Proofs: A Long-Form Mathematics Textbook is available on Amazon here.
The book's table of contents, as well as a sample from the book, can be found at the bottom of this page.
Some hints and select solutions to the exercises can be found here.
Some readers want a solution to every exercise. However, some professors who use the book don't want any solutions posted. As a compromise, I chose 10 problems each chapter and gave a complete solution to those problems. This should provide self-studiers with enough exercises to challenge themselves, while leaving enough problems without solutions for professors who want to assign problems without easily-attained solutions.
For a list of errors in the book, click here.
I'm working on collecting other things for you to read or watch. Stay tuned.
LaTeX is the program that nearly all mathematicians use to type up their work. It was used to make every modern math book you've ever read and every math research article published this year.
Bates College has a great introduction to LaTeX, packed with templates and lessons. You can access their webpage here. You can use the website Overleaf.com to typeset your documents into beautiful PDFs, or you can download a program to your computer. More can be found at the Bates website here.
Some consider The Not So Short Introduction to LaTeX to be one of the ultimate reference guides.
Now, inevitably problems will arise. When they do, Google is of course your friend. Most of what I've learned has been through troubleshooting with Google, which usually directs me to some page on tex.stackexchange.com in which someone many years ago asked the same thing. I'll link to tex.stackexchange here but I'll admit that despite visiting their site literally thousands of times, until right now I've never gone to their homepage; Google always suggests the right page for me. Another good reference site is https://en.wikibooks.org/wiki/LaTeX. Google will also direct you there, but their main page also has an organized list of topics for you.
Another nice tool is Detexify, in which you can draw a symbol that you would like to type, and it will tell you the command for it.
If you would rather learn in a language other than English, check out learnlatex.org. They have good stuff in Català (Catalan), Deutsch (German), English, Español (Spanish), Français (French), मराठी (Marathi), Português (Portuguese), and Tiếng Việt (Vietnamese).
Lastly, here are some documents that might be helpful to search through:
My books talks a lot about definitions in math. Mathematicians also use a lot of jargon, and the good folks at MathVault have collected The Definitive Glossary of Higher Mathematical Jargon for you all. Check it out here.
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 25 colleges or universities which have used this textbook in their classes: Sacramento State University, Sonoma State University, Soka University of America, American River College, Oglethorpe University, the University of Cleveland, the University of Massachusetts Amherst, Northern Kentucky University, the University of Albany, CSU Northridge, Lee University, the University of Memphis, East Carolina University, Sarah Lawrence College, Belhaven College, the University of Scranton, Southern Methodist University, the Technical University of Munich, Lusófona University, CSU San Marcos, Michigan State, Penn State, SUNY Old Westbury, CSU Stanislaus and Bandung Institute of Technology. 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.