This book is the culmination of a set of notes that has been growing and fermenting in a bottom drawer of my desk for over 30 years. The goal was that after the right amount of pedantic aging, they would be ready to uncork the bottle and share with the world. It is my hope that the finished product harbors the perfect blend of intuition and rigorous preciseness, theory versus pragmatism, hard‐nosed mathematics, and an enjoyable learning experience.

I must confess that when the entrails of this book were taking form, the intended audience was the 25 or so students in my Introduction to Higher Math class, but now that they have been released into the wild, my hope is they will find a wider audience.

• College students who have taken courses in calculus, differential equations, and perhaps linear algebra, may be unprepared for the more advanced courses of real analysis, abstract algebra, and number theory that lie ahead. (If you cannot negate a logical sentence, you are probably not ready for prime time.) Although the basic calculus et al. sequence is important for developing a rounded background in mathematics as well as developing problem‐solving skills, most calculus courses are not intended to prepare the student for advanced mathematics. Few students, after going through the basic sequence, are even familiar with the basic language of mathematics, such as the sentential and predicate logic. If a student is to develop skills for reading, analyzing, and appreciating mathematical arguments and ideas, knowledge of the basic language of mathematics is a must.
• Mathematics teachers and math education students will hopefully find this book a valuable aid for teaching an inspiring and exciting introduction to mathematics that goes beyond the basics. The large and varied collection of historical notes and varied problems at the end of each section should be worth the “price of admission” alone.
• Bright high school students with good backgrounds and a strong interest in mathematics will hopefully fall in love with this book, all the way from the “Important Note” inserts to the exciting problem sets.
• Scientists, engineers, and out‐of‐practice others in the professional workforce, who are discovering the mathematics they learned long ago is not adequate for their current needs, will benefit by spending some time with this book. This book might even show self‐study individuals just how uncomplicated and enjoyable mathematics can be, possibly at variance from their college days, and in fact might develop newly found mathematical confidence.

Although this book is intended to raise the mathematical thought process of the reader, it is not intended for the reader to come away thinking that mathematics consists simply in reading and proving theorems. The goal is to blend a nuanced amalgam of inductive and deductive reasoning. Utmost in the goal of the book was to avoid the dogmatism of page‐after‐page of theorems and proofs, to prompt the reader in thinking about the ideas presented.

The book is loaded with informative sidebars, historical notes, and tons of graphics, which hopefully provides an enjoyable atmosphere for a constructive learning process.

Over the years, I have managed to track down, dig up, sniff out, and even make up a few fascinating problems myself. Each section of the book is packed with loads of problems for readers of all interests and abilities. Each section is filled with a broad range of examples intended to add clarity and insight to abstract concepts. The book takes time to explain the thinking and intuition behind many concepts.

Chapter 1: Logic and Proofs

This book is no Principia Mathematica by Alfred North Whitehead and Bertrand Russell, who famously prove 1 + 1 = 2 after 378 arduous pages in their seminal 1910 work on the foundations of mathematics. That said, there are many mathematical proofs in this book, and each and every one of them is intended to act as a learning experience. The first and foremost, of course, being that before anything can be proven true or false, mathematics must be stated in a precise mathematical language, predicate logic. Chapter 1 introduces the reader to sentential and predicate logic and mathematical induction. After two introductory sections of sentential logic and the connectives: “and,” “or,” “not,” “if then, if and only if,” we move up the logical ladder to predicate logic and the universal and existential quantifiers and variables. Sections 1.4 and 1.5 are spent proving theorems in a variety of ways, including direct proofs, proofs by contrapositive, and proofs by contradiction. The chapter ends with the principle of mathematical induction.

Chapter 2: Sets and Counting

Sets are basic to mathematics, so it is natural that after a brief introduction to the language of mathematics, we follow with an introduction to sets. Section 2.1 gives a barebones introduction to sets, including the union, intersection, and complements of a set. Section 2.2 introduces the reader to the idea of families of sets and operations on families of sets, tools of the trade for more advanced mathematical subjects like analysis and topology. Section 2.3 is an introduction to counting, including permutations, combinations, and the pigeonhole principle. Sections 2.4, 2.5, and 2.6 introduce the reader to the basics of Cantor's discoveries on the cardinality of infinite sets. Brief discussions are included on the continuum hypothesis and the axiom of choice, as well as the Zermelo–Frankel axioms of set theory.

Chapter 3: Relations

As the English logician Bertrand Russell once said, mathematics is about relations. This chapter introduces order relations, e.g. partial, strict, and total order, followed by equivalence relations and the function relation. Section 3.5 introduces the idea of the image and inverse image of a set, concepts important in analysis and topology.

Chapter 4: The Real and Complex Number Systems

Sections 4.1 and 4.2 show how the real numbers can either be defined axiomatically, or constructed all the way from the natural numbers, to the integers, to the rational numbers, and finally to the real numbers using equivalence relations introduced in Chapter 3. The Dedekind cut is introduced in Section 4.2, which defines the real numbers in terms of the rational numbers. Finally, in Section 4.3, a brief tutorial on the complex numbers, a subject often overlooked in the undergraduate curricula.

Chapter 5: Topology

This chapter introduces a number of sides of topology, starting with Sections 5.1 and 5.2 on graph theory. Section 5.3 introduces some basic ideas of geometric topology, such as homeomorphisms, topological equivalence, Euler's equation for Platonic solids, and a hint at its verification. Finally, Section 5.4 introduces the basic ideas of point‐set topology on the real line, including open and closed sets, limit points, interior, exterior and boundary of a set, and so on.

Chapter 6: Algebra

Chapter 6 provides a brief introduction to symmetries and abstract groups, rings, and fields. Section 6.1 finds the symmetries of geometric objects and shows who a composition of symmetries gives rise to the algebraic structure of a group. Sections 6.2, 6.3, and 6.4 introduce the symmetric and cyclic groups of permutations, along with the idea of a subgroup. Finally, Section 6.5 gives a brief summary of rings and finite fields.

Conclusion

May the ghosts of past students of this book be with you as you find your way to a mathematical utopia. In proper hands, this book can be used in a one‐semester course, covering the entire book or portions of the book. For a more leisurely pace, an instructor can pick and choose depending on one's preferences. If rushed, one may pass over portions of the book, depending on one's preferences.

And here let me insert a parenthesis to insist on the importance of written exercises. Compositions in writing are perhaps not given sufficient prominence in certain examinations. In the École Polytechnique, I am told that insistence on compositions would close the door to very good pupils who know the subject, yet are incapable of applying it in the smallest degree. The word “understand” has several meanings. Such pupils understand only in the first sense of the word, and this is not sufficient to make either an engineer or a geometrician.

—Henri Poincaré

Keeping a Scholarly Journal

One cannot help but be impressed with the huge number of important English naturalists who lived during the nineteenth century. In addition to Darwin, there were Wallace, Eddington, Thompson, Haldane, Galton, and others. One characteristic that permeated their work was the keeping of detailed journals where every observation and impression was recorded. In addition to recording data, a journal provides a way to organize one's thoughts, explore relationships, and formulate ideas. In fact, they stimulate learning through writing.

Journal keeping has declined in the twentieth and twenty‐first centuries, but readers of this book have the ability to recapture that important tool of learning‐through‐writing.

Entries in your (leather‐bound) journal can be entered daily or in conjunction with each section of the book. It is useful to date entries and give them short descriptive titles. While there are no specific rules on what to include in the journal or how to write them, you will eventually find your “voice” on what works best. From the habit of rereading old entries each time you write new ones, see how your grasp of material grows. After years have passed, you will be impressed on the value you give past journal entries.

This book is accompanied by a companion website:

www.wiley.com/go/farlow/advanced-mathematics

The famous mathematician, George Polya, once remarked that mathematics is not a spectator sport, and to learn mathematics, you must do mathematics. Otherwise, he argued, you are simply a mathematical spectator.

In the context of this book, Polya’s observation means that while reading the book, the reader is a mathematical spectator, after which the reader develops serious mathematical skills and becomes a mathematician by solving, untangling, playing with, etc., yes, even enjoying the problems in the book.

Solving mathematical problems is similar to other endeavors in life, even activities far afield from mathematics, like swimming or playing a musical instrument. We acquire skills in these activities by trial and error, by practice, by making observations, throwing out what does not work, and keeping what does. After a while, we become adept, even proud of our skills, and more often than not, enjoy such activities, that earlier we might have disliked, even abhorred.

I seriously hope the reader will enjoy and develop mathematical skills from the problems in this book, as much as I enjoyed making them.