This is the beta version of a book intended primarily for students who have studied calculus or linear algebra and who now wish to take courses that involve theorems and proofs in an essential way. The book is also for students who have less background but have strong mathematical interests.

We have written the text for a one-semester or two-quarter course; typically such a course has a title like "Gateway to Mathematics" or "Introduction to proofs" or "Introduction to higher mathematics." Our book is shorter than most texts designed for such courses. Our belief, based on many years of teaching this type of course, is that the roles of the instructor and of the textbook are less important than the degree to which the student is invited/requested/required to do the hard work.

Here is what we are trying to achieve:

• To show the student some important and interesting mathematics.
• To show the student how to read and understand statements and proofs of theorems.
• To help the student discover proofs of stated theorems, and to write down the newly discovered proofs correctly, and in a professional way.
• To foster for the student something as close as feasible to the experience of doing research in mathematics. Thus we would want the student to actually discover theorems and write down correct and professional proofs of those discoveries. This is different from being able to write down proofs of theorems that have been pre-certified as true by us (in the text) or by the instructor (in class).

Many books intended for a gateway course are too abstract for our taste. They focus on the different types of proofs and on developing techniques for knowing when to use each method. We prefer to start with useful mathematics on day one, and to let the various methods of proof, definition, etc., present themselves naturally as they are needed in context.

Here is a pdf copy of the beta version of our book. We would welcome comments, especially from instructors who have tried this material in their classes!

Copyright 2009 by the authors. All rights reserved. This is a beta version of a book that will eventually be published in printed form. The beta version of this book may be freely reproduced and distributed, provided that it is reproduced in its entirety from the most recent version. This book may not be altered in any way, except for changes in format required for printing or other distribution.

"Logic moves in one direction, the direction of clarity, coherence and structure. Ambiguity moves in the other direction, that of fluidity, openness, and release. Mathematics moves back and forth between these two poles. [...] It is the interaction between these different aspects that gives mathematics its power."
William Byers (How Mathematicians Think, Princeton University Press, 2007)