This book is an offspring of two beliefs which the authors have held for many years: it is worthwhile for the average person to understand what mathematics is all about; it is impossible to learn much about mathematics without doing mathematics. The first of these convictions seems to be accepted by most educated people. The second opinion is less widely held. Mathematicians teaching in liberal arts colleges and universities are often under pressure from their colleagues in the humanities and social sciences to offer short courses which will painlessly explain mathematics to students with varying backgrounds who are seeking a broad, liberal education. The extent to which such courses do not exist is a credit to the good sense of professional mathematicians. Mathematics is a big and difficult subject. It embraces a rigid method of reasoning, a concise form of expression, and a variety of new concepts and viewpoints which are quite different from those encountered in everyday life. There is no such thing as ''descriptive'' mathematics. In order to find answers to the questions ''What is mathematics?'' and ''What do mathematicians do?'', it is necessary to learn something of the logic, the language, and the philosophy of mathematics. This cannot be done by listening to a few entertaining lectures, but only by active contact with the content of real mathematics. It is the authors' hope that this book will provide the means for this necessary contact.