Preface
This is a mathematical coming-of-age book, for students on the cusp, who are maturing
into mathematicians, aspiring to communicate mathematical truths to other mathematicians
in the currency of mathematics, which is: proof. This is a book for students who are
learning—perhaps for the first time in a serious way—how to write a mathematical proof.
I hope to show how a mathematician makes an argument establishing a mathematical truth.
Proofs tell us not only that a mathematical statement is true, but also why it is true, and
they communicate this truth. The best proofs give us insight into the nature of mathemat-
ical reality. They lead us to those sublime yet elusive Aha! moments, a joyous experience
for any mathematician, occurring when a previously opaque, confounding issue becomes
transparent and our mathematical gaze suddenly penetrates completely through it, grasp-
ing it all in one take. So let us learn together how to write proofs well, producing clear
and correct mathematical arguments that logically establish their conclusions, with what-
ever insight and elegance we can muster. We shall do so in the context of the diverse
mathematical topics that I have gathered together here in this book for the purpose.
What is a proof, really? Mathematicians are sometimes excused from jury duty, it is said,
because according to the prosecutors, they do not know what it means to prove something
“beyond a reasonable doubt,” which is the standard of evidence that juries follow for con-
viction in US criminal courts. Indeed, a mathematician’s standard of evidence for proof is
very high, perhaps too high for the prosecutors to want them on the jury.
Mathematical logicians have a concept of formal proof, which is a detailed form of proof
written in a rigid formal language. These proofs, often intended to be processed com-
putationally, understood and verified by a machine, can irrefutably establish the validity
of the theorems they prove, yet they are often essentially unreadable by humans, usually
providing us with little mathematical insight beyond the truth of the raw statement that
is proved. Despite this, the emerging field of automated theorem proving may in future
decades profoundly transform mathematical practice.
In high-school geometry, students often learn a standard two-column form of proof, in
which certain kinds of statements are allowed in the left column, provided that they are