Math Begins Where Intuition Ends
November 30, 2025
A couple of weeks back, I was at school studying for an upcoming Real Analysis exam when an old friend of mine happened to come by. I often get questions about what I study from non-Math majors, since my subject is kind of mysterious to many, unfortunately. My friend took one look at my work on the whiteboard and said It can't be that complicated.
On the board were some facts about continuous functions, with the proofs included for a few. What my friend was asking me is something I've heard a lot, and it's very natural. Classes like Calculus 1-3, or Linear Algebra, Math classes that many people have taken, are normally taught with an intuitive approach, where we rely mostly on intuitive or natural understanding and leave the hard facts and details aside. But I think this approach leads to gaps and edge cases in your knowledge.
To prove my point, I asked my friend a question: Define a continuous function. As expected, he gave an intuitive answer. A function is continuous if you can draw it as a line without lifting your pen. Many of you who've taken some Calculus classes in high school or college are probably already familiar with this definition, but take a second to think about it, and you'll notice the flaw. This definition assumes you can draw every function. You can't. Take this function, for example:
$$ f(x) = \begin{bmatrix} 0 & x \\ -x & 1 \end{bmatrix} $$
What this function does is take some number $x$ and return a matrix. You'd need a five-dimensional paper and pen to draw a function like this. Without drawing this function, how would we know it's continuous? This is where the naive definition collapses, where intuition ends.
In Math, we use non-ambiguous, precise definitions that don't rely on subjectivity or physical metaphors, though we use these to inspire and guide the process of creating a definition. The idea of a continuous function is undoubtedly inspired by drawing a line without picking up your pen, but to be effective, it has to be more precise than that.
Definitions in the math sense don't really exist in the real world, so being unfamiliar with them makes sense. Think of it like this: if we assume Person A is human, we still wouldn't be able to assume much. Does this person have hands? Possibly not. Feet? Maybe. But with Math, we need definitions that give us something to work with.
If we assume that a natural number $x$ is even, that means it is divisible by 2, then we can take some natural number $n$ such that $x = 2n$. At the same time, to prove that a number is even, we need to show that it can be described as $2$ mutliplied by another natural number. This is useful since we can now prove that the sum of even numbers is even, for example:
Proof. Suppose two natural numbers $x, y$ are even. Then we can take two natural numbers $n, k$ such that $x = 2n$, $y = 2k$. Observe that $x + y = 2n + 2k = 2(n + k)$. Since $x + y = 2(n + k)$ for a natural number $n + k$, we say that $x + y$ is even by definition. □
The entirety of mathematics sits on precise, effective definitions that allow us to make statements about different kinds of objects with confidence. These definitions are almost always inspired by real-life observations, but are sharper and more detailed. A world without these definitions is a world without math.