Proofs: A Long-Form Mathematics Textbook

by Jay Cummings

This textbook is designed for students. 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. This book covers intuitive proofs, direct proofs, sets, induction, logic, the contrapositive, contradiction, functions and relations. The text aims to make the ideas visible, and contains over 200 illustrations. The writing is relaxed and conversational, and includes periodic attempts at humor. This text is also an introduction to higher mathematics. This is done in-part through the chosen examples and theorems. Furthermore, following every chapter is an introduction to an area of math. These include Ramsey theory, number theory, topology, sequences, real analysis, big data, game theory, cardinality and group theory. After every chapter are "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. Also, after each chapter's exercises is an introduction to an unsolved problem in mathematics. 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. ~Jay Cummings

Genres: Mathematics, Textbooks, Nonfiction, Science, Logic, Academic, Computer Science

511 pages, Paperback

Published January 19, 2021

Reviews

[Ralph Stoever · Rating 5 out of 5]

A beautiful math book by any standard. My math classes at university are 29 years away and school is 35 years ago. This book was a welcome reminder of the incredible capabilities of clear and deeply structured thought. It's undoubtedly useful if you are learning mathematics, but it applies in every domain. I am delighted that this is the first mathematics book I have read in almost 3 decades. It's intellectually inspiring and will reawaken anyone's thirst for knowledge.

[Kieran Lele · Rating 4 out of 5]

I did not like it very much. I am probably going to fail my midterm. Having said that though, he actually did a pretty good job of explaining things in a clear manner. That is why I had to give it four stars despite my distaste for the material.

[Kyle · Rating 5 out of 5]

A great book for a refresher on a variety of math topics at the upper undergraduate level. The author does a good job of explaining how proof-based math works, and has a lot of great examples and exercises. I wish more books took a leaf out of this book and gave slightly longer proofs that were clearer rather than going for as concise as possible while still understandable. Sometimes a little bit more really helps with clarity.

If you feel like a math novice, then this probably isn't a great choice, as it does assume familiarity with at least high-school level math and facility with algebra. A motivated high-school student who has completed algebra could probably get through it, but I would recommend at least a thorough comfort with algebra to go through the book.

[Joshua · Rating 5 out of 5]

Proofs: A Long-Form Mathematics Textbook is one of the best textbooks on the market today, there are so many reasons why this textbook is a great purchase, most notably, its incredible pedagogical style. Cummings has done a fantastic job making the transition to higher-level mathematics fun and interesting while not compromising on rigour. A few things stand out to me that are unique to this book. Firstly, at the end of each chapter, there is a brief introduction to different areas of mathematics. These consist of Ramsey theory, number theory, topology, sequences, real analysis, big data, game theory, cardinality, and group theory. This is valuable to the student/reader because it gives them a taste of higher mathematics and each subject follows very naturally from the chapter that it proceeds. Furthermore, each chapter has “pro tips” which Cummings has complied as things he wish he knew when he was taking the class, as well as an open question, which is an unsolved problem in mathematics. This encourages the future generation of mathematicians to ponder these questions. The book also contains plenty of helpful diagrams and annotations to aid with understanding. The book is also packed with humour, which is great when studying a hard subject. Overall this book is incredible for anyone interested in studying proofs and is great for self-study.

Video review: https://www.youtube.com/watch?v=ZJ9EU...

[Seth Loudermilk · Rating 5 out of 5]

Very well written, clear, and conversational. This book goes deep while not losing sight of the big picture.

[mila · Rating 5 out of 5]

I had so much fun

[Hamzah Ali · Rating 4 out of 5]

I really enjoyed the writing style of this book and I feel that more than any other introductory proofs book I've seen for mathematics, this book is definitely the most "fun". The author has wonderful diagrams, clear explanations, and a great sense of humor.

The only thing about this book that I'd say I didn't like was that the order of introduction for the various topics of the book felt very strange to me. The book covers the following topics in the given order
1. Pigeonhole Principle (Various Contexts)
2. Direct Proofs (Proof by Cases, some basic Number Theory results)
3. Naive Set Theory
4. Proof by Induction
5. Basic Logical notions
6. Proof by Contrapositive
7. Proof by Contradiction
8. Functions
9. Relations
Pedagogically, I think this is an extraordinarily bizarre introduction to the topic and in that respect, I really think this is better as a course textbook or "return through math-logic" book as opposed to a self-study textbook. I personally would approach this topic in a very different order but I think other than this point, the book is exceptional.

[Dylan Zachrich · Rating 3 out of 5]

There's something sort of indescribably unlikable about sarcastic humor in a "textbook" that just makes me want to beg the author to shut the hell up and just talk to me about the reason I am reading the book in the first place. It doesn't come off as charming, and I legitimately do not understand who it is written for. This book awkwardly stumbles through being both an educational resource/collection of AI generated Calvin and Hobbs jokes at it's lowest points, and fails to be funny when it's distracting the reader away from the subject. In spite of this... The proofs are well explained and informative, it's a legitimately competent resource, if somebody could edit this book to have any trace of the authors pathetic attempts at humor and general personality completely redacted then it would be a pretty beautiful and terse book that explains the nature of mathematical proofs in an intuitive way.

[yakyak · Rating 5 out of 5]

Seriously fantastic. I'm using this book as a supplement to the assigned textbook for my Intro to Abstract Math class (an intro proofs class). It is hilarious, highly accessible to the non-math-genius, and perfectly compatible with any Intro to Proofs college class. Not only does it teach you proof-writing form nothing, it also introduces advanced math topics.

The writer has a fantastic sense of humor and I've never laughed so much at footnotes (we're talking David Foster Wallace tier footnoting). This textbook is pure joy! Have I ever felt joy for any of my math textbooks until now?? I'm so grateful that this book exists.

Highly recommend for math majors who are not math geniuses and CS majors who have to take Discrete Math.

[Yumeko (blushes) · Rating 5 out of 5]

I love this so much honestly. He's very good at introducing someone to new ideas, or showing the usefulness of ideas you might already know. I didn't read the book cover to cover because I was a bit too familiar with the contents, so I couldn't always bear rereading, but he's amazing with introducing new ones regardless (especially in Pro tips). I'm very happy with how unconventional it was, I pick up math textbooks all the time, and I never imagined how different something could look (though ofcourse it's also much looser in many ways). I learned way more than I expected to from this.
Oh and I realized I used to follow Jay on twitter lmao.

[Nik · Rating 5 out of 5]

A really good first introduction to university-level mathematics. Easy to-read and often funny, yet still tackles the hard part of the material.

[Karl · Rating 5 out of 5]

AMAZIN X

[Lucille Nguyen · Rating 5 out of 5]

A good refresher on proof, logic, and undergraduate mathematics. Has good overviews of group theory, combinatorics, and higher level mathematics throughout.

[Jack Shao · Rating 5 out of 5]

A fantastic supplement to any math undergraduate student with an appreciation for mathematics beyond just the assigned problem sets. Explanations and details are verbose.

[Q* := Q - {0} · Rating 5 out of 5]

This is a very good text for students being introduced to rigorous proofs for the first time. It could benefit from a discussion of other introductory techniques rather than a hurried survey of famous proofs from “The Book” (of Erdös fame), but it accomplishes its primary goal admirably.

Students interested in further study would benefit from “How to Read and Do Proofs” by Daniel Solow.

[Emanuel · Rating 5 out of 5]

I purchased this book along with its counterpart, "Real Analysis: A Long-Form Mathematics Textbook," during the first year of my undergraduate studies. It was only available on Amazon in America, but the price tag made it a reasonable purchase nonetheless. I felt sad that there was no hardback version, but most people probably will not keep the book for long.

The value of a book should be judged with respect to its purpose. And this book is the best introduction to higher or intermediate mathematics I have encountered. It is not a hardcore mathematics textbook delving into a topic; rather it introduces many different ones. These are all used to explain and motive proofs/proof methods. The book is interspersed with humorous elements and the topics change rapidly. However, there's a clear, cohesive thread connecting each chapter to the next. It is a cohesive whole.

I particularly enjoyed the narrative style and the comedic elements. It felt like I was exploring mathematics alongside the author. This approach could be beneficial for anyone who feels intimidated by mathematics.

A personal note—on the accessibility of the book:
A little while after receiving the book, a friend of mine was joining the military. We decided to do some problems together before he left. He had borrowed the book beforehand and read a bit. This was the last time we had done math problems together since primary school. Later, I gifted him both books in the series for his birthday, shortly before he began his training to become a sergeant. This book let me seize an opportunity I had long thought passed. I sincerely hope it wasn't the last.

[A]

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[Callie C · Rating 5 out of 5]

This is like Proof Writing for Dummies, if it existed.

[James Millikan · Rating 5 out of 5]

Clear, humorous, and eminently readable, Proofs: A Long-Form Mathematics Textbook is the best book I have found for self-studying higher mathematics. I especially appreciated the "scratch work" which allows you to see how the author's proofs took form, as well as the the creative use of graphics and geometric diagrams. I'm looking forward to reading Real Analysis: A Long Form Mathematics Textbook next. Highly recommended.

[Latchezar Tomov · Rating 5 out of 5]

A great book for first course in the art of proof, very different from the traditional approach. The author really does his best to explain things clearly and to give you the reasoning behind the proofs that most often proceeds in the reverse direction (from what we want to prove, step by step until reaching the proper methods and prerequisites for proofs)