Dr Katie Chicot is about to take thirty wide-eyed schoolchildren on an unforgettable journey into the uncharted outer reaches of infinity. The Open University tutor's sightseeing tours of mathematical space form part of her extensive outreach work at schools across the country, which she carries out alongside her role in the Department of Mathematics & Statistics.
To the untrained mind, algebraic formulas can seem wholly perplexing and intimidating at first appearance. Enlisting young converts to the wonders of mathematics therefore requires someone with an evangelical zeal for the subject (which Katie describes as "Compulsive and awe-inspiring"). She is channelling her infectious enthusiasm for numbers to help open up young, inquisitive minds to the possibilities that can be unlocked through an understanding of mathematics. She admits that part of the appeal of explaining such mind-bending concepts to children is "Because it gives me a chance to show off, really!"
In previous interviews, Katie has hailed mathematics as 'The queen of the sciences.' Whilst science seeks evidence in support of hypotheses in order to advance understanding, "Mathematics is special. It's a unique discipline where we have incontrovertible proofs. Science has advanced over time, rendering some concepts taught in Classic Greek science as obsolete. But Pythagoras' Theorem remains as true today as it was when it was first conceived."
Youngsters are particularly receptive to the concept of infinity, Katie explains. "I think that children have a good intuition for infinity. They can imagine something that goes on forever and is without boundaries. I begin by introducing students to 'real' infinity across the vast reaches of outer space. Cosmologists ponder whether the universe is infinite in size, whilst physicists propose the existence of an infinite number of parallel universes."
Within minutes of entering the classroom, the students have begun to engage with the ultimate nature of reality via mathematics. Suddenly and excitedly they are beginning to comprehend how mathematics has the potential to take human minds on a rollercoaster journey to the very outer reaches of human understanding. It frequently entails soaring through the cosmos at the speed of light and indulging in a bit of time travel for good measure.
Outer space is a realm where conceptual, mathematical notions of infinity encounter 'real' infinity. "My original, slightly naïve view used to be that space was just something that went on forever, and there was some 'stuff' inside it – planets, stars, galaxies and the like," Katie remembers. "I was happy for that 'stuff' to be finite, but not comfortable with the idea of space being finite. After all, what would the boundary of the universe be? But it seems that these naïve ideas of boundaries and the shape of the universe are a long way from the real picture."
Once children have accepted that infinity is really out there, Katie guides them along as they unpack the mathematics involved. "We start by establishing that there are infinitely many counting numbers," she explains. In short, if you try to think of the biggest number conceivable, you could always add one to it and create an even bigger number. So we can quickly conclude that the counting numbers must go on forever."
Once you have a handle on infinity, it's possible to explore the exciting and sometimes perplexing paradoxes that it presents us with. The mischievous Hilbert's Infinite Hotel paradox, conceived by the German mathematician David Hilbert, illustrated the notion of an infinite set of numbers. It's the tale of a luxurious seaside resort with a novel way of accommodating new guests. At first glance, the hotel is full up, because its infinite number of five-star rooms are occupied by an infinite number of holidaymakers. Yet, as absurd as it seems, there is always room for more guests.
When a new holidaymaker checks in, the proprietor offers them the keys to room one, renowned for its spectacular sea view. In a scheme which would likely have earned the hotel scathing reviews on TripAdvisor, the manager insists that the guest who had previously resided in room one moves along the corridor into room two, with the guest in room two moving along to room three, and so on. In all, an infinite number of guests relocate to the next room up. The new guest spends the night in room one and all other guests kindly acquiescing to the manager's request to pack up their things and move next door. In fact, Hilbert's Hotel is paradoxically able to make room for an infinite number of new arrivals, despite the fact that every single room is always occupied.
Considerations of infinity lead mathematicians down mind-bending and often counter-intuitive paths. Philosophers through the ages have also struggled to think about infinity in a robust way. Consider the famous paradox of Achilles and the Tortoise, or the question of whether the past might be infinite.
It's with the use of such intriguing questions that Katie engages her audience.
Dabbling in these ideas gives rise to considerations of greater layers of complexity, including notions of multiple layers of infinity. "We have an infinite number of even numbers and an infinite number of odd numbers. How can we know which set of infinite numbers is the biggest?" muses Katie. "And what does 'size' even mean when you're dealing with two infinite sets of numbers!? It's great to see the students try to answer these questions for themselves, grappling with the same concepts that have enchanted mathematicians for centuries."
The cynical observer might be inclined to ask, "So what's the point of it all?" Contemplating the wonders of infinity (not to mention the infinite wonders of infinite infinities) might be intellectually stimulating and even fun, but what good comes from gazing up at the stars and pondering our infinitesimally insignificant existence in the great expanse of time and space?
Katie is quick to explain. "Contemplating the complexities of the infinitely small eventually gave rise to Calculus," she contests. "Calculus is the study of change, any change. Since all scientific study is the study of change, Calculus is used in every realm of the physical sciences. It's the single greatest application of mathematics to the sciences and has, in my (possibly biased) view, made a wider contribution to science than Darwin's theory of evolution or the sequencing of DNA. Unfortunately, mathematics exists somewhat in the shadows. It's the silent partner in scientific research and few people are aware of the vital role it plays as a foundation for the sciences."
Knowledge of the infinite has proven lucrative for many. One of the largest employers of mathematicians in the world is Pixar, the renowned animation studio responsible for the Toy Story series and several other classic children's films. Inventive use of an algorithm based on fractals inspired a paradigm shift in animation, enabling CGI artists to generate highly complex and realistic depictions of natural landscapes which had seemed impossible before. Fractals are geometric patterns of infinite complexity, found in mathematical models as well as the natural world, where they are evident in everything from the spiral formations of galaxies to the branching of trees. These patterns retain their complexity at all scales of magnification, from the infinitely large to the infinitesimally small.
The concept of infinity has met with considerable controversy throughout the ages. The German mathematician Georg Cantor's formalisation of the concept of infinity in mathematical formulae was fiercely opposed by his contemporaries. He met with philosophical objections from religious quarters, who perceived his concept as a heretical challenge to the uniqueness of the absolute infinite in the nature of God. His former professor, the influential mathematician Kronecker, denounced Cantor as a "corruptor of youth" because of his maverick ideas.
When we think about infinity, there are some ideas that are so utterly counter-intuitive to the human mind that it's almost impossible to visualise them. "Even the finest mathematicians of the time struggled with the idea of infinity, especially 'bigger' infinities," Katie insists.
"Our intuition simply isn't built to deal with these realities. It's incredibly challenging to visualise something which is so far outside of the realms of human experience. However, if we really want to get to the bottom of infinity then we have to do the maths and accept the conclusions that we reach."