The purpose of this book is to introduce you to the culture, language and thinking of mathematicians. We say "mathematicians", not "mathematics", to emphasize that mathematics is, at heart, a human endeavor. If there is intelligent life in Erewhemos, then the Erewhemosians will surely agree that \(2+2=4\). If they have thought carefully about the question, they will not believe that the square root of two can be exactly given by the ratio of two whole numbers, or that there are finitely many prime numbers. However we can only speculate about whether they would find these latter questions remotely interesting or what they might consider satisfying answers to questions of this kind.
Mathematicians have, after millennia of struggles and arguments, reached a widespread (if not quite universal) agreement as to what constitutes an acceptable mathematical argument. They call this a "proof", and it constitutes a carefully reasoned argument based on agreed premises. The methodology of mathematics has been spectacularly successful, and it has spawned many other fields. In the twentieth century, computer programming and applied statistics developed from offshoots of mathematics into disciplines of their own. In the nineteenth century, so did astronomy and physics. The increasing availability of data make the treatment of data in a sophisticated mathematical way one of the great scientific challenges of the twenty-first century.
In this book, we shall try to teach you what a proof is - what level of argument is considered convincing, what is considered overreaching, and what level of detail is considered too much. We shall try to teach you how mathematicians think - what structures they use to organize their thoughts. A structure is like a skeleton - if you strip away the inessential details you can focus on the real problem. A great example of this is the idea of number, the earliest human mathematical structure. If you learn how to count apples, and that two apples plus two apples make four apples, and if you think that this is about apples rather than counting, then you still don’t know what two sheep plus two sheep make. But once you realize that there is an underlying structure of number, and that two plus two is four in the abstract, then adding wool or legs to the objects doesn’t change the arithmetic.