Mathematical Thinking and Writing

One of Euclid’s geometry students asked a familiar question more than 2000 years ago.
After learning the first theorem, he asked, “What shall I get by learning these things?”
Euclid didn’t have the kind of answer the student was looking for, so he did what anyone
would do—he got annoyed and sarcastic. The story goes that he called his slave and said,
“Give him threepence since he must make gain out of what he learns.”1
It is a familiar question: “So how am I ever gonna use this stuff?” I doubt that anyone
has ever come up with a good answer because it’s really the wrong question. The first
question is not what you’re going to do with this stuff, but what this stuff is going to do
with you.
This book is not a computer users’ manual that will make you into a computer
industry millionaire. It is not a collection of tax law secrets that will save you thousands
of dollars in taxes. It is not even a compilation of important mathematical results for you
to stack on top of the other mathematics you have learned. Instead, it’s an entrance into
a new kingdom, the world of mathematics, where you learn to think and write as the
inhabitants do.
Mathematics is a discipline that requires a certain type of thinking and communicating
that many appreciate but few develop to a great degree. Developing these skills
involves dissecting the components of mathematical language, analyzing their structure,
and seeing how they fit together. Once you have become comfortable with these principles,
then your own style of mathematical writing can begin to shine through.

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