a setting like our own world, but different. The points of contrast, whether serious or humorous, are an important aspect of what makes these stories compelling to readers in this world.
Some characterizations of science fiction describe it as a literature of “what if,” or perhaps “alternate science.” Stories in this vein ask questions like “what if life evolved on a neutron star where the force of gravity is billions of times stronger than on Earth?” (Dragon’s Egg, by Robert L. Forward.) From this speculation, Forward, an astrophysicist, imagined further developments. What would these beings’ sense of time be like? What about direction? What would that mean for an intelligent society? In this framework, SF is built to ask, “what if the observed laws of physics, biology, chemistry, and so on, were slightly different than we know them today; what would happen next?”
Yet, in practice, many works of SF don’t perfectly fit this model. Tropes such as “aliens come to visit Earth” or “humans use faster-than-light technology to travel the galaxy” have become sufficiently familiar that we readily suspend disbelief, without stopping to ask “what kind of atmosphere or nutrients or gravitational force are these aliens used to from their own planet, and how have they adapted to survive on Earth?” A narrative that gets too caught up in explaining these details can struggle to incorporate all the other elements of a story, particularly in shorter forms where word count is deliberately constrained.
Mathematics is sometimes described as the most pure of the sciences, and at other times, so abstract as to not be a science at all (in the sense of relying on experimental data). In that framework, is it even possible to write a good science fiction story about “what if mathematics were otherwise”? Should we try?
“Division by Zero,” by Ted Chiang, posits one answer to the question. The story is told in three strands; one part summarizes the axiomatic approach to mathematics, the second is from the point of view of Renee, who attempts suicide after discovering a fundamental inconsistency in the laws of math, and the third focuses on Renee’s husband Carl, who also has a history of suicidality. In an elaborate structural metaphor, Chiang arranges these interweaving parts like the lines of a false proof.
As literature, “Division by Zero” may succeed in its portrayal of a failing marriage. Perhaps it may even be illustrative for non-mathematicians as a vivid evocation of just how shattering such an inconsistency result would be for people who have devoted their lives to the subject, even if it didn’t call any practical, day-to-day arithmetic into question. But does it succeed as a story about math? This may be a professional bias, but I would argue: no. The fundamental premise of the story is the inconsistency Renee has discovered, but it’s impossible to depict because it contradicts the axioms of mathematics. I can imagine an alien life form with five legs and eight eyes, or a beverage that causes the people who drink it to grow wings, or a visitor from a thousand years in the future showing up at my office, even if all of these things would be highly improbable or perhaps better classified as fantasy than science fiction. But for a fictional proof like Renee’s, it’s hard for me to suspend my disbelief about “there was a hole in the laws of mathematics.”
Mathematics is sometimes described as the most pure of the sciences . . . In that framework, is it even possible to write a good science fiction story about “what if mathematics were otherwise”? Should we try?
A different extreme is illustrated by Cixin Liu’s “The Circle” (this is actually one of the chapters from The Three-Body Problem, rewritten as a standalone short story). The story consists of a fanciful depiction of how a digital computer might have been created twenty-two centuries ago. After all, the fundamental unit is not the electronic transistor itself, but the algorithms that manipulate bits, and this could be done (if orders of magnitude too slowly to be practical) using analog tools.
“The Circle,” then, obeys the known laws of mathematics, and could more properly be defined as an alternate history (or a “secret history,” as the technology is lost in the end and the pace of technological development gets back on track with the world we know). However, I suspect it may appeal to other mathematicians because of the playful illustration of how bit algorithms work, stripped of the physical context where we normally see them implemented. This style of SF, like Dragon’s Egg, often emphasizes the “what if” narrative at the expense of complex characterization.
Does any of this matter? Only to the extent that, as a mathematician, I’d like to be able to point to portrayals that illustrate some of the pleasures of doing mathematics. (As opposed to frustrating academic politics or geniuses who cause problems for everyone unfortunate enough to be in their social life, of which there are plenty in the realms of literary fiction.) There are fictional depictions of other kinds of art that give some sense of the joy their practicioners experience, so why not math?
In Douglas Hofstadter’s Gödel, Escher, Bach, one of the fictional “dialogues” interspersed with the nonfictional expository chapters describes the Subjunc-TV, in which characters watch an instant replay of what would have happened in a football game if one player had not stepped out of bounds. Hofstadter writes: “it was inspired by a completely ordinary, casual remark made to me by a person sitting next to me at a football game. For some reason it struck me and I wondered what made it so natural to slip that particular thing, but not, say, the number of the down, or the present score.” The TV even has a setting for what the play would have looked like if thirteen were not a prime number! Although we readers sadly don’t get to see that play, Gödel, Escher, Bach also adopts the strategy of mixing whimsical narratives, and traditionally fantastic tropes like talking crabs and tortoises, with serious mathematical exposition.
Another approach eschews fiction and concentrates on the literary features of mathematical communication. In the introduction to 99 Variations on a Proof, Philip Ording warns, “The oft-repeated “beauty” and “elegance” may be important components of mathematical taste, but they fail to convey its range or subtlety or how it relates to literary and aesthetic experiences beyond mathematics.” He goes on to depict a wide variety of approaches to solving a cubic equation—some simple, some ornate, some comic—emphasizing the diversity of approaches to the same underlying problem. While this book presents many delights for a mathematically-inclined reader, I’m not sure how much appeal it would hold to non-mathematicians. It may be that the best way to appreciate math is to find puzzles that appeal to you and engage with them directly, rather than watching from the outside in.
Of course, no story can be all things to all people; individual works of fiction will differ widely in how the author chooses to balance plot and character, depicting worlds unlike ours or reflecting on the present, and many other variables. But I hope that, rather than having to construct new axioms from the bottom up, authors and readers will continue to find novel outlets to depict the many kinds of beauty that we see in the mathematical world.
The Astounding Analog Companion 
Try to locate a collection called THE MATHEMATICAL MAGPIE, edited by Clifton Fadiman….
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