u/OkGreen7335

So I know about this subreddit, MSE and MO.
I don't know about other platforms where math ppl gather.

AI is widely hated and just wondering was there ever a "new" that was that much(or more) hated in the past?

Sometimes I reach a theorem near the end of a chapter or course, and I can follow the proof completely. I understand every line, every implication, and I can explain why each step is valid.

But at the same time, I still feel like I don't really understand it.

It's hard to describe. It's not that I think the proof is wrong. It's more like my intuition expected a completely different kind of argument. For example, I might expect a computational proof, but the actual proof is very abstract, or vice versa. Even though I can follow the proof, it doesn't feel "Correct"

After reading it, I usually need to spend a long time thinking about it on my own, asking myself "Why does this approach work?" or "Why wasn't my intuition correct?" Until then, I have this strange feeling that I haven't fully accepted or internalized the result. And I have this feeling of unacceptance

Is this a common experience when learning mathematics?

One of my friends says that you don't need to read a lot of books to reach a master's-level understanding of mathematics. He claims he's only read about 15 math books in total. I definitely haven't read that many.

I've probably read around 5 or 6 books. For example: Calculus by Ron Larson, Baby Rudin, Complex Analysis by Eberhard Freitag and Rolf Busam*, Linear Algebra* by Kenneth Hoffman and Ray Kunze I also read an ODE book but don't remember the author name.

By "read," I don't necessarily mean reading a book cover to cover and doing every exercise. If you've read a chapter, a section, or even used a book as a reference for a topic you were studying, I'd still count that.

I'm curious to see what all of you have read. I know some people have read parts of dozens of different books, and it might be hard to remember every single one, but list as many as you can.

Whenever I have free time and nothing I need to do, sometimes I don't reach for my phone, scroll social media, or play games. Instead, I sometimes just sit and stare at a wall for a while.

I don't fully understand why I do it tbh. It doesn't demand anything from me, doesn't push me toward another task, and doesn't constantly grab my attention the way most things do. I can just sit there and exist for a while.

I was talking to a friend who is struggling with calculus. He said that one thing he hates about mathematics is how everything is connected. If you don't properly learn something from a previous year, it can come back and affect you later. He also said that some concepts that seem very basic when you first learn them end up playing a much deeper role in more advanced mathematics, he was talking about the slope of a line might seem completely straightforward when he first encounter it in geometry, but later it becomes the idea of rate of change in calculus.

That's probably not a particularly deep example to people who have studied a lot of mathematics, but that comment got me wondering.

What are some elementary concepts that seem simple, obvious, or uninteresting when you first learn them, but later turn out to have a much deeper interpretation in advanced mathematics?

By "elementary," I don't necessarily mean elementary mathematics. I mean a concept that is easy to learn and encountered early in whatever subject it belongs to. The concept could come from anywhere: geometry, algebra, analysis, topology, number theory, etc where an idea initially feels straightforward but later reveals unexpected depth or significance.

I was talking to a friend who is struggling with calculus. He said that one thing he hates about mathematics is how everything is connected. If you don't properly learn something from a previous year, it can come back and affect you later. He also said that some concepts that seem very basic when you first learn them end up playing a much deeper role in more advanced mathematics he was talking about the slope of a line might seem completely straightforward when he first encounter it in geometry, but later it becomes the idea of rate of change in calculus.

That's probably not a particularly deep example to people who have studied a lot of mathematics, but that comment got me wondering.

What are some elementary concepts that seem simple, obvious, or uninteresting when you first learn them, but later turn out to have a much deeper interpretation in advanced mathematics?

---

By "elementary," I don't necessarily mean elementary mathematics. I mean a concept that is easy to learn and encountered early in whatever subject it belongs to. The concept could come from anywhere: geometry, algebra, analysis, topology, number theory, etc where an idea initially feels straightforward but later reveals unexpected depth or significance.

To be clear, I'm not denying any aspect of World War II or questioning that it happened. What I'm saying is that the reality of it seems so extreme, bizarre, and unbelievable that if it didn't happen, many people would probably dismiss it as unrealistic fiction and can't happen.

Imagine an alternate universe with humans just like us, but where Earth is very different. There is no Europe, no Germany, no Russia, no America, no familiar countries, languages, cultures, or historical figures, they have different languages, cultures and geography, continents, nations, and histories.(just to not offend a single country so this "fictional story" won't be banned ). This Earth never had a global conflict before.

Now imagine I write a completely "original" novel for that audience. Every country, city, leader, and ideology has a "fictional name" (which are just the real ones from this world). The setting is entirely alien to them. "So I will just take to them the real WWII as a fiction novel".

How would readers react?

I honestly think many would call it ridiculous.

They would say the characters (Mainly Hitler and Stalin) are cartoonishly evil. They would accuse me of creating dictators so extreme and too unrealistic that they feel more like comic-book antagonists than actual human beings.

The scale of violence is so absurd. They would ask why an entire state is running an industrialized extermination program against millions of people. They would say no government could realistically devote that many resources to genocide, and no population would tolerate it for long.

They would complain that the war's scale is absurd. A defeated and humiliated nation somehow rearms and threatens an entire continent. A second totalitarian empire suffers catastrophic losses yet emerges as one of the world's dominant powers. Entire cities are reduced to rubble from the air. The story keeps escalating until it ends with two weapons so destructive that they sound more like something from speculative fiction than from a serious historical drama.

Reviewers would probably call it exaggerated. They would accuse the author of constantly raising the stakes for shock value. They would say the plot lacks restraint and that the atrocities are so extreme that they stop feeling believable.

Some readers would probably argue that human psychology simply doesn't work that way.

Yet all of this actually happened.