Why do we need to do maths? It's a question asked in classrooms around the world and the usual response is "because it's useful". While this is true, on its own it is not entirely convincing since most pupils know that they and their parents rarely apply algebra and geometry in everyday life.
Exploring a little further with pupils is worthwhile, and looking to the history of maths provides many reasons why established societies have needed maths. Mathematical activity takes place in three areas. The first could be called social arithmetic, which we all use in our everyday activities. This seems a very natural process and it is easy to see why this level of activity has never been questioned. The market place in modern England is not so very different from the market place in ancient Cairo or Athens. Arithmetic has prompted some deeper thinkers to explore below the surface and attempt to generalise arithmetic and its processes.
This has led to algebra and a further exploration of ideas. This is the research area and it is where maths is created.
Most people find it difficult to imagine how research can take place in maths. It takes place by searching for truths relating to established mathematical results. Research takes place first in pure maths and then with applications of maths to real problems. Between this level and social arithmetic is an area which uses established maths to create algorithms, which are processes that can easily be used, or to solve problems in many areas essential to modern life: in science, statistics, engineering and the financial world. Original maths rarely comes directly out of this area, but it instigates fundamental research in maths.
Financial maths is an area which is of concern to everybody. Increased wealth in Europe during the 16th and 17th centuries led to a concern about investments and the possible return on them. It led to the idea of an annuity and life assurance.
Various Bills of Mortality were published in England (Gaunt, 1662) and Holland (de Witt, 1671) and Edmond Halley (of comet fame) produced tables of annuities for single and double lives.
Major contributions were also made by Abraham de Moivre, who also worked in the more abstract area of complex numbers. The fundamental work of Halley and de Moivre created the basis for actuarial science.
Of less concern to most people in their daily lives is the maths of science. Maths developed in tandem with science over the centuries, with physics and astronomy particularly strong driving forces. It is also true that maths was often ahead of any application to the physical world. The need to model the physical world and the universe led to the creation of the necessary tools. The calculus of Leibniz and Newton led to the development of differential equations, an immensely powerful tool, which was applied with enormous success in modelling the physical world.
So successful were differential equations, in fact, that mathematicians fell into the trap of believing them almost infallible and began to make assumptions that would fit the required result.
A new area of maths evolved, however, which took into account the fact that the behaviour of a system changes over time. This is non-linear maths and includes fractals, chaos and catastrophe theory. They provide powerful new tools for work on areas as diverse as meteorology and schizophrenia.
Maths is a discipline with a long history and it continues to grow. There is a strong relationship between abstract, pure maths and applications to real problems. As the language of technology, it is essential in modern society.