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2022-04-20

Math is a Simulation of Reality

I have hated maths for the longest time, from school all through college. I loved it until \(x\) and \(y\) showed up. I never really understood how abstract mathematics would help me in real life, hence never gave any effort to the subject.

I rediscovered math while watching 3blue1brown on Youtube and the detailed visual explanations helped me understand mathematics in a far deeper level that I ever could in school. Going down that path has helped me realise and love what mathematics really is, a simulation of reality.

Maths is the Physical World on Paper

Although I never loved math, I was intensely curious about it. Math had consistent properties and patterns that always intrigued me, how equations can accurately predict physical outcomes and how certain quantities relate beautifully to each other.

Consider the famous equation \( E=mc^2 \). When we were deriving this formula in school, We would start out with the formula for one dimensional motion, and after derivations and integration we would arrive at the formula for making Nuclear Bombs. Even though this was fascinating, I never really gave it any thought in school.

Everytime I came across equations and numbers Fermats Library that are remotely interesting, the thought of how one mathematical symbol flows effortsly to another comes to mind.

During my zoning out in the middle of the day ritual, I was thinking about numbers and wanted to know exactly why \(2+2 = 4\), and what is the deeper meaning of mathematics. I came to the conclusion that \(2+2=4\) is so because \(4=2+2\).

Consider an apple. Now consider another apple. How many apples do we have now? We have \(2\) apples. So what is \(2\) then? \(2\) symbolically represents the amount of apples we have when we take one apple and another apple. Let's say we have another \(2\) apples, so the total number of apples we have is \(4\). So in that respect if we take one apple, then another, then another, and finally another, \(4\) denotes the thing that we have now.

Maths is then an abstraction on top of the physical world. It's a way for us to work with symbols that can be easily manipulated on paper as our primary medium. Math is then a language for representing the physical world. It's nothing more. But that is the sole reason why it reflects the physical world so beautifully. It's a language as close to the metal as possible. It is the shallowest usable abstraction that we have.

When we are working with maths, we are working in a reduced set of the world with clearly defined rules. We are manipulating a clearly defined model of the physical world.

Consider the formula.

\[ \begin{aligned} F &= ma \\ a &= \frac{v - u}{t} \\ F &= m(\frac{v-u}{t}) \end{aligned} \]

What this formula is saying is that Force \(F\) is what we get when we throw a mass \(m\) with an accelration \(a\). Whatever that is required to throw a mass with \(m\) with accelration \(a\) is denoted by \(F\). Accelaration \(a\) is whatever the change in final velocity \(v\) and initial velocity \(u\) over a period of time \(t\). So force \(F\) becomes the effort requried to stop a moving body in a given time \(t\). That is the conceptual model of force. Hence \(F\) denotes that quantity, which is also denoted as Force in the English language. We can then mathematically calculate that quantity on paper, instead of physically moving that object and measuing it.

This is the reason why we have formulas that makes sense and meshes so well with the real world. The real world does not obey the rules of mathematics, mathematics is the real world in abstraction.

Maths is beautiful

Maths hence is beautiful. Math is what asembly is to the CPU. Not binary but close enough to be tremendously useful. Maths makes sense, it's the only true language. It reduces complex physical interactions into easily manipulatable symbols without the chaos of the physical world.

We need to teach students that maths is nothing more than an abstraction, but an abstraction that mirrors the real world in it's entirety. Having mastering over it, not only helps you understand it, but gives you a sandbox to test your theory that you can use it to bend reality. Maths is now very, very interesting, go I wish I knew this earilier.