0.4: Advice to the Student
Welcome to higher mathematics! If your exposure to University mathematics is limited to calculus, this book will probably seem very different from your previous texts. Many students learn calculus by quickly scanning the text and proceeding directly to the problems. When struggling with a problem, they seek similar problems in the text, and attempt to emulate the solution they find. Finally, they check the solution, usually found at the back of the text, to "validate" the methodology.
This book, like many texts addressing more advanced topics, is not written with computational problems in mind. Our objective is to introduce you to the various elements of higher undergraduate mathematics - the culture, language, methods, topics, standards and results. The problems in these courses are to prove true mathematical claims, or refute untrue claims. In the context of calculus, the mathematician must prove the results that you freely used. To most people, this activity seems very different from computation. For instance, you will probably find it necessary to think about a problem for some time before you begin writing. Unlike calculus, in which the general direction of the methods is usually obvious, trying to prove mathematical claims can feel directionless or accidental. However it is strategic rather than random. This is one of the great challenges of mathematics at the higher levels, it is creative, not rote. With practice and disciplined thinking, you will learn to see your way to proving mathematical claims.
We shall begin our treatment of higher mathematics with a large number of definitions. This is usual in a mathematics course, and is necessary because mathematics requires precise expression. We shall try to motivate these definitions so that their usefulness will be obvious as early as possible. After presenting and discussing some definitions, we shall present arguments for some elementary claims concerning these definitions. This will give us some practice in reading, writing and discussing mathematics. In the early chapters of the book we include numerous discussions and remarks to help you grasp the basic direction of the arguments. In the later chapters of the book, you will read more difficult arguments for some deep classical results. We recommend that you read these arguments deliberately to ensure your thorough understanding of the argument and to nurture your sense of the level of detail and rigor expected in an undergraduate mathematical proof.
There are exercises at the end of each chapter designed to direct your attention to the reading and compel you to think through the details of the proofs. Some of these exercises are straightforward, but many of them are very hard. We do not expect that every student will be able to solve every problem. However, spending an hour (or more) thinking about a difficult problem is time well-spent even if you do not solve the problem: it strengthens your mathematical muscles, and allows you to appreciate, and to understand more deeply, the solution if it is eventually shown to you. Ultimately, you will be able to solve some of the hard problems yourself after thinking deeply about them. Then you will be a real mathematician!
Mathematics is, from one point of view, a logical exercise. We define objects which do not physically exist, and use logic to draw the deepest conclusions we can concerning these objects. If this were the end of the story, mathematics would be no more than a game, and would be of little enduring interest. It happens, however, that interpreting physical objects, processes, behaviors, and other subjects of intellectual interest, as mathematical objects, and applying the conclusions and techniques from the study of these mathematical objects, allows us to draw reliable and powerful conclusions about practical problems. This method of using mathematics to understand the world is called mathematical modelling. The world in which you live, the way you understand this world, and how it differs from the world and understanding of your distant ancestors, is to a large extent the result of mathematical investigation. In this book, we try to explain how to draw mathematical conclusions with certainty. When you studied calculus, you used numerous deep theorems in order to draw conclusions that otherwise might have taken months rather than minutes. Now we shall develop an understanding of how results of this depth and power are derived.