In many American universities there are courses known universally, if not officially, as `Maths for poets'. This book does not belong to that genre. It is intended, rather, as `Maths for mathematicians' --- for mathematicians who know very little mathematics as yet, but who, perhaps, will one day give lectures which the present author attends open-mouthed with admiration [When Wiles was a humble graduate student at Cambridge I had already reached the dizzying heights of a lectureship. Twenty years later I was in the back row when he announced his solution of the three centuries old Fermat problem.].
I hope that this book will also be enjoyed by my fellow professionals and by those general readers who value mathematics without fearing it [The desire for large sales is shared by authors and publishers. David Tranah, to whom, like many other CUP mathematics authors, I owe an immense debt of gratitude, suggested that a different title would help. For once, I did not take his advice and the title `The Joy of x' remains available.]. Both groups will have to indulge in some judicious skipping (the professionals will not need to have Cantor's diagonal argument explained to them or to be told that Kolmogorov was a great mathematician; the general reader may tiptoe past the scarier algebra). My colleagues are perfectly capable of deciding by themselves whether or not to read this book, but two analogies may be of assistance to the general reader.
First she might like to consider why `fly on the wall' documentaries showing life in medical school or on a warship are interesting to many who are not doctors or sailors. Listening to a mathematician talking to mathematicians about things that interest mathematicians may well be more enlightening than listening to mathematicians speaking to non-mathematicians about things that they hope may be interesting to non-mathematicians. Alternatively she may consider the choice facing tourists in some exotic city. They may dine in fine restaurants where everything is scrupulously clean and the service is excellent but the cuisine has been modified to suit international tastes. Alternatively they can go to a local taverna where the waiters are rushed off their feet and, in any case, speak very little English, where glimpses of the kitchen fail to inspire confidence in its hygiene, and some of the dishes look very strange indeed. This book is the local taverna. The disadvantages are genuine but so is the cooking.
Even those for whom this book is intended should not necessarily expect to understand all of it. I have pitched the level of exposition at the level I would expect of beginning students of mathematics at Trinity Hall, and, if I were talking to them, I would not be surprised to have to give extra explanations on points not taught to this or that student at this or that school. Only an exceptionally well taught (perhaps even overtaught) 14 year old could expect to understand everything in this book (though I hope that any persevering 14 year old should end by understanding much of it). Professional mathematicians consider a mathematics book worthwhile if they understand something new after reading it and excellent if they understand a fair amount that is new to them; anything more would lead them to suspect that the material was too easy to be worthwhile.
If there is something that you do not understand you should (if you can) ask someone like your school teacher about it. If there is no one to ask, read on and perhaps understanding will come. If this does not work, try some other part of this book. Although the book is complete without the exercises, I hope that you will glance at them. Some, like Exercise 9.2.3, are simple commentaries on the text. Others, like Exercise 11.4.14 and Exercise 16.2.3 make use of mathematics met near the end of a school or near the beginning of a university course; they require substantially more knowledge than is assumed in the body of the book. I believe that such exercises have been clearly sign-posted. Appendix B may help if you are baffled by my notation.
Montaigne feared that `some may assert that I have merely gathered here a big bunch of other men's flowers, having furnished nothing of my own but the string to hold them together'. This book is such a bunch of flowers and I hope that the reader may be sufficiently intrigued by some topic or quotation to explore the garden from which that particular flower came [Hence `the habit of fitting out the most trivial quotation with a reference as though it were applying for a job'. Most of the references, including this one, will be found in Appendix C.] In any case I hope that she will see that the characteristic sound of mathematics is not that of a chorus speaking in solemn unison but the babble of individual voices.
Equally, since I have addressed the reader as I would my own students or colleagues, I have not sought to hide the fact that I hold opinions on many topics. In this and several other respects I have chosen to write a book which is intended for schoolchildren but not suitable for them.