The classic story of science and mathematics was that reason could reach through to some pure, complete understanding of the world. But from Lewis Carroll’s work as a mathematician to Gödel’s incompleteness theorems, mathematics, and therefore the science it underlies, tells a different story. As Minh-Hoang Nguyen argues, the world, like that of Wonderland, is fundamentally incomplete. The notion that we live in a closed system that can be fully explained was not just unachievable; it was a philosophical mistake.

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.

Mathematicians start off with as small a collection of assumptions as possible, known as axioms. From this they then aim to construct an entire conceptual universe, be it to represent just mathematics or the physical world. Traditionally starting out with a simple notion of a set, increasingly complex structures emerge: numbers, functions, geometry, topology, and many of the abstract ideas that occupy contemporary research questions. Mathematicians, like architects or city planners, build up from a handful of principles to construct roads, boulevards and streets governed by internally consistent rules.

At the beginning of the twentieth century, the mathematical consensus was that we were reaching the end of the project. The success of Cantor’s set theory, further developed by Zermelo and Fraenkel, led the field to close in around Zermelo–Fraenkel set theory with the axiom of choice (ZFC). This system was thought to provide a secure and elegant system through which every mathematical truth could be proven, and through its formal structures no contradictions could lead us to falsehood.

And yet, in 1931, Kurt Gödel devastated many and inspired more by showing that the dream of a perfectly consistent and sound mathematical model was impossible. According to his theorems on incompleteness, no system of mathematics can prove all the true things that you can express in that language. Worse still, you could not disprove all the untrue things within that system either. Consistency, then, cannot ensure we can express all the truths we think we know about mathematics or the world.

Thinkers from Russell and Whitehead to Frege found themselves in a brand new situation. Their whole projects had been shown not only to be impossible, but for the very nature of mathematics to be flawed. So, no matter how we aim to construct mathematics, the language which forms the basis of our most assured conclusions, there will always remain truths and falsehoods we can never answer.

To many, this seemed like a complete failure. Human intuition craves the certainty of an absolute truth. But now we had mathematical proof that such a solution was impossible.

Yet the work of famed author Lewis Carroll in his Alice’s Adventures in Wonderland suggested a different possibility. Lewis Carroll holds a particular and important view on coming to grips with this absurdity. Caroll was in fact known to his friends as the Oxford mathematician Charles Dodgson. And Dodgson had experienced a similar collapse of meaning for mathematics. As a mathematician, Dodgson had sought to fend off the paradoxes and absurdity of a new mathematics, one steeped in new concepts and abstractions. The Euclidean framework, which had been the foundation of mathematics for thousands of generations, was being thrown off for something new.

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Wonderland resembles mathematics itself. The modern mathematical field frequently embraced concepts that initially appear impossible or nonsensical.

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