Imagination: The usefulness of incompleteness

Minh-Hoang Nguyen

ISR, Phenikaa University

28-05-2026

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Mathematics is often mistaken for a discipline of quick calculations and precise answers. Yet mathematics is not fundamentally about arithmetic. At its deepest level, mathematics is an exercise of world-building (Bischoff, 2026).

Mathematicians begin with a small collection of assumptions, known as axioms, and construct entire conceptual universes from them. Starting with simple notions such as sets, increasingly complex structures emerge: numbers, functions, geometry, topology, and many of the abstract ideas that occupy the frontiers of contemporary research. Like an architect designing a city from a handful of principles, mathematicians create worlds governed by internally consistent rules.

By the early twentieth century, many mathematicians believed they were close to completing this grand project. The dominant framework, known as Zermelo–Fraenkel set theory with the axiom of choice (ZFC), appeared capable of providing a secure foundation for all mathematics. The dream was elegant: establish a system that was both complete, meaning every mathematical truth could be proven, and consistent, meaning no contradictions could arise (Bischoff, 2026).

Then came a surprise that bursts the dream.

In 1931, the young logician Kurt Gödel demonstrated that this dream was impossible. His incompleteness theorems revealed that any sufficiently powerful mathematical system necessarily contains statements that can neither be proven nor disproven within the system itself. Even more unsettling, such a system cannot prove its own consistency using only its internal rules (Gödel, 1931).

The implication was profound. No matter how carefully a mathematical world is constructed, there will always remain truths that lie beyond its reach.

At first glance, this may seem like a failure or bad outcome. As human intuition often craves certainty, a complete system appears safer, cleaner, and more satisfying than one containing unanswered questions.

Yet Lewis Carroll's Alice's Adventures in Wonderland suggests a different possibility (Carroll, 1865).

Behind the pseudonym Lewis Carroll stood Charles Dodgson, a mathematician at Oxford University. Dodgson lived during a period when mathematics was undergoing a dramatic transformation. New concepts such as imaginary numbers and symbolic algebra were challenging traditional understandings of logic and quantity. To many mathematicians, these innovations opened exciting new directions for exploration. To Dodgson, however, they often appeared absurd (Bayley, 2009).

Unlike the geometry of Euclid, which connected mathematical reasoning to tangible spatial reality, symbolic algebra allowed mathematicians to manipulate entities that seemed detached from physical experience. Negative numbers, imaginary numbers, and increasingly abstract symbols operated according to rules that were internally consistent but not always intuitively meaningful.

Unable to win these debates through academic argument alone, Dodgson transformed them into fiction.

Wonderland is frequently interpreted as a realm of nonsense, but much of its nonsense can be understood as a satirical reflection of mathematical abstraction. Alice repeatedly encounters situations that violate ordinary expectations. Her size changes unpredictably. Arithmetic behaves strangely. Logical conversations spiral into paradox. Familiar rules cease to function (Bayley, 2009).

From the perspective of ordinary experience, Wonderland appears irrational.

Yet it is not chaotic.

Its inhabitants follow their own peculiar logic. The Caterpillar is perfectly comfortable with Alice growing and shrinking. The Mad Hatter's conversations obey patterns invisible to outsiders. What appears absurd from one perspective becomes sensible within another framework.

In this respect, Wonderland resembles mathematics itself.

Modern mathematics frequently embraces concepts that initially appear impossible or nonsensical. Imaginary numbers, non-Euclidean geometries, and higher-dimensional spaces once seemed nonsensical because they violated familiar assumptions about reality. Yet by accepting new axioms and new rules, mathematicians discovered coherent worlds where these strange ideas became not only meaningful but indispensable.

The history of mathematics therefore reveals a fascinating paradox. The failure to establish a complete and final foundation did not halt intellectual progress. Instead, it created opportunities for entirely new forms of reasoning. Gödel showed that every sufficiently rich system contains unanswerable questions. Rather than ending mathematics, these gaps encouraged mathematicians to explore alternative foundations, alternative perspectives, and alternative worlds.

Wonderland embodies a similar process. The incompleteness of one framework gives rise to another. What appears nonsensical from the outside becomes intelligible once its underlying rules are understood.

This feature extends far beyond mathematics.

Human societies often seek perfectly ordered systems: complete explanations, universal truths, and definitive answers. Yet reality repeatedly resists such ambitions. Scientific theories remain provisional. Philosophical questions persist across centuries. Social systems generate unexpected consequences. Uncertainty never entirely disappears (Khuc & Nguyen, 2026).

However, this incompleteness should not always be feared.

Without unanswered questions, there would be no exploration. Without conceptual gaps, there would be no imagination. Without the limits of one world, there would be no motivation to envision another.

In this sense, Alice's Adventures in Wonderland can be a demonstration that the boundaries of reason can become the starting points of imagination. As part of life, we are not constituents of closed systems prone to the order depicted in mathematical and algorithmic worlds; rather, we do math and use mathematics to navigate an uncertain world (Vuong, 2025; Nguyen, 2026).

If the incompleteness of a sensible world gives rise to a nonsensical world that ultimately renders the unsensible sensible, then that incompleteness is useful.

References

Bayley, M. (2009). The mathematical meaning of Alice in Wonderland. New Scientist, 204(2739), 38-41. https://doi.org/10.1016/S0262-4079(09)63322-4

Bischoff, M. (2026, May 26). Why some mathematical theorems will always be unprovable. https://www.scientificamerican.com/article/how-the-mathematician-goedel-proved-that-not-everything-can-be-proven/

Carroll, L. (1865). Alice's Adventures in Wonderland. Macmillan.

Khuc, V. Q., & Nguyen, M. H. (2026). Cultural Additivity Theory. Available at SSRN 6767760. https://ssrn.com/abstract=6767760

Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik Und Physik, 38(1), 173-198. https://doi.org/10.1007/BF01700692

Nguyen, M.-H. (2026). Ayn Rand and Kingfisher on zero-carbon bombs and a sustainable future. Visions for Sustainability, 25(13474), 1-13. http://dx.doi.org/10.13135/2384-8677/13474

Vuong, Q. H. (2025). Wild Wise Weird. AISDL. https://books.google.com/books?id=C5dDEQAAQBAJ