Geometric Parable from 1884 a must-read for Digital Natives
Trapped in Our Own Dimension
Humans are hardwired to resist contradictory evidence. We cling to established beliefs even when facts suggest otherwise. Psychologists call it confirmation bias.
Whether we're dismissing scary climate data or rejecting medical consensus because it cramps our lifestyle, our minds instinctively filter reality for maximum cognitive comfort.
Only 23% of Americans trust scientists, down from 39% in 2020.
Digital life reinforces our natural biases through algorithmic curation—it’s come to be called echo chambers.
Large language models (LLMs) inherit cognitive biases from training data, similar to social biases. Studies show LLMs are influenced by biases like the threshold priming effect, which affects relevance judgments.
This very contemporary perceptual crisis was eerily anticipated in 1884 by Edwin A. Abbott in his novella Flatland.
Abbott's protagonist—a square living in a two-dimensional world—cannot conceive of a third dimension until directly experiencing it. Even after his revelation, his fellow Flatlanders reject this "impossible" knowledge, with authorities imprisoning him for spreading dangerous ideas that threaten their ordered society.
Just as Flatlanders physically cannot perceive the third dimension, we struggle to see beyond our cognitive boundaries, not even realising they exist.
Our mental shortcuts, once essential for survival, have become our own worst handicaps.
Are we modern Flatlanders, convinced we see the whole picture while missing entire dimensions of reality?
You can read Flatland in the public domain here.
Abbott: A Geometric Revolutionary
"I know not how to convince you. But I tell you that I have seen it." — Abbott's protagonist after witnessing the third dimension
Edwin A. Abbott (1838-1926) lived in a society as rigidly stratified as the geometric shapes in his masterpiece. Victorian England was obsessed with social hierarchy—where birth determined destiny and stepping outside one's prescribed role invited ridicule or worse.
As headmaster of the City of London School at just 26, Abbott witnessed this system from within yet maintained a quietly progressive outlook, advocating for women's education and social reforms.
Abbott's intellectual world was equally bounded. Victorian science was dominated by Newtonian physics and Euclidean geometry—both presenting reality as fixed, knowable, and three-dimensional.
Yet in the late 19th century, mathematicians like Gauss and Riemann proposed space might have more than three dimensions. The radical idea was filtering into popular discussion, blending with spiritualism and theological speculation about unseen realms.
Flatland: A Romance of Many Dimensions was published in 1884 under the pseudonym "A Square." The novella depicts a society of two-dimensional geometric shapes whose perception is literally limited to their plane of existence. When the protagonist Square encounters a Sphere from the third dimension, his understanding of reality shatters—but his attempts to share this revelation lead to persecution.
On its surface, Flatland was a mathematical curiosity that cleverly illustrated how higher dimensions might work.
It was also a multilayered critique of Victorian society. The rigid hierarchy of Flatland—where circles rule over polygons, who in turn lord over triangles, while women exist as mere lines—satirized the absurdity of Victorian class and gender divisions.
As Einstein's theories later popularized the concept of spacetime as a fourth dimension, Abbott's work gained new appreciation. Today, it's recognized as a remarkably prescient text that anticipated developments in physics and our modern understanding of how perception shapes reality.
Square prefigured our challenge of communicating across isolated information ecosystems. His depiction of a society that violently rejects inconvenient truths mirrors our polarized discourse where facts themselves have become tribal tokens.
While Victorians saw Flatland as an ingenious mathematical fantasy, Abbott had crafted a parable about the limitations of perspective that would only grow more relevant with each passing decade.
The Dangers of Intellectual Certainty
A Square: "In One Dimension, did not a moving Point produce a Line with two terminal points? In Two Dimensions, did not a moving Line produce a Square with four terminal points? In Three Dimensions, did not a moving Square produce...a Cube with eight terminal points? And in Four Dimensions shall not a moving Cube...result in a still more divine Organization with sixteen terminal points?"
The Sphere: "Nonsense. Nonsense, I tell you. No such thing!"
Even the "enlightened" resist paradigm shifts.
The Sphere, who patiently taught the Square about the third dimension, immediately dismisses the possibility of a fourth. It’s how scientists and experts often reject revolutionary ideas outside their framework—from continental drift to quantum physics.
The Sphere demonstrates conceptual conservatism: the tendency to preserve existing knowledge even in the face of new evidence.
From consciousness studies to climate science, established authorities dismiss ideas that challenge their dimensional thinking.
The Cost of New Knowledge
The Council of Flatland: "Resolution of the Council...enjoining the arrest, imprisonment, or execution of anyone who should pervert the minds of the people by delusions, and by professing to have received revelations from another World."
A chilling declaration shows the threat to power when fundamental assumptions are challenged. In Abbott's time, challenging religious or scientific orthodoxy could indeed lead to ostracism or worse. Today we see the "professional execution" of scientists and thinkers who challenge dominant paradigms.
Whistleblowers face retaliation, climate scientists receive death threats, and researchers who question established theories find funding and publication opportunities cut off. The Council's reaction demonstrates what psychologists call worldview defense—the aggressive rejection of information that threatens one's core beliefs about how reality operates.
The Paradox of Perception
A Square: "You, who are blessed with shade as well as light, you, who are gifted with two eyes, endowed with a knowledge of perspective, and charmed with the enjoyment of various colors, you, who can actually see...the inside of all things...—how I could make clear to you the extreme difficulty which we in Flatland experience in recognizing one another's configuration?"
The Square struggles to explain to the Sphere how difficult recognition is without depth perception.
Studies of implicit bias demonstrate that we carry unconscious prejudices we cannot directly observe. Similarly, research on the Dunning-Kruger effect shows that those with the least knowledge in a domain are least able to recognize their ignorance.
Abbott captured this paradox: the truly limited perceiver cannot perceive their own limitations.
Ripples through Science & Culture
When Einstein's theories of relativity introduced time as a fourth dimension in the early 20th century, Flatland suddenly appeared prophetic.
In 1920, physicist William Garnett wrote to the journal Nature specifically praising Abbott's foresight about the importance of additional dimensions in explaining physical phenomena.
Beyond physics, Abbott's work shaped how we teach complex spatial concepts. His method of using lower-dimensional analogies to help readers grasp higher dimensions became standard in mathematics education.
Carl Sagan famously used Abbott's approach in his Cosmos television series to explain higher dimensions, introducing millions to the concept of "dimensional thinking" without many viewers realizing the Victorian origins of this pedagogical technique.
Flatland also quietly influenced philosophy of mind and epistemology. Thomas Nagel's influential 1974 paper "What is it like to be a bat?" wrestles with the same fundamental problem as Abbott's Square: the impossibility of truly comprehending experiences outside our cognitive framework.
In literature, Abbott created a micro-genre of "dimensional fiction." Sequels and homages appeared throughout the 20th century, including Dionys Burger's Sphereland (1965) and A.K. Dewdney's The Planiverse (1984), each extending Abbott's fundamental insights into new scientific and social contexts.
Perhaps most significantly, Flatland introduced a powerful metaphor for intellectual humility that continues to resonate in cognitive science.
The novella reminds us that just as a square cannot directly perceive a cube's full reality, we may be structurally incapable of grasping certain truths—a humbling lesson for an age when complex systems challenge the limits of intuitive understanding.
Breakout of the Perceptual Prison
Abbott's Square eventually breaks free from his two-dimensional mindset. Could it help us escape our own cognitive confines?
The key insight from Abbott's work is that awareness of our limitations is the first step toward transcending them. The Square's journey begins with discomfort—the jarring experience of encountering something that doesn't fit his existing framework. This "cognitive dissonance" signals a clash between perception and reality, creating an opportunity for growth rather than a threat to be suppressed.
Abbott suggests several practical approaches to expanding our mental dimensions.
First, he emphasizes the value of analogical thinking—using what we know to grasp what we don't. The Sphere helps the Square understand the third dimension by drawing parallels to lower dimensions: just as a point moving creates a line, and a line moving creates a square, a square moving "upward, not northward" creates a cube. This method of progressive understanding—building from the familiar to the unfamiliar—offers a model to approach seemingly incomprehensible new ideas.
Second, Flatland demonstrates the importance of perspective-taking. The Square gains insight not just through abstract reasoning but through direct experience—being physically lifted out of his plane to see his world from above.
Intellectual understanding alone is insufficient; we need experiential knowledge that challenges our default viewpoint. In modern terms, this means deliberately seeking perspectives from different cultures, disciplines, or life experiences to triangulate a more complete picture of reality.
Finally, Abbott emphasizes the courage required to maintain unpopular truths. The Square continues to advocate for the third dimension despite ridicule and persecution, showing that expanding our mental frameworks requires moral courage as much as intellectual openness.
By recognizing our perceptual limitations, using analogical reasoning to grasp unfamiliar concepts, actively seeking diverse perspectives, and maintaining the courage to face uncomfortable truths, we can begin to escape our mental Flatlands and glimpse higher dimensions of understanding.
Interested in Flatland? Check out…
"What Is the Fourth Dimension?" by Charles Howard Hinton (1880s) – A contemporary of Abbott, Hinton wrote essays exploring the concept of higher dimensions that likely influenced Flatland. Hinton coined the term "tesseract" and, like Abbott, used the idea of lower-dimensional beings trying to comprehend higher dimensions as a teaching tool.
The Country of the Blind by H.G. Wells (1904) – This short story depicts a sighted man in a valley of the blind who cannot convince its inhabitants that sight exists. Wells' exploration of the clash between different perceptual realities parallels Abbott's themes about the difficulty of communicating across experiential divides.
We by Yevgeny Zamyatin (1924) – Though written decades later, this dystopian novel continues the tradition of using geometric metaphors (the characters have numbers instead of names) to critique social conformity and limited perception. Zamyatin's citizens live in glass apartments in a strictly regulated society that punishes imagination—a three-dimensional take on the repressive aspects of Flatland.
Beyond the Plane
In a world increasingly divided by incompatible worldviews, where facts themselves seem tribal, Flatland's message is urgently relevant: we are all, in some sense, two-dimensional beings trying to comprehend a multidimensional reality.
The most powerful aspect of Abbott's vision is its humility. The Square eventually realizes that even after grasping the third dimension, he may still be limited—there could be fourth, fifth, or infinite dimensions beyond his comprehension. At a time when certainty is often valued over curiosity, Abbott reminds us that intellectual progress begins with acknowledging our limitations.
As the Square writes at the end of the book from his prison cell: "I exist in the hope that these memoirs... may find their way to the minds of humanity in Some Dimension, and may stir up a race of rebels who shall refuse to be confined to limited Dimensionality."
Abbott's 141-year-old call to dimensional rebellion is an invitation to break free from our perceptual prisons and glimpse realities beyond our comfortable plane.
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