5 Must Read Math Books to Learn Mathematical Thinking

In the past few weeks, many of my readers have asked me for book recommendations about mathematical proof. Below, I have listed some of the books that my teachers have recommended to me in my academic career and that I have had the opportunity to read. I hope that these resources will reinvigorate your brain to think more mathematically.

Ali
5 min readFeb 24, 2022

People sometimes find themselves wanting to learn about math. When they explore this curiosity by opening and trying to read a math book, however, they feel like they are stranded in the middle of an ocean. That is because most mathematicians assume their audience is at the same level as them and thus write their math learning books for those who are already really good at it. This leads many readers to silently put away the book after just a couple of pages.

Like I mentioned before, man cannot love something or someone that they do not know. Therefore, it is ridiculous to expect someone who is not familiar with mathematics to enjoy and embrace it. The question of how to learn mathematics arises, then. The answer to that is quite simple, actually, you first have to learn mathematical proof techniques, the core of math itself.

Furthermore, mathematical proof is a type of artwork. Just how sitting across from a Picasso for hours, trying to understand and encompass the painting, is incredibly enjoyable, learning about and internalizing one of Euclid or Gauss’s proofs is similarly enthralling and enjoyable.

I have also compiled a list of mathematicians’ favorite proof books in Abakcus. You can find that list here.

Book of Proof

Book of Proof by Richard Hammack | Check the reviews on Amazon.

Book of Proof should be a bedside book for anyone trying to get into mathematical thinking and should be on every math student’s bookshelf. I can confidently say that in the 36 years I have been alive, I have never seen a book that is as well organized and structured as this one.

Dr. Hammock teaches mathematics from the very beginning. He first teaches the fundamentals; clusters and logic, he then builds on them as if building a pyramid in every chapter that follows. That is why if you skip around in the book, you might find yourself needing to consult the parts of the book that you skipped.

This isn’t just a reference book on a bookshelf, however, it is also a book that you can open every now and then and read over and over again.

The Mathematical Universe: An Alphabetical Journey Through the Great Proofs, Problems, and Personalities

The Mathematical Universe by William Dunham | Check the reviews on Amazon.

In this book, William Dunham takes us on a historical journey. This book is especially brilliant as it takes us through the history of theorems that make our modern world tick. While understanding the proof of a theorem is important, it is even more incredible to understand how mathematicians years before came up with such proofs. It is truly eye-opening to delve into the mind of mathematicians.

It is incredibly motivating to learn about how Archimedes found the area of a circle, and how he, without the help of modern computers, discovered the digits of pi. 

Learning of how brilliant minds overcame struggles due to lacking resources is one of the most motivating experiences in the world. For example, when Einstein was just a kid he was introduced to Euclid’s Elements and became enthralled with mathematics, and we know where that led him. In short, learning the story behind the discovery of a mathematical theorem opens brand new doors for us that we didn’t know existed.

Proofs: A Long-Form Mathematics Textbook

Proofs: A Long-Form Mathematics Textbook by Jay Cummings | Check the reviews on Amazon.

Cats are fun animals and therefore a book with cat pictures will likely be fun as well. A lover of math, Jay Cummings has prepared a book in the most simplified and fun way possible in Proofs: A Long-Form Mathematics Textbook. In his book, you can tell that he is a man who has devoted immense effort to teaching and making math enjoyable for everyone. By adding questions without answers next to sample problems with solutions, Cummings encourages the reader to think deeper and perfect their understanding. It really is an incredible book to learn mathematical thinking.

Proofs from THE BOOK

Proofs from THE BOOK by Martin Aigner | Check the reviews on Amazon.

Proofs from THE BOOK might be the second greatest book Springer Publishing has released, after Naive Set Theory. This book is a treasure chest of several beautiful mathematical proofs. Spend a few hours calmly reading this book on a weekend and I guarantee you will come out of it with great enjoyment and understanding.

A Beginner’s Guide to Mathematical Logic

A Beginner’s Guide to Mathematical Logic by Raymond M. Smullyan | Check the reviews on Amazon.

Perhaps humanity has never needed leaders who could only make logical decisions like Abraham Lincoln. This book should be used as a textbook in logic classes to teach students how to put a stop to those who continuously make bad decisions that make our lives harder. Maybe then everyone would approach life logically and make the right decisions based on fact. Raymond M. Smullyan has truly helped humanity by writing this book.

** Note: I might get commissions for purchases made through links in this post.

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Ali
Ali

Written by Ali

Math Teacher. Content Curator. Soccer player. Maradona fan. Mostly write about the lectures I love to learn better. alikayaspor@gmail.com

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