DiBeos

If you’ve studied math or physics (or if you’ve ever done serious research in these areas) there’s a pretty high chance that, at some point, you’ve sat in front of a page full of symbols, completely lost, just thinking to yourself: “Maybe I’m just not smart enough for this. Maybe this is too much for what my brain can process.” You’re not alone in that. Actually, if you never had that thought, you probably just didn’t get to that point yet. But impostor syndrome is much more common than what you may think. We will see exactly what impostor syndrome means and I don’t know if you will relate to it, but I’m pretty sure that many of you struggle with it, because I myself have struggled this way many times, and I still do. 

For example, we have a video on our channel called “The Core of Differential Forms”, which we published a while back. But, honestly, you guys have no idea how hard it was for me to make that video. To write the script, I had to somehow take a lot of important things in the subject, deeply understand them, and then narrow them down to the most essential topics. Understanding everything took me a very long time. And I got stuck many times, especially when I had to figure out how to visually represent the difference between covariant and contravariant bases. I felt stupid, I felt like an impostor. I read many books and articles online, watched videos on YouTube, asked AI bots, and I just couldn’t get it… That was the point when I felt like giving up on the video idea, because I couldn’t even get it, how was I supposed to actually teach it online… But I pushed through and eventually things clicked. I found a reasonable way of representing the main differences between them.

Obviously, the video is far from perfect and I think I’ve gotten much better at explaining things nowadays than I did a year ago, for example. But the point that I am trying to make with this story is that learning advanced math is really painful. Sometimes, it feels like it is easy for everyone else, everyone else gets it, but you just can’t get it. And then you end up feeling like an impostor.

Impostor syndrome in advanced math is a lot more common than most of us would like to admit, and most of us end up having occasional moments of feeling like we don’t belong here. This is partly because we’re all quietly pretending that we’ve got it all together, when in reality, we’re just trying to survive the next proof, or the next assignment. And what makes this worse is that this field attracts people like us, who grew up being “the smart ones in the class”, the kids who always knew math more than others. But then the higher you go in math, the more your identity gets crushed, really.

You’re good at math. You’ve always been. And then suddenly, nothing clicks anymore. You read the same sentence five times and feel nothing. You try to solve an exercise and realize you don’t even know where to start. And when that happens, your mind doesn’t say, “Wow, this is a really hard topic.” Instead, it says, “I’m a fraud. I’m just stupid.”

And the thing is, many people can relate to that feeling. There’s a deep, almost philosophical, weight in realizing that you will never fully succeed in grasping this field. One commenter in our channel put it perfectly: “The amount of knowledge all around that you will never be able to understand in total in your lifetime is a great burden.” And that’s just the truth. Mathematics is endless, and no one will ever fully understand all of it. Because yeah, there is a lot of it. But also because, even to understand a little piece of mathematics, you need to put in a lot of mental effort, and it’s super frustrating. Even though it is worth it.

So, what’s the point of all of it?

Here’s the straight answer: You’re in the right place. Frustration is what studying math feels like, and that’s how it is supposed to be.

Any person who says things like “this is so easy, why don’t you get it?!”, doesn’t understand different levels in the learning process. If people around you keep on saying that something is easy, but you seem to not get it at all, no matter how much you try, then stop listening to these people. Just think about it: if it were that easy, by definition, you wouldn’t really be learning. The delta between what you know and what you want to know in the future would be really small, probably it would be an epsilon actually. If you find yourself struggling, remember: you’ve reached what you wanted! You’re at a point where you reached the end of your knowledge in a specific field, and you’re about to expand your limits.

Learning advanced mathematics is like a cycle, you have a new task that you have no idea how to solve. You use all of your resources and think through it, cracking your head open trying to come up with solutions. You fail, again and again. But eventually you figure it out. And now, you’ve learned something, until that something brings you a new problem to figure out.

So, impostor syndrome is the tendency to doubt your own accomplishments and feeling like a fraud, like you don’t belong. You basically feel like an impostor.

Actually, you are just at the beginning of the cycle. You don’t need to fix your confidence before you keep going. You just need to not walk away. If you’re feeling behind, take action, revisit the gaps. If your self-esteem is crashing because you’re not performing how you used to, recognize that being stuck just means that you hit the next level, and all of us hit our own levels eventually. If you don’t, it probably means that you are not learning.

Don’t let your brain trick you into mistaking those moments as evidence that you somehow don’t belong. Struggling with math is not a sign that you’re an impostor. It’s the strongest evidence that you’re doing math seriously, so keep on persevering.