Images of Mathematics Viewed Through Number, Algebra, and Geometry

FOREWORD

    Does the world need another book introducing the non-specialist to the world of mathematics beyond that found in typical high school presentations? There are indeed myriads of books: some have a recreational approach; some are specialty books with a particular focus (books on π, e, 0, etc.); and some books are without (or almost without) any mathematical formulas, meant to give a flavor of the allure and scope of mathematics. Alternatively, there are books of an encyclopedic nature, or books focusing on details of specific episodes and techniques in the development of mathematics which may leave the reader amongst a beautifully luxuriant thicket, but without giving a sense of the broader directions of mathematics. In this book, using secondary school math skills as a base, I wish to add to the literature which focuses on some of the broader themes of mathematics. I hope to engage those of you who are just learning high school mathematics or those who have completed it, moved on, and still wonder what mathematics is all about. In both cases, I believe that by taking advantage of the math training that you have already acquired, broad mathematical themes can be explored and experienced, creating a greater appreciation of one of humanity’s greatest creations.

    Basics of number systems, algebra, and geometry are being put or have been put into your tool kit in high school with much of the motivation for these subjects supplied through problem solving. Moreover, it is explained that the development of mathematical fluency is an absolute requirement for those seeking careers in science, engineering, or other quantitative disciplines. And so it is. But mathematics is also, in its own right, one of the great creative and intellectual achievements of humanity. I believe that the typical introductory math training provides the skills necessary to allow the fundamental sources and directions of mathematics to be developed and illustrated in a systematic way. Furthermore, I believe this can be accomplished by focusing primarily on illustrations of concepts rather than on the many detailed calculation methods required for problem solving.

    Unlike many other books for the non-specialist of mathematics, I have included in my presentation lots of equations, proofs, and abstract concepts. I have selected this approach because mathematics at its core is abstract with symbolic representation as its language. Developing a comfort with such an approach is essential to enjoying and understanding the meaning of mathematics. My hope is that this book will provide an opening to the world of mathematics beyond just a qualitative sense of wonder and to inspire and enable you to continue on to the wider, more advanced literature of mathematics such as exemplified by the referenced works or available through a search of subjects on the internet. I firmly believe that mathematics is not just for the practitioners, just as the arts are not for the artist alone. Independent of its extraordinary usefulness, mathematics can provide unique pleasures and insights for the mind.

    For those of a more practical nature, a better acquaintance with some of the great themes of mathematics will provide motivation beyond that typically encountered in school. Also, a broader view along with the skills acquired will facilitate an understanding of mathematical details that are otherwise often only understood as sets of rules.

    The subject matter presented here is at an introductory level in keeping with the mathematics of an inquisitive student in high school. At this level, the subjects presented, although selected to illustrate major themes in mathematics, are necessarily very preliminary and incomplete in nature in that entire books have been written covering just individual sections. You should view this book as a friend telling you about things to look out for on your upcoming trip. Thus, this book is not meant to replace more specialized works, but to give you an entrée as a self-study guide to those works which are typically aimed at those who have already begun to specialize in mathematics.

    Topics explored here include: the relationship of the real numbers to the integers and rational numbers, the uncountability of the irrational numbers, complex numbers, logic and proofs in math, the world of non-Euclidean geometries, vectors, tensors, matrices, and paths from sets of the real numbers to the exotic world of topology. These topics would lead in the twentieth century to controversies over the meaning of mathematics, the subject of my final chapter. I emphasize that these topics are developed starting with only introductory math as a basis without assuming prior knowledge of advanced mathematics techniques. New techniques and concepts are developed, building from chapter to chapter.

    In addition, to the subjects mentioned above, I have included an introduction to the calculus. This is a subject which typically is only taught to those considered gifted in mathematics. I do not see why all students should not be familiar with the basic concepts (again, in contrast to the complex manipulations) as the calculus can be motivated with geometric approaches and is the gateway to much of science. I have not included the subjects of probability, statistics, and discrete mathematics which have added their own important themes to mathematics, but do not fit as smoothly into unity of the other topics as presented here.

    This book is meant to be read with care, but mostly with the pleasure that comes with understanding and new insights. A careful reading is meant to provide all the necessary details to follow the conceptual developments. By working through all of the arguments, my hope is to give you not only the feel of the extraordinary scope of mathematics, but to actually experience it. In writing this book, I have frequently made use of phrases such as, “we will now see” or “we can determine . . . .” By “we”, I mean that you are following along with me in my line of reasoning. In cracking the hard nut of an argument, I hope you, like Archimedes, will experience your own eureka moments. If you are moved in this way and this book makes it possible to continue your journey in mathematics as one of life’s pleasures, then my efforts will have served their purpose.