Mathematical Proof and the Principles of Mathematics/Introduction/The reason this book was written
In the late 1800s and early 1900s there was a movement to put mathematics on a more sound basis than it had been on previously. This turned out to be an extremely long and difficult process, but its effects on the progress of mathematics were extremely important. One of the results of this effort was Principia Mathematica, an attempt by Alfred Whitehead and Bertrand Russell to formalise the foundations of all of mathematics in terms of logic. Though a monumental work, the aims of the work were destined to fail for several reasons. First, although mathematics relies on logic intimately, it seems that it's impossible to cover the full scope of mathematics using logic alone as a foundation. In other words, some of the fundamental assumptions of mathematics have an inherently mathematical rather than logical character. Second, one reason for trying to use logic as the sole basis of mathematics was to establish the certainty of mathematical truths at the same level as logic itself. However, in the 1930s 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. Not only that, but there is no way to establish that mathematics is free of contradictions which render the entire subject invalid.
In a similar vein, beginning in 1935 a group of mathematicians who collectively called themselves Nicolas Bourbaki began writing an axiomatic treatment of much of pure mathematics. While their encyclopedic tomes are surely of great use to professional mathematicians, they are written at the highest possible level of abstraction and are not useful as an introduction to mathematics. Similarly, most of Principia Mathematica consists of symbolic notation only legible to experts in formal logic.
Relearning the basics
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 might have laughed and said "1 + 1 is 2, and that's that!". If they were especially good teachers they might have tried to tried to convince you by giving an example. This explanation might go:
Here I have one apple, and over here I have one apple. I put them together like so, and now I have two apples. So one and one make two.
But even though the statement 1 + 1 = 2 seems simple and familiar, it really hides some deep philosophical questions, some of which have not been fully resolved even today:
- What, in fact, is a number?
- What is 'one' and what is 'two'?
- What does it mean for two things to be equal?
- What is addition?
Famously, Principia Mathematica proves 1 + 1 = 2 after hundreds of pages of cryptic symbols.
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. Although we have by early childhood a subconscious knowledge of what verbs, nouns and adjectives are and how to put them together to make a sentence, it requires the study of grammar to truly understand them.
Just as a writer should have a firm understanding of basic grammar, a mathemaician should have firm understanding of what might be called the grammar of mathematics. The purpose of this book is to teach you this grammar. To do that, we need a more formal perspective that reexamines the basics you've already learned. If we do that too formally, we end up with another Principia or Boubaki. (Not that it would be a bad thing, but it's not the purpose of this book.) But the informal approach which you may have seen until now will not be enough to progress to the level of a mathematician.
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.