I did not have any formal mathematical training beyond high-school. So, I was completely jarred by the contrast of my mental model of what a real number intuitively is, and how difficult it is to precisely define what real numbers are. This realisation happened when I read Terrence Tao's Analysis (which is a great textbook in two parts), and came across the construction of the reals in this video (also see this).

Here, I want to write down some reminders for future me, reminders to Tao's beautiful words that are worth cherishing. As Tao poetically says multiple times in his book that the preliminary ideas we learn in high school are scaffoldings necessary for the construction of the building. We need the flawed concepts we learn in high school to build the pure and logical theories of Real Analysis. But we must ultimately abandon almost everything we learned as scaffolding, which seems like a waste, but it isn't a waste since there was no other way to build. I find comfort knowing that my life hasn't been a complete waste. What I find poetic about Tao's words is that he gives hope when your whole world comes crashing. His words are like the bittersweet moments of several Ghibli films that dabble in juxtaposition of order and chaos with themes like humans destroying nature (Princess Mononoke), nature destroying humans (Nausicaä, Pom Poko), or just humans destroying humans (Porco Rosso). In these Ghibli movies, every character is gray, instead of being black or white. And with Tao's words, you feel like a Ghibli character yourself, impure, incapable of holding the full knowledge that might make a human a pure math genius, and knowing that you never will.

Maybe I strayed too far in putting what I personally felt learning about Real Analysis, and maybe not everyone relates to it. But what I feel that everyone should take away from this is that nothing you know is absolute, and there's always scope to learn more. You should be wary about the things you think are absolute truths that do not change no matter how much contrary evidence is given. You should be open to change. And you should study math to train your mind to become comfortable with change and being proven wrong often.