Understanding Maths as a Language

While taking calculus 1 and linear algebra in university. A realization that made these courses "click for me" was that Mathematics is a language we created to describe the world. It's built on logical definitions that make sense within the world we live in. The definition that really helped this idea click in for me is this: every polynomial of degree n has n solutions.

Take this equation, for example:

$$x^2 + 1 = 0$$

If we try to solve this using only the real number system, we won’t find any solutions. But once we define a new number say, $i$, where $i = \sqrt{-1}$, we can solve it. This extension leads to the complex number system, and now the equation has two solutions: $x = i$ and $x = -i$.

What’s powerful here is the realization that there’s nothing inherently "real" about real numbers or even complex numbers. We define them to solve problems and describe the world. We can define more numbers if we wanted to. In fact, developers working in 3D graphics do exactly this when they use quaternions, which extend complex numbers further to help with 3D rotations.

This is realization may seem philosophical to some. As self taught software developer who worked on defining my own programming language, this idea resonates with me. Mathematics seems to me overly similar to a programming language, although maths comes with more complexity.