Mathematics underpins the world around us. It allows us to understand the workings of the microworld to an extent that we can manipulate atoms and molecules and design materials that process information, make phone calls, medical probes, and aircraft.

The esoteric properties of prime numbers lie at the heart of the best coding systems. Logic programs computers. Fractals have led us to new ways to store huge amounts of information. Our appreciation of what is reliably predictable in Nature hinges upon our mathematical understanding of chaos.

The nature of complex systems and the delicate balance that is often required to maintain them, whether they are flames, ecosystems, traffic systems, or economies, requires a mathematical understanding of their dynamic stability. Our statistical understanding of chance and risk is essential if more technologies are to be safe.

The ubiquity of mathematics in science and human affairs derives from the fact that mathematics is the study of the collection of all possible patterns. Some of these patterns occur between shapes, some between numbers, some are in sequences of events, in the clustering of galaxies, or in the interactions between the most elementary particles of matter.

Others occur where we expect to see patterns, in artistic designs or in music. Just as in music and other creative arts, there is room both for an appreciation of the mathematics behind the world of patterns as well as a detailed professional understanding of it.

**Accessibility**

Not all mathematics is easy. No professional mathematician today works with more than a small fraction of the whole. It is the job of mathematicians and mathematical scientists to provide an accessible and appealing way in to their subjects. Unfortunately, mathematics has become something that many people are proud to be bad at. This is not the case in many other cultures.

Traditionally, mathematics has been seen as algebra, arithmetic, and geometry. Today the immediacy of good computer graphics and the web allow us to show where this leads. There is huge interest amongst the public at large and young people in the developments at the frontiers of sciences like astronomy and particle physics.

Alongside these modern tools there should be a place for games and puzzles. The success of toys like the Rubik Cube reveal something of many people’s fascination for puzzles. At root they are part of a larger body of problems that are both instructive and fun.

But they have a serious side as well. The development of intuition about space, about pattern, quantity, and probability, provides a foundation from which to build a fuller understanding and appreciation of the world around us.

For the dedicated pragmatist we should stress that mathematics also leads to a wide range of vital, interesting and rewarding jobs.