Proofs: A Long-Form Math Textbook

This textbook is designed to provide students with a more comprehensive and engaging learning experience compared to the traditional definition-theorem-proof-repeat format. Instead of a terse approach, the text includes extensive commentary, motivation, and explanation, with the goal of fostering a deeper understanding of the material.

The proofs presented are not concise but rather aim to guide students through the thought process, often preceded by "scratch work" or a proof sketch to give them a big-picture view and an explanation of how they might arrive at the solution on their own.

The topics covered in the book include intuitive proofs, direct proofs, sets, induction, logic, the contrapositive, contradiction, functions, and relations. The text places a strong emphasis on making the ideas visually accessible, with over 200 illustrations throughout.

The writing style is relaxed and conversational, with periodic attempts at humor, creating a more enjoyable and approachable reading experience for the students.

This text also serves as an introduction to higher mathematics, achieved through the carefully selected examples and theorems. Following each chapter, there is an introduction to a specific area of mathematics, such as Ramsey theory, number theory, topology, sequences, real analysis, big data, game theory, cardinality, and group theory.

After every chapter, the text includes "pro-tips" - short insights and reflections on the material, study tips, historical notes, and comments on mathematical culture, providing students with valuable guidance and context.

Additionally, at the end of each chapter's exercises, there is an introduction to an unsolved problem in mathematics, further expanding the students' understanding of the field and its ongoing challenges.

The book also includes three appendices: the first discussing additional proof methods, the second containing a collection of particularly beautiful proofs, and the third providing writing advice for students.

Overall, this textbook offers a unique and engaging approach to teaching introductory mathematics, with a focus on fostering a deeper understanding of the concepts through extensive commentary, visuals, and interactive elements.

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