By Steven Gimbel, Ph.D., Gettysburg College
Ever since Isaac Newton’s discoveries, mathematics had been viewed as an absolute bedrock on which we could construct a completely firm structure of understanding. New findings in the early 20th century, though, shattered this bedrock. Just as literature reveals much about collective cultural anxieties, so the depiction of math in literature, particularly Alice in Wonderland and Flatland, illuminates the resulting chaos as our fundamental understanding of reality was called into question.
Mathematical Crisis Reflected in Literature
During the Enlightenment, the world was viewed as an orderly place in which everything operated by precise mathematical principles. Beginning with the Romantic backlash, though, and unfolding into the early 20th century, rationalism in mathematics, as well as the concept of absolute truth, was called into question.
This change came about through startling discoveries in the field of mathematics, revealing that what we thought were fundamental principles in both geometry and arithmetic were not always true in every situation.
This is a transcript from the video series Redefining Reality: The Intellectual Implications of Modern Science. Watch it now, on Wondrium.
Interestingly, this collapse of certainty—and the resulting wide-scale wreckage to the foundations of our reason—was reflected in a pair of notable and related works of fiction coming out of Britain in the late 1800s: Lewis Carroll’s Alice’s Adventures in Wonderland, and Edwin Abbott Abbott’s Flatland: A Romance of Many Dimensions.
Alice in Wonderland—as the tale has come to be known—and its sequel, Through the Looking-Glass, were written by Lewis Carroll, the pen name of Charles Lutwidge Dodgson, a mathematical logician at Oxford.
Because of the ascent of non-Euclidean geometry, and the attempts to find a firm foundation for arithmetic in set theory, the world of mathematics had turned its attention largely to logic in hopes that an analysis of the nature of mathematical reasoning would yield the needed justification to keep mathematics as the hardcore basis of all that was certain.
But mathematicians are a strange lot. While they understood the gravity of the circumstances, they also found themselves drawn to the paradoxes that could be created when these foundations were examined creatively.
The heart of traditional logic is the law of the excluded middle—the claim that either a sentence or its negation, but not both, must be true. Either I have a brother or I don’t; I can’t both have a brother and not have a brother.
If we know that one claim is true, we know the other is false. And if we know one is false, then we know the other is true. A paradox is a sentence or set of sentences that contradicts itself. That is, it must be true, but then its truth implies its falsity.
Since the law of the excluded middle holds that a sentence can’t be both true and false, we have an affront to the basis of logic itself. Logicians like Dodgson were examining purported paradoxes generated by the logical system itself. If authentic, such paradoxes would undermine the underpinnings of our most rigorous form of thought.
Learn more about math concepts explored by literary figures like Kurt Vonnegut
The Collapse of Logic in Alice in Wonderland
Testing paradoxes were not limited to Dodgson’s professional published work; it’s also what he was playing with in his famous work for children. Think of the opening scene in which Alice spies the White Rabbit: he takes a pocket watch out of his waistcoat, declares that he’ll be late, and then he dashes down a hole into which Alice follows.
Carroll famously wrote that “Either the well was very deep, or she fell very slowly, for she had plenty of time as she went down to look about her and wonder what was going to happen next.”
Now, remember that Dodgson was British and the most celebrated influential figure in Britain was Isaac Newton. Newton’s laws of motion explained Galileo’s finding that all objects close to the surface of the Earth fall at exactly the same rate.
If we’re taking seriously the possibility that Alice is falling slowly, then we’re taking seriously the possibility that the immutable laws we take as part and parcel of the working of the universe no longer apply.
We have entered a realm where the Enlightenment presupposition of a well-behaved universe whose rules are accessible to our rational faculties can be reasonably denied. Reason implies nonsense.
Reason is not ultimately self-justifying, but ultimately self-defeating. Wonderland represents the death of the rationalist project.
Think of Alice’s encounter with Humpty Dumpty in Through the Looking-Glass. The two are discussing birthdays and birthday presents when Humpty Dumpty exclaims, “There’s glory for you.” Alice replies, “I don’t know what you mean by glory.” To quote the passage further:
Humpty Dumpty smiled contemptuously. “Of course you don’t—till I tell you. I meant ‘there’s a nice knock-down argument for you!’”
“But ‘glory’ doesn’t mean ‘a nice knock-down argument’,” Alice objected.
“When I use a word,” Humpty Dumpty said, in rather a scornful tone, “it means just what I choose it to mean—neither more nor less.”
“The question is,” said Alice, “whether you can make words mean so many different things.”
“The question is,” said Humpty Dumpty, “which is to be master—that’s all.”
In Greek, a language that Dodgson spoke, the word for “word” is logos, which also means, in some contexts, logic. When he has Humpty Dumpty redefine the word “glory,” it’s not a random sense but rather a nice knock-down argument, that is, the goal of logic.
When Alice protests, Humpty Dumpty replies that the real question is who is to be master: humans or words? That is, which ought to be the measure of reality: our experience, our freedom, our lives, or logic? Should we be subservient to our reasoning as the Enlightenment-influenced rationalists would have?
Or should we take command over logic, words, and logos? If we follow logic, do we disappear down a rabbit-hole—something that seemed possible given the paradoxes that mathematical logic was generating?
Learn more about the universe as changing and unstable
Re-Examining Rationality in Flatland
A similar challenge is found in Abbott’s Flatland, which takes place on a two-dimensional plane. It’s a flat world populated by shapes, the narrator being a lowly square. In Flatland, the more sides you have, the higher your social position.
The square is visited by a sphere, a three-dimensional figure that appears to the square at first as a point, then a circle of increasing diameter, then a circle of decreasing diameter, and then a point—finally, a disembodied voice. The sphere tries without any initial success to convince the square of the existence of the third dimension until he finally flings the square out of his planar world into the space above it.
Upon returning to Flatland, the square becomes evangelical about convincing his fellow flatlanders about the existence of this third dimension that is upward not northward—a dimension they haven’t seen.
He’s arrested and charged with heresy by the high priest, and at his trial, he is asked to provide any evidence for the existence of this third dimension of which he so passionately speaks.
The square’s argument is mathematical. If we can take a point and move it, we get a line. If we take a line and move it parallel to itself, we get a plane. If we take that plane and move it parallel to itself, we get space.
The priest asks for physical, rather than mathematical, reasoning. The square can provide none, so the priest offers his argument: He asserts that there is no reason to think this mathematical talk is anything but trickery, with no relation to anything real.
The argument is compelling; all the while the reader knows it’s wrong. But it’s reflective of the strange results coming out of mathematics at the time. They threatened to undermine our comfortable certain basis for rationality.
But should they be accepted? Has reason led to nonsense, or is there a foundation for rational thought to be found in rational thought?
Learn more about the paradoxical subject of quantum mechanics
For more than a thousand years, we accepted Euclid’s axioms and postulates as true because they were self-evident, seemingly self-justifying. But at the dawn of the 20th century, mathematics—our most secure and definitive science—was in turmoil.
We were forced to re-examine the basis of what we thought reality would be like. Lewis Carroll’s Wonderland was about to be discovered here in our world, when the science of the 20th century forced us to reconsider reality itself.
Common Questions About Math in Literature
Lewis Carroll, whose real name was Charles Dodgson, was indeed a mathematician. At the time, groundbreaking new mathematical concepts were coming out such as imaginary numbers. Dodgson, whose views on math were rather traditional, considered these new concepts to be absurd, and the world of Alice’s Adventures in Wonderland mirrors this absurdity.
The scientific name for Alice in Wonderland syndrome is dysmetropsia, which is a neuropsychological condition in which one’s perception is distorted. In Lewis Carroll’s novel, Alice experiences a distorted sense of her body and physical surroundings, which reflects the anxiety many people were experiencing at that time as new mathematical concepts were introduced which changed how they viewed the world.
In the book Flatland, Flatland is a world occupied by flat, two-dimensional objects (circles, squares, etc.). Space, on the other hand, contains three-dimensional shapes such as spheres and cubes. The objects in Flatland can’t comprehend that this three-dimensional world exists because all they know is their own world.
Flatland and Lineland are similar in that each land consists of lines and points. Unlike the world of Spaceland, Lineland is familiar to Flatland because they share common elements.