Part I — Mathematical Foundations for Understanding LLMs

Introduction

To understand how Large Language Models (LLMs) “think,” we must begin not with code, GPU clusters, or neural network diagrams, but with mathematics. Part I builds the essential mathematical language that allows us to describe, analyze, and reason about LLMs with clarity. This is not mathematics for its own sake—these are the core principles that define how modern AI systems operate.

Many readers can use LLMs, and some can even fine-tune or deploy them in production. But far fewer can explain why these models behave as they do: why they generate one word rather than another, why scaling up size improves performance, or why attention mechanisms work so effectively. These questions cannot be answered by intuition alone. They require first principles—probabilities, information, linear algebra, and representation theory.

Part I provides this foundation. It is written for readers who value conceptual clarity and want to understand LLMs as structured mathematical objects, not mysterious black boxes. The goal is simple: to give you the conceptual compass that makes every later chapter far easier to understand.

Why Mathematics Comes First

In AI, architectures and APIs evolve quickly, but the mathematics underneath does not. Probability distributions, entropy, vector spaces, and optimization are universal ideas. They remain stable whether you are working with GPT-4, Claude 3, Gemini 1.5, Mistral models, or any architecture yet to be invented.

Mathematics is the invariant layer of AI. Once you grasp these principles, LLMs no longer appear magical—they become understandable, predictable systems governed by quantitative rules.

What You Will Learn in Part I

Chapter 1 — Mathematical Intuition for Language Models

We begin by building the basic vocabulary of language modeling: tokens, probability distributions, random variables, conditional probability, entropy, and information. Each term is introduced gently and intuitively. By the end of this chapter, you will understand that LLMs are probabilistic predictors—engines that transform sequences of tokens into probability distributions over the next token.

Chapter 2 — LLMs in Context: Concepts and Background

Next, we step back to understand what a Large Language Model is from a structural and conceptual perspective. We explore pretraining, parameters, scaling laws, and why Transformers revolutionized language processing. This chapter connects abstract mathematics to concrete architecture.

Chapter 3 — Essential Math: Probability, Statistics, and Linear Algebra

Finally, we complete the mathematical toolkit needed for the rest of the book. Concepts such as distributions, expectations, variance, covariance, log-likelihood, vector spaces, matrices, dot products, and high-dimensional geometry are explained clearly and applied directly to language modeling. This prepares you for the more advanced ideas that appear in later parts.

Why These Chapters Matter

Part I is not optional background—it is the conceptual foundation for everything that follows. The ideas introduced here reappear throughout the book: attention mechanisms (Part II) rely on dot products and probability distributions; optimization (Part III) depends on log-likelihood and gradients; representation learning emerges from vector spaces and linear transformations. Without these fundamentals, the mathematical heart of LLMs remains hidden.

Where We Go Next

Now that the stage is set, we begin with the most natural entry point: how to express language mathematically. The next chapter answers a key question:

How can we represent text—something inherently human—in a form that mathematics can understand and compute with?

Let us begin with Chapter 1 — Mathematical Intuition for Language Models.