Have you ever actually, truly, genuinely pondered equality?
What it actually means, as a concept, and not just a symbol?
I don’t mean gender equality or some other abstract, social notion. I’m talking about mathematical equality.
“As I discussed before, ‘equality’ is a rather stringent notion, and really not very many things are genuinely, rigorously equal to each other – you can only be equal to yourself.” – Eugenia Cheng, How to Bake Pi, p. 226.
As a physicist, it’s mind-blowing to me how two, simple, little parallel lines can actually be so profound. Those two little lines say, ‘here is this quantity, this concept, this physical thing I would like to understand and explore on a deeper level’. So, what do you do? You expand it into two parts. Three. Four. Maybe even more parts.
And how do you explain those parts? You use many other mathematical concepts that translate measurable, real-life quantities into things often found in nature, transformable into numbers, and generalised by your friendly neighbourhood mathematician, so you can open your maths toolkit and say marvellous things, like:
‘Velocity is distance over time.’
‘Distance is velocity times time.’
‘Momentum is mass times velocity.’
You take ideas and generalise them. You take ideas and break them down into smaller concepts. And how do you say this? How do you do this?
Two, simple, parallel lines.
In mathematics, equality is a very stringent notion, a very strict notion, as proclaimed by Dr. Eugenia Cheng in her breath-taking book, How to Bake Pi. Thanks to something she said (as quoted above and many more concepts around it/leading up to it), I came to the conclusion that equality is one of the most powerful tools we have at our disposal. I will also add, I saw this concept more clearly when doing an online course on algebra to hone my fundamentals, and she really made it all click and hit home.
It is so mind-blowing to me that we can just declare what something actually is. What it is made of. What its parts actually are. And without mathematical rigour, this would be impossible to prove. Yet, when I read papers on physical topics I’m interested in and I see their equations, I am now realising that when they tell me something like ‘this parameter = a whole bunch of symbols and mathematical processes’, it is them telling me that, hey, it turns out, when you break this thing down into its parts, we find these quantities. To make them make sense, we multiply, divide, use the exponential function… whatever is necessary. The operations, the processes, they all so elegantly and eloquently encapsulate the real world and what happens to those quantifiable realities.
Are you feeling what I’m feeling right now? This sense of profound wonder at the fact that maths lets us take the world and say nearly endless things about it, using concepts we express in concise, accurate manners, through rigorous logic? And so, so, so, so much of it is possible due to the simple concept of equality.
This symbol says ‘whatever is on the right side equals the left side’. It’s like putting your picture on one side and your picture also on the other side, because, mathematically speaking, only you are equal to you. That’s another thing I read in the book, another thing that really conveyed what mathematics is trying to tell you when it says =.
I have never seen mathematics in physics and mathematical equations and concepts in physics in a better, clearer, more profound light. They are just ideas reworded and put under a microscope of logic. They are not scary, random nonsense that you try to grasp at. They are telling you the secrets of the world!
And how do you hear those secrets? It’s simple, really. You have to understand the underlying, fundamental concepts governing maths itself.
To end this post: I highly recommend reading “How to Bake Pi – Eugenia Cheng”.