Let’s not delve into the cliche of extolling the beauty of mathematics and berating those who fear it for being irrational. I won’t be speaking like mathematical geniuses to whom quantitative abstract thinking came naturally when they were four years old. The truth is, fearing any subject is not entirely baseless; it’s human. Mathematics, in particular, seems to evoke trepidation, not because it’s inherently scary but perhaps due to its peculiar mode of expression and the abstract nature of symbols and notations. Reflecting on my own relationship with mathematics, I can say I don’t fear it now. Yet, I can vividly recall the torturous thoughts it induced when I first encountered some of the concepts. Over time, the apprehension diminished not because mathematics changed its inherent nature but because I gradually became more accustomed to its intricacies.

This apprehension isn’t exclusive to mathematics alone; it extends to many sciences and beyond. The issue lies in how we learn these subjects in a formal education setting. The disjointed presentation often obscures the interconnectedness of concepts. In reality, many ideas evolve through multiple failed attempts and revisions by researchers from various fields over decades, if not centuries. Expecting to accept it all gleefully in a two-hour lecture is akin to landing in the 100th episode of season 5 of a TV show without having seen the preceding seasons and episodes. There’s no shame in feeling afraid. It should be scary. Imagine finding yourself amid a complex plot without having witnessed the buildup. The goal is to acknowledge the absurdity, training our minds to move back and forth in this intricate setting through reason and gaining knowledge, filling up the gaps and hence becoming a part or perhaps even an active contributor to a process greater than life and the living.

When we think of mathematics, our minds often imagine complicated equations and symbols, almost akin to witchcraft. Yet, if we delve deeper, we realize it’s just another language - much like English. It has rules (akin to grammar), fundamental components expressed through notations (like words), and the creative freedom to use these to communicate ideas. Regardless of the natural language in which a mathematics textbook is written, the equations remain constant. It highlights mathematics' power as a universal language, efficiently communicating ideas quantitatively and globally. When we start viewing mathematical notations as words and equations and expressions as sentences, it becomes less alien, resembling any other language. The intimidation, however, persists, rooted in the abstract ideas it conveys and the inherent complexities of quantitative thinking.

The abstraction in mathematics is often deemed unnatural and disconnected from the real world. But consider this: even the simple words we use are abstractions of real-world objects, and the sounds often lack any logical relation to the entities they represent. Humans have always found ways to abstract the real world for communication. It’s when this abstraction ventures into the unknown that it becomes unsettling.

Mathematics, in this respect, commits the ‘crime’ of having fewer restrictions. Beyond a set of basic rules, it relies entirely on deductions, delving as deep and abstract as one’s thoughts allow. This freedom, however, comes at the cost of giving up on immediate real-world associations many times. The beauty lies in the fact that such abstract thinking, sometimes seemingly divorced from reality, might one day explain what’s unknown today. However, mathematics is not just a tool designed solely for the real world or its future, waiting to be useful someday. It’s a language of ideas expressed quantitatively, governed by rules that allow boundless exploration. When current rules fall short, we might have to refine or add onto them carefully without running into contradictions. It’s a venture in itself - an exquisite set of ideas expressed efficiently and precisely. Mathematics transforms our qualitative understanding into quantitative reasoning, deriving and communicating patterns and associations, often also aiding in making the world a more logically manageable place to be in.

Three Very Different Things Called “Mathematics”

One reason math anxiety persists is that we conflate three genuinely distinct activities under a single label.

The first is computation: carrying out arithmetic, manipulating expressions, running an algorithm. This is the mathematics of school drills. It requires accuracy and practice, but not deep understanding. A calculator does it better than you, and that’s fine.

The second is formal reasoning: understanding why a result is true, not just that it is. This is the mathematics of proofs. When you prove that $\sqrt{2}$ is irrational, you are not computing anything - you are constructing an airtight logical argument. No calculator can do this. A student who only drills computation and then encounters a proof course often feels they’ve entered a foreign country, because they have.

The third is creative mathematics: discovering new patterns, conjecturing, finding the right formulation of a problem. This is rare and requires genuine insight. But you do not need to be here to benefit enormously from mathematics. You need the second layer - formal reasoning - far more than most curricula admit.

The tragedy of standard education is that it spends almost all its time on layer one, introduces layer two fleetingly and confusingly, and gestures vaguely at layer three when describing “mathematical talent.” Students conclude that if computation feels tedious (it often is), they must not be “math people.” This is a category error.

Mathematics as a Language for Precise Thought

Consider the claim: “if you keep halving the distance to a wall, you never reach it.” Informally obvious. But to say it precisely, to prove things about it, requires the language of limits. The statement $\sum_{n=1}^{\infty} \frac{1}{2^n} = 1$ is not just a fact about numbers - it is a precise, verifiable, and somewhat surprising claim about infinite processes, stated in a language that admits no ambiguity.

This is what mathematics is: a language designed for maximal precision. When an ordinary sentence is ambiguous, we argue about meaning. When a mathematical statement is ambiguous, we fix the definitions and the argument disappears, replaced by a proof or a counterexample.

For those working in machine learning or computer science, this is not optional background knowledge. Gradient descent, backpropagation, generalization bounds, the convergence of neural network training - these are mathematical objects. You can use them without understanding them, in the same way you can drive without understanding combustion. But you cannot design better engines without knowing the physics.

A Taste of What Precision Buys You

Consider the question: are there infinitely many prime numbers? This is not obvious. Primes become sparse as numbers grow large. Maybe they eventually stop. Euclid settled this around 300 BCE with one of the cleanest proofs in all of mathematics.

Theorem (Euclid): There are infinitely many prime numbers.

Proof: Suppose, for contradiction, that there are only finitely many primes: $p_1, p_2, \ldots, p_n$. Consider the number $N = p_1 \cdot p_2 \cdots p_n + 1$. Either $N$ is prime, or it has a prime factor. But $N$ is not divisible by any of $p_1, \ldots, p_n$ (dividing gives remainder 1 each time). So either $N$ itself is a new prime, or $N$ has a prime factor not on our list. Either way, our list was incomplete. Contradiction. $\blacksquare$

No computation. No approximation. A logical argument from definitions. And it is certain - not in the way scientific theories are certain (pending future evidence), but in the way that $2 + 2 = 4$ is certain, because it follows necessarily from the rules.

Now consider the identity $e^{i\pi} + 1 = 0$. This connects five fundamental constants of mathematics - $e$ (the base of natural logarithms), $i$ (the imaginary unit), $\pi$ (the ratio of circumference to diameter), $1$ (the multiplicative identity), and $0$ (the additive identity) - in a single equation. It is not a coincidence or an approximation. It follows from Euler’s formula, $e^{i\theta} = \cos\theta + i\sin\theta$, evaluated at $\theta = \pi$. The beauty here is not aesthetic in the superficial sense. It is the shock of discovering that objects defined completely independently turn out to be profoundly related. Mathematics is full of this.

The Spectrum from Symbol to Proof

There is a progression in how one engages with mathematics, and it is worth naming it honestly.

At the base: symbol manipulation. Differentiate $x^3 \sin x$. Apply the product rule. Get the answer. This is useful and necessary, but it is the grammar drills of mathematics - a means, not the end.

In the middle: understanding definitions and theorems. What is a derivative, precisely? What does “continuous” mean, exactly? When you can state the $\varepsilon$-$\delta$ definition of a limit and explain why each quantifier is there, you have crossed a threshold. This is where most of this blog series lives.

At the top: proof and construction. Given a new problem, can you figure out whether a statement is true and why? This requires creativity, but it is a learnable skill, not an innate gift.

The books that changed how I think about mathematics are those that refuse to skip the middle layer. Axler’s Linear Algebra Done Right banishes determinants until they can be defined properly. Apostol’s Calculus starts with the integral before the derivative, grounding everything in area. These are not harder - they are cleaner. When you build from definitions, nothing is arbitrary, and nothing needs to be memorized without reason.

What This Series Is

This series is an attempt to build the middle layer, carefully. We will cover limits, derivatives, integrals, and the ideas that underpin machine learning - not by telling you the rules, but by deriving them from definitions. We will state theorems formally, prove the important ones, and be honest about what we are skipping and why.

By the end, you will be able to read a mathematical argument without flinching at quantifiers. You will know what it means for a function to be continuous, differentiable, or integrable - not as a vague intuition but as a precise condition. And you will see why automatic differentiation in neural networks is not magic, but a careful application of the chain rule.

The fear, if it persists, will at least be rational.

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