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tiagomonteiro0715/The-Math-Behind-Artificial-Intelligence-A-Guide-to-AI-Foundations

A book on the mathematical foundations of AI from an engineering perspective.

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The Math Behind Artificial Intelligence

The Math Behind AI Book Cover

A comprehensive guide to AI's mathematical foundations from an engineering perspective

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Table of Contents

What is this?

Most AI math books are written by mathematicians for mathematicians.

However, this one is completely different.

Written from an engineering perspective (think in terms of building blocks), it connects theory to practice with visualizations, Python code, and real life applications.

Whether you are a student, self-taught dev, or practitioner, you'll find clear explanations of complex math topics that make complex concepts accessible.

You will explore not just what the math is but why it matters and where it is applied in real life.

For example, how derivatives make the backpropagation algorithm possible. Which in turn enables neural networks, which enable all LLMs.

In less than a week it got over 20K views, and since then it is only growing!

What's included:

  • The Architecture of Mathematics: How math connects from foundations to AI, including Gödel's paradoxes and Einstein's relativity
  • The Field of Artificial Intelligence: From Control Theory to modern AI, understanding symbolic vs. non-symbolic AI approaches
  • Linear Algebra: Vectors, matrices, determinant, eigenvalues, and transformations that show geometry of data in machine learning
  • Multivariable Calculus: Limits, Derivatives, integrals, and how change in many directions powers backpropagation
  • Probability & Statistics: Bayesian methods, distributions, and Markov models for learning from uncertainty
  • Optimization Theory: Gradient descent, Adam optimizer, and how machines learn step by step
  • Real-World Applications: A lot of Python code examples, animated visualizations, and practical examples of where the math is applied

Preview

See how math come to life with visualizations!

Integrals

Integrals Animation

Derivatives

Derivatives Animation

Linear Regression Visualization

Linear Regression Animation

Getting Started

Prerequisites

  • Basic programming knowledge (Python recommended)
  • High school-level mathematics
  • Curiosity about how AI really works

Table of Contents

Chapter 1: Background on this Book

  • The Objective Here
  • Why is This Book About AI Different?
  • Let Me Introduce Myself
  • Prerequisites

Chapter 2: The Architecture of Mathematics

  • The Tree of Mathematics: How Everything Connects
  • A Quick History of Mathematics: From Counting to Infinity
  • Foundations of Relativity: How Einstein Used Math to Understand Space and Time
  • Gödel's Biggest Paradox: Can Math Explain Itself?
  • What About Applied Math and Engineering?
  • Code Examples: Analytical and Numerical Approaches
  • The Impact of a Grand Unified Theory of Mathematics
  • A Final Lesson From History

Chapter 3: The Field of Artificial Intelligence

  • What is Artificial Intelligence?
  • Symbolic vs. Non-symbolic AI: What's the Difference?
  • Before AI: Control Theory as the "First AI"

Chapter 4: Linear Algebra - The Geometry of Data

  • What Are Matrices and Why Do They Simplify Equations?
  • Vectors and Transformations: Moving in Multiple Directions
  • Linear Independence, Dependence, and Rank: Why It Matters
  • Determinants: Measuring Space and Scaling
  • What Are Mathematical Spaces and How Do They Simplify Calculations?
  • Eigenvalues and Eigenvectors: Unlocking Hidden Patterns
  • Applications of Linear Algebra in AI and Control Theory

Chapter 5: Multivariable Calculus - Change in Many Directions

  • Limits and Continuity: Understanding Smooth Change
  • Why are limits important to understand derivatives and integrals?
  • Derivatives: How Things Change and How Fast
  • What About Integral Calculus?
  • Applications in AI and Control Theory: Calculus in Action

Chapter 6: Probability & Statistics - Learning from Uncertainty

  • Mean, Median, Mode: Measuring Central Tendency
  • Variance and Standard Deviation: Measuring Spread
  • What Is the Normal Distribution? The Bell Curve of Life
  • How the Central Limit Theorem Helps Approximate the World
  • Bayes Theorem: Learning from Evidence
  • What Are Markov Models? Predicting the Next Step, One Step at a Time
  • Applications in AI and Control Theory: Making Decisions Under Uncertainty

Chapter 7: Optimization Theory - Teaching Machines to Improve

  • What is Optimization Theory?
  • Why Optimization Drives Learning in AI
  • Simple Optimization Techniques: How Machines Learn Step by Step
  • What is Adam? The Most Popular Way AI Models Finds the Best Learning Path
  • Applications in AI and Control Theory of Optimization Theory

Conclusion: Where Mathematics and AI Meet

  • Mathematics is the Foundation of AI
  • The Future: On Device AI and the Democratization of AI
  • Final Reflections
  • Acknowledgements

About the Author

Code Examples

Each chapter has its own folder with Python and latex code.

# Clone the repository
git clone https://github.com/tiagomonteiro0715/The-Math-Behind-Artificial-Intelligence-A-Guide-to-AI-Foundations.git

# Navigate to the directory
cd The-Math-Behind-Artificial-Intelligence-A-Guide-to-AI-Foundations

# Install uv (fast Python package installer)
# On macOS/Linux:
curl -LsSf https://astral.sh/uv/install.sh | sh

# On Windows:
powershell -c "irm https://astral.sh/uv/install.ps1 | iex"

# Create a virtual environment with uv
uv venv

# Activate the virtual environment
# On macOS/Linux:
source .venv/bin/activate

# On Windows:
.venv\Scripts\activate

# Install all required packages using uv
uv pip install numpy scipy pandas matplotlib seaborn scikit-learn torch lightning hmmlearn notebook

Contributing

This book is actively being improved based on reader feedback! I am open to any suggestions!

How to Contribute

  • Found an error, have a suggestion or want to discuess a topic? Reach out at monteiro.t@northeastern.edu
  • Enjoyed the book? Star the repository!

Built With

  • Python - Code examples and demonstrations
  • UV - Fast Python package installer
  • Ray.so - For beautiful code snippets and visualizations
  • LaTeX - Mathematical typesetting

Contact

Tiago Monteiro

License

This project is licensed under the MIT License - see the LICENSE file for details.

If you find this book helpful, please consider starring the repository!
Your support helps others discover this resource.

Current Edition: 1.0 | Status: Actively Maintained

Languages

Python71.7%Prolog28.3%

Contributors

TI
MIT License
Created December 31, 2025
Updated February 22, 2026
tiagomonteiro0715/The-Math-Behind-Artificial-Intelligence-A-Guide-to-AI-Foundations | GitHunt