Mathematics is often misunderstood as a subject of formulas, calculations, and memorization. However, "Everything You Always Wanted to Know About Mathematics (But Didn’t Even Know to Ask)" by Brendan W. Sullivan, written with Professor John Mackey, completely changes that perspective. Rather than teaching students how to solve equations mechanically, the book teaches them how mathematicians think, reason, and construct proofs. It is a comprehensive guide for anyone transitioning from computational mathematics to abstract mathematical thinking.
Whether you're an undergraduate mathematics student, a computer science enthusiast, or someone preparing for advanced mathematics courses, this book serves as an exceptional bridge between elementary mathematics and rigorous proof-based mathematics.
Free PDF Download: Everything You Always Wanted To Know About Mathematics* (*But didn’t even know to ask)
Book Overview
This nearly 700-page textbook is divided into two major parts:
- Part I – Learning to Think Mathematically
- Part II – Learning Mathematical Topics
Instead of overwhelming readers with definitions, the authors gradually develop mathematical intuition before introducing formal concepts. The book emphasizes understanding why mathematical statements are true, not simply accepting them.
One of its strongest messages appears right at the beginning:
Mathematics is not about performing calculations—it's about discovering truths and proving them.
This philosophy remains consistent throughout the entire book.
Why This Book Is Different
Many mathematics books jump directly into theorems and formal proofs.
This book starts with a far more important question:
What actually is mathematics?
The opening chapter explains that mathematics is fundamentally about
- logical reasoning
- discovering patterns
- proving universal truths
- communicating ideas clearly
The authors even compare mathematics with experimental sciences, explaining why checking millions of examples can never replace a mathematical proof. They use examples like the Goldbach Conjecture to illustrate why experimentation alone is insufficient.
This approach immediately changes how readers think about the subject.
Learning Proofs the Right Way
One of the greatest strengths of this book is its treatment of proof writing.
Instead of presenting perfect proofs from the beginning, the authors show:
- correct proofs
- incomplete proofs
- misleading proofs
- common logical mistakes
For example, the discussion surrounding the Pythagorean Theorem examines multiple "proofs," encouraging readers to judge whether each argument is logically sound and clearly written. This teaches not only mathematical correctness but also the importance of clear mathematical communication.
Readers gradually learn
- direct proof
- contradiction
- counterexamples
- logical reasoning
- mathematical rigor
without feeling overwhelmed.
Topics Covered
The book offers a remarkably broad foundation in discrete and abstract mathematics.
Major topics include:
- Mathematical reasoning
- Writing mathematical proofs
- Logic
- Sets
- Mathematical induction
- Relations
- Functions
- Cardinality
- Modular arithmetic
- Combinatorics
- Proof strategies
- Counting principles
- Infinite sets
- Pigeonhole Principle
- Inclusion-Exclusion Principle
An extensive appendix summarizes important definitions, theorems, proof techniques, and mathematical notation, making the book a valuable long-term reference.
Excellent Learning Style
Unlike traditional textbooks that often present theorem after theorem, this book uses an engaging teaching style.
Each chapter generally includes:
- motivation
- learning objectives
- intuitive examples
- visual illustrations
- exercises
- puzzles
- chapter summaries
- look-ahead sections
The progression feels natural.
Rather than memorizing mathematics, readers gradually develop mathematical maturity.
Ideal for Computer Science Students
Computer science students often struggle when transitioning into theoretical courses because they have little experience writing proofs.
This book addresses that challenge perfectly.
Concepts such as:
- recursion
- induction
- logic
- sets
- functions
- relations
- combinatorics
form the mathematical backbone of many computer science topics including:
- algorithms
- data structures
- artificial intelligence
- graph theory
- compiler design
- cryptography
Students preparing for these subjects will find this book especially valuable.
A Strong Focus on Thinking
Perhaps the most refreshing aspect of the book is its philosophy.
Instead of asking,
"Can you solve this problem?"
it asks,
"Can you explain why your solution must always work?"
This subtle shift transforms mathematics from a computational subject into an intellectual discipline.
Readers begin to appreciate that mathematics is not merely about finding answers but about building convincing arguments.
What Makes This Book Stand Out
Clear explanations
Complex topics are introduced gradually with strong intuition before formal definitions.
Excellent proof instruction
Few books teach proof writing as effectively and patiently.
Large number of exercises
Exercises range from introductory questions to challenging problems that deepen understanding.
Reader-friendly writing
The conversational tone makes difficult topics approachable without sacrificing rigor.
Comprehensive coverage
It provides a complete introduction to abstract mathematics suitable for multiple university courses.
Who Should Read This Book?
This book is ideal for:
- Undergraduate mathematics students
- Computer science students
- Engineering students
- Data science learners
- Competitive exam aspirants
- Future researchers
- Anyone interested in mathematical reasoning
Even experienced programmers who never formally studied proofs will benefit greatly.
Pros
- Outstanding introduction to proof writing
- Highly readable and engaging style
- Covers nearly every foundational abstract mathematics topic
- Excellent balance between intuition and rigor
- Rich collection of examples and exercises
- Great reference book for future study
Cons
- The book is extensive, spanning nearly 700 pages, so it requires commitment.
- Beginners without a basic algebra background may find some later chapters challenging.
- Since it focuses on reasoning rather than computation, readers expecting a traditional problem-solving textbook may need time to adjust.
Final Verdict
Everything You Always Wanted to Know About Mathematics (But Didn’t Even Know to Ask) is far more than a mathematics textbook—it is a guide to thinking logically, writing clearly, and understanding the true nature of mathematics. By emphasizing proofs, reasoning, and communication, it equips readers with skills that extend well beyond mathematics into computer science, engineering, and analytical problem-solving.
If your goal is to move beyond formulas and truly understand why mathematics works, this book is one of the best resources available. It encourages curiosity, develops rigorous thinking, and builds the confidence needed to tackle advanced mathematical ideas.
Rating: ⭐⭐⭐⭐⭐ (5/5)
A must-read for anyone who wants to master mathematical thinking rather than simply learn mathematical techniques.
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