In the Spring of 1962, I was a first year student at Loyola University of Los Angeles, studying math. I was fairly new to math (and to study, for that matter), but Father Wallen, as we knew him, took a few of us under his wing in a special course, taught with his own notes. He took us through the development of the number system, from the Natural Numbers to the Complex Numbers, one step at a time. Along the way we followed a generally historical path, seeing the introduction of new ideas and becoming familiar with axiom systems and their implications.
This course gave me the best idea of the nature of mathematics that I would gain until late in graduate school. Three of us took the course, and I believe that we were the first to do so. All of us, myself, Alexander Hahn, and Michael Cullen, all went on to earn doctorates and to teach the rest of our careers. Mike went back to Loyola to teach Math and become department head. He died several years ago. Alex was head of the Mathematics Honors Program at Notre Dame where he has spent most of his adult life. I switched to Computer Science and taught that for most of my career, never losing the abstract view that I learned from Clancy, as his friends knew him (though we never dared to utter it).
While the concept of number doesn't completely cover the idea of mathematics, it has been an important component. Nearly everyone today learns something about numbers and a bit about math itself, but few people truly understand the fundamental idea of mathematics. So, what is Mathematics? Really? Here we hope to shed some light on that question.
We, you and I, will use the development of number systems to try to get a handle on what it is that mathematicians do. They don't memorize facts like you did at age 10, or so. They don't just prove theorems as you may have done in secondary school or university. It is actually quite different. Before one can prove a theorem they need to discover what the theorem actually is. It may not be involved with number at all. But mathematicians do prove theorems: statements that can be derived logically from simpler known truths. And here you will have an introduction to theorem proving if you haven't seen it previously. But that isn't the essence of math.
Some of what mathematicians do is deadly boring, but necessary. Some of it is wildly exciting. The boring stuff comes when you need to verify some rules and prove theorems using already–understood techniques. The exciting stuff happens when you find a question that no one has answered sufficiently and begin to explore it. It is a search for mathematical truth that may begin in a very dark place. None of your existing tools shed any light on the topic you'd like to explore. Often frustrating. Occasionally exhilarating. Historically it has even been controversial, inviting ridicule from prominent mathematicians of the day, and only later becoming generally accepted.
The development we will use here is primarily a product of 19th and 20th century thought. Prior to that, the tools used by mathematicians, logicians, and philosophers were more informal. We will trace the development from the counting numbers, the Natural Numbers, through the Complex Numbers. The latter are needed to do the engineering used in your smartphone. We will use a sequence of extensions starting with an axiomatic base. The axioms are supplemented by definitions of interesting things such as operations, like addition, and relations, such as less than. We will stick pretty close to this path, but will occasionally point to additional topics that depend on or are related to the ideas of the main thread.
Mathematics is central to a classical education. In the Medieval university there were seven basic subjects, all taught in Latin: Arithmetic, Geometry, Astronomy, Music Theory, Grammar, Logic, and Rhetoric. Liberal Arts colleges such as Loyola try to give students a grounding in most of these as well as more modern ideas. I studied all but Astronomy as an undergraduate, though not all to the same depth. More advanced scholars in the university of year 1200 or 1300 would study Moral Philosophy, Metaphysics, and Physics. I too studied these in the 20th century. But it was the rigorous way of thinking provided by the basic subjects, especially maths, that enabled the later studies.
This book has ideas drawn largely from arithmetic and logic, but will provide excursions to other areas of math. It does not, however, give a complete treatment of any single topic. We will build on ideas established early to develop later topics and so will focus on essentials needed for that development. Along the way we will leave much for you to explore. Think of it as a long hallway with many, many, rooms off to the side. We will spend time in some of these rooms, but will only glimpse a few of the others. Some of those rooms lead to other hallways, each with additional rooms. You will be able to explore those elsewhere.
1 Mathematics: Misconceptions
2 Early Counting – Speculation
3 Natural Numbers and Axiom Systems
4 An Introduction To Sets And A Bit Of Logic
5 Zero. A First Extension – The Whole Numbers
6 The Integers
7 The Rational Numbers
8 Infinite Sequences
9 The Real Numbers
10 Polynomials
11 The Complex Numbers
12 Mathematics
13 Ideas for Studying Mathematics
14 Further Reading