Proofs: A Long-Form Mathematics Textbook is available on Amazon here.
Some hints and select solutions to the exercises can be found here. There are more to come.
I have received many requests from self-studiers for complete solutions to a handful of exercises from each chapter so that they can check their work; in response to this I do plan to provide some complete solutions. For many other problems I will provide hints.
I am working out a way to do this in such a way that the solutions do not come up in Google searches, as a number of exercises in my book also appear in others' books, and professors around the world have reasonable concerns about the proliferation of solutions online.
I'm working on collecting other things for you to read or watch. Stay tuned.
I'm working on a some documents to help you learn LaTeX. Stay tuned.
This book is the second of a series of textbooks which I am calling “long-form textbooks.” Rather than the typical definition-theorem-proof-repeat style, this text includes much more commentary, motivation and explanation. The proofs are not terse, and aim for understanding over economy. Furthermore, dozens of proofs are preceded by "scratch work" or a proof sketch to give students a big-picture view and an explanation of how they would come up with it on their own. The writing is relaxed and conversational, and includes periodic attempts at humor.
The book covers intuitive proofs, direct proofs, sets, induction, logic, the contrapositive, contradiction, functions and relations. In the first appendix we discuss some further proof methods, the second appendix is a collection of particularly beautiful proofs, and the third is some writing advice.
Each chapter ends with exercises and an open question, as well as ``pro-tips,'' which are short thoughts on things I wish I had known when I took my intro-to-proofs class. They include finer comments on the material, study tips, historical notes, comments on mathematical culture, and more.
This text is also an introduction to higher math. This is done in-part through the chosen examples and theorems. Furthermore, after every chapter is a 6-8 page introduction to an area of math. These include Ramsey theory, number theory, topology, sequences, real analysis, big data, game theory, cardinality and group theory.
I also believe that learning mathematics has become too expensive and commercialized, and textbooks in particular are far overpriced. I am endeavoring to help change this. This 500-page book is currently available on Amazon for just $16, and can be purchased on Amazon here.
I am aware of five colleges or universities which have used this textbook in their classes: Sacramento State University, Sonoma State University, Soka University of America, American River College and Oglethorpe University. If you know of another school which has used it, please let me know!
Traditionally, math classes were taught in the "sage on the stage" style, in which a professor lectures at students, keeping tightly to the material with little to no discussion. This is slowly changing, and research shows that it is changing for the better.
Proof-based textbooks are traditionally a dry list of definitions and results, with terse proofs of those results and not much else; "sage on the page," if you will. These textbooks are typically written by the foremost research experts in the field. I have great respect for these experts, and one of the most challenging tasks in writing a book is the determination of what to include and in what order to present it. These experts did a tremendous job on this, and without their work authors like myself would be unable to write the books we do.
That said, I believe a shift is beginning to occur in the textbook market. As prices continue to climb and mega-corporations are consolidating larger and larger slices of the industry, something has to give. Moreover, upper division math textbooks seem to be written for only the strongest and most self-motivated students. There are about 20,000 bachelor degrees given each year in mathematics and statistics at US institutions. And according to the AMS, there are only about 2,000 PhDs given in these fields per year. So even if one argues that reading terse math is proper because math research articles are (regrettably) written tersely, that argument ignores 90% of undergraduates for whom we want to understand and appreciate the material. Furthermore, it is always worthwhile to remind ourselves that only about half of the 10% that do go on to graduate school end up working in academia, so as pointed out by Stan Yoshinobu---professors are peculiar, and we shouldn't fall into the trap of only teaching to the students we were.
I have argued that there is a need for a new style of textbook, and fortunately I came to this realization just as the gatekeepers of publishing were critically weakened by the advent of self-publishing and online marketplaces. I am not an analyst. I really like analysis, have taken half a dozen semesters of analysis and have self-studied on top of that, but I have never published a single paper in the field nor do I claim to be an expert. A decade ago there is no way that Springer would approach someone like me to write an analysis textbook, however today Amazon's self-publishing arm couldn't be happier to print my book. A decade ago I would have had enormous difficulty bringing my book to the attention of others, but with nearly a hundred million Amazon users in the US and even more worldwide, the only gatekeepers are my readers (please rate and review!). I expect this reality to change the publishing market in the next couple decades, as more educators self-publish their own takes on the classic textbooks.