Introduction to the Chapter
This chapter will briefly introduce many of the key mathematical terms and concepts that I shall be using throughout the book. The material in this chapter should, I hope, be thoroughly familiar to anyone who has studied mathematics at university or even at sixth form level, and such readers can probably skip most of it altogether, and refer to the glossary if my use of any particular term later on does not quite accord with your understanding of it. Intelligent, but less mathematical readers, who may be feeling nervous already, will, I hope, find that this chapter is not really that difficult. My main aim will be to introduce you to a number of words that will probably seem familiar, but which are used in mathematics with a different meaning from their common one.
In case of difficulty, remember that you will usually be able to find much fuller explanations than are give here on mathematical sites on the internet.
One difficulty I have in presenting this material is knowing where to begin. It would be nice to be able to proceed like Euclid, by beginning with definitions and axioms, from which everything else followed in a strictly logical way. Unfortunately, this is not at all easy to do. Although most of us probably feel we have a good understanding of what such terms as "point", "straight line", "between", and "bigger" mean, defining them is not at all an easy task. In fact the more we think about basic concepts the harder it will prove to pin them down, and in some cases our everyday intuitions will turn out not to be unconditionally true at all, so that the notion of an axiom as a "self-evident" truth, will prove sadly wanting. Attempting to put everything on a firm foundation would result in our having to consider difficult philosophical issues, which I am ill-equipped to explain, and which are irrelevant to my aim.