What is 5 minus 8? You could probably do the math before you hit puberty, but for centuries this problem was considered “unsolvable.” After all, if you have 5 apples and take away 8, what does a negative apple even look like?

Negative numbers appear surprisingly late in civilisations’ histories. An accounting system using negative numbers appeared in China around the 3rd century, rules about negative numbers in India in the 7th, and the concept didn’t arrive in Europe until the 15th century. Even after their importation, negative numbers continued to be resisted into the 19th century as absurd, impossible and even sinister.

## Using the impossible to solve the unsolvable

Now consider another “unsolvable” equation:  x multiplied by x = -1. We were all taught that any number multiplied by itself results in a positive number, so this x could never exist (just like a negative apple).

The x is incredibly useful, however, to anyone who has to determine what will happen when more than one force is at play – like an electrical engineer working with alternating current (voltage, current, resistance, amplitude, direction, phase shift, etc.). The ‘x’ was rechristened ‘i’ for ‘imaginary,’ although it’s no more imaginary than negatives. The calculations of electrical engineers, air-traffic controllers and meteorologists are possible without ‘i,’ but they’re much simpler with it.

## Imagining worlds to sharpen insights into our own

There are two (interrelated) approaches to mathematical modelling today. One is to create formulae that depict the physical world and predict how things will act. The other is to imagine a world through mathematics, an entirely new concept (like the negative apple) that could resolve some of todays’ biggest questions.

 Fawad Hassan, Senior Lecturer in Theoretical High Energy Particle Physics and Gravitation in the Physics Department, discusses how current mathematics faces a challenge in describing certain physical phenomena like gravity, and how a new mathematical theory could well lead to a ‘new physics.’

The issue that Dr Hassan is pointing toward is one of the biggest challenges in theoretical physics: the missing link that turns the ‘theory of almost everything’ (aka Standard Model) into the ‘theory of everything’ (aka Unified Field Theory). The ‘almost’ missing from the Standard Model is gravity – the fundamental force that a toddler gleefully exploits at dinner time has not been connected to the other three forces.

## Gravity at the quantum and cosmic levels

Nothing we currently know can unite gravity with the other forces at every scale – we might say it’s ‘unsolvable.’ However, this challenge may well be solved like the other issues in mathematics – it might just be the perfect opportunity for a proverbial negative apple.

And maybe ‘solving the unsolvable’ means redefining gravity itself – a ‘massive gravity’ instead of the currently accepted ‘massless’ one. The breakthrough is a creative leap from the cliff, not knowing beforehand exactly where you’ll land.

Mathematical Theory Development and Modelling is a much broader research area than just theoretical physics. Modelling pervades the natural sciences (notably in astronomy and meteorology), and is also used extensively in the social sciences (e.g. economics). Theorists and practitioners collaborate extensively across disciplinary boundaries, enhancing the toolbox of mathematics as well as developing ‘new worlds’ within pure maths.