Principles of Mathematics/Introduction/Why I write this book
Why we write this book
In the 1910's Alfred Whitehead and Bertrand Russell began publishing Principia Mathematica in an attempt to formalise the foundations of all of mathematics. However, in the 1930's Kurt Gödel showed that any such effort was destined to fail. No matter how much foundational material one generated, there was yet more that could be added. Unfortunately, Principia Mathematica is very hard to follow.
Beginning in 1935 a group of mathematicians who collectively called themselves Nicolas Bourbaki began writing an axiomatic treatment of much of pure mathematics. Their encyclopedic tomes are surely of great use to professional mathematicians, but because they are written at the highest possible level of abstraction, they are not useful as an introduction to mathematics.
What these efforts both show is that writing a rigorous introduction to mathematics is actually very difficult.
When we were young, our teachers taught us that 1 + 1 = 2. But probably none of us asked, "Why?". If we had done so, our teachers would have laughed and said "1 + 1 is 2, and that's that!". If you also laugh, then this book is not for you. This book is intended for readers who are interested in logical reasoning and mathematical exploration.
To show that 1 + 1 = 2 is not that simple. Mathematics is analogous to language. Despite the fact that we all speak at least one language natively, that does not mean we are experts in how languages work. It turns out that the most basic of things, such as how languages work, or why 1 + 1 = 2, are difficult to explain, even though they lie at the very foundation of a skill we all have.
So it is with the foundations of mathematics. Before we can even begin to explain why the language of mathematics works, we figuratively need to become experts in language. To do that, we need a language to describe the language of mathematics, and we need an abstract perspective that sits at a higher level than mathematics itself. If we do that too formally, we end up with another Principia or Boubaki. But if we do it too informally, we never get to ask, "Why?".
This book will guide you through the fundamentals of mathematics. It does not start from counting apples or from learning multiplication tables. It starts from logic, a friend and tool throughout this book.