**I.** Mathematics is an exact language for description, calculation, deduction, modeling, and prediction — more a systematic way of thinking than a set of rules.

Mathematics is the language in which it is impossible to make a nebulous or imprecise statement.

Using a legal analogy, mathematics is a language for writing contracts with Nature that Nature accepts as legally binding.

**II.** The practical importance of mathematics lies in its ability to describe the real world.

The real world consists of what matters. The word “matter” as a noun is used for what the physical world is made of. But if we ask, “What’s the matter with Anne?” we may be asking about a physical ailment, or we may be asking about an idea that is causing Anne to behave strangely. Ideas matter.

The whole point of mathematical education is to make ideas real for students, ideas that were not real for

them before. Ideas like fractions, for example. The fact that 2/3 is smaller than 3/4 matters in the real world.

**III.** Mathematically educated people are stem cells of a technologically advanced society. Because of the universality of mathematics, mathematicians and well educated users of mathematics are flexible in applying and inventing tools for work in technological environments which never existed before.

**IV.** Learning mathematics involves the profound assimilation of intellectual and aesthetic criteria as well as practically orientated ones. The very difficulty in learning mathematics makes it a personality-enhancing experience.

[With contributions and borrowings from David Corfield, Tony Gardiner, Michael Gromov, Niall MacKay, Henri Poincare, Frank Quinn, David Pierce.]

I once heard myself say in a presentation:

‘….and I feel that the World would be a far, far better place if there were many more mathematicians – and many fewer lawyers….’

This led to an interesting exchange with an international lawyer, who just happened to be present…..

Mathematics is what EVERYBODY does when they think clearly.

I am teaching a new first-year module about what mathematics is to nearly 300 students. I went to the trouble of looking up the dictionary definition of mathematics. My Chambers dictionary says:

Mathematics,n singandn pl, the science ofmagnitude and number, the relations of figures and forms, and of quantities

expressed as symbols.

This may tell you what mathematics is, but says nothing about how you do it (the main thing I am trying to get across to them). I prefer Paul Erdos’ statement:

“The purpose of life is to prove and to conjecture”.

This demonstrates both sides of the subject, the logical and the creative.

While I was writing this part of the notes, I had a long conversation with a Big Issue seller near Baker Street station. He was interested in the fact that I was a mathematician, but thought of mathematics as being entirely logical, and was very struck by my demonstration that it has a creative side to it.

Intriguing. I would like to know who said what. I guess that IV is from Tony Gardiner, but I very strongly agree anyway – though I will add that failure in attempting to understand mathematics, can, dually, be personality-destroying.

I wish to query I: mathematics is NOT a language – mathematical language is a language, albeit a strange one, which ranges in register from the extremely precise to registers which are scarcely more precise than everyday English – and as everyone knows actual mathematical papers as published in journals are not written in the very precise language of (some form of) symbolic logic.

This is an important point: too many pupils, and people, already believe that mathematics is nothing but a set of rules for solving recognisable problems. If they are to appreciate that mathematics is an extremely rich and complex creative activity, as well as a source of extremely powerful methods and techniques, then this confusion between the activity of mathematics and particular registers of mathematical language, used in specific circumstances, must be killed off.