Proofs: A Long-Form Mathematics Textbook (The Long-Form Math Textbook Series)

Written in a conversational tone that exudes the joy of mathematics, this is the perfect book for anyone looking to enter into the world of higher math. It gives an excellent and very accessible introduction to all the essential general proof methods (direct proof, contradiction, contrapositive, induction, etc.) And prepares the reader for math at the college level by introducing them to key ideas that are generally applicable.

I am a mid-career engineer with a need to have a better understanding of mathematical proofs for projects in work. I’ve just finished chapter 1. Normally, I prefer my mathematical instruction to come through YouTube as I find most textbooks are good at giving information, but not teaching it. This text is entirely different. The author has kept it engaging and understandable. The information isn’t just presented, but taught and taught well.Hopefully I will update this as I go through the book.

Jay effectively wields a sincere and conversational approach as he takes on the role of Sherpa to guide the reader on the arduous journey into the rigor and jargon-filled of formal mathematics. There are humorous footnotes, quirky diagrams/pictures, and insightful intuitions - all in an effort to acclimate the reader to a thinkspace where abstractions can crystallize into concepts more easily.He isn't shy about math being a tough and discouraging journey. The moment I knew I fell in love with this book is when I saw that Exercise 1.1 was to read Carol Dweck's "The Secret to Raising Smart Kids" - an insightful article that dispels the myth of people self-sorting into "good at math" and "bad at math". He constantly stresses that the ability to understand math is overwhelmingly a function of diligence, not some innate von Neumann-ian trait. He encourages confronting earnest failure/adversity and understanding that though frustration is discouraging, it is an indicator of progress and development. Jay is contagiously optimistic in guiding the reader through the journey and it cultivates more profound fulfillment when concepts finally "click".He has solutions to chosen (not all) exercises on his website - I can see why people are frustrated at this, because it can feel akin to having no 'supervision' to train your burgeoning 'neural math model', which can spark self-doubts as to whether you're navigating a problem correctly or not. However, for some other people, it's akin to pouring the alcohol down the drain for the alcoholic. I realize I an answer-aholic who often lets my frustration tempt me to look at the solutions to retro-verify whether my thinking was correct (and defeat the definition of an "exercise"), so I don't mind the lack of solutions as much. This just means it forces me frustration threshold to be a lot higher before I neurotically google/search for solutions online (which is my own shortcoming!).This is all to say that this book is an optimistic and humorous guide in approaching the fundamentals of formal math.

I am a Y2 math undergrad and I really enjoyed this book. It provides clear explanations how to write a proof, i.e. the book gives you blue-prints how to approach and deal with proofs and what kind of types of proofs there are.The writing style is conversational and often even entertaining. You also (re-)learn a lot about math methods, conventions and standards, some can appear rather basic - but no one might have ever told you before. The book also helped me to mentally organize many concepts I have learned previously, i.e. things I already knew but could not put into the appropriate context.A few things to keep in mind:- the author self-published the book - our benefit is the fair price (thanks Jay), the downside might be the rather poor paper quality (it's okayish, I have seen worse paper - but any highlighter leaks rights through the pages). The book is printed in black-and-white, i.e. no colors - but this is perfectly fine.- the book has lots of (good) exercises but no solutions. Some hints are published on the accompanying website, but proper solutions would greatly enhance the self-study experience and make this book and ever more valuable study tool.Overall: if you are an undergrad STEM major: buy this book, you won't regret it!

Not only does this textbook introduce proof-writing in a wonderfully clear and organized way, the text is also full of wildly amusing footnotes. I've laughed out loud MULTIPLE times while reading this text as a supplement to the very staid text assigned for my Intro to Abstract Mathematics class.This proofs book feels like my friend. Seriously. It's weird. I love it so much. I truly, genuinely look forward to reading each section.If you are a math or CS major, buy this book. It will help mold your intuition. It will make math feel fuzzy and warm and exciting and delightful and fun. It will be a wonderful complement to the (probably) staid and formal textbook you have been assigned for your proofs book.THANK YOU, Professor Cummings!

Done college water geology classes. Seen shows on this subject. Seen like it in class, very interesting and understandable parts. Gets deep and a skillful math approach's in itself as tools.

Very useful for learning proofs and the intuition. An essential part of any budding mathematics education.

This is the best written text book I have read (have a masters degree and have read in the vicinity of 100), and it isn't close. The style of writing is like a lecture, not like the terse examples you might be used to, so it is particularly well suited to self study.