
"Advanced" is a loaded word, and therein lies some interesting observations.
Most of the math I learned, I learned more or less on my own from books. To be sure, I was a math major... an "advanced" one, perhaps. I took a bunch of math classes. But most of the time, lecture was the second time I saw something, not the first.
Reading from books without having other classmates or colleagues around to compare yourself to affects your perception. When you see your classmates struggle with a concept (... or hear from those who have come before you how difficult a particular concept is), then you think "okay, this is an advanced concept." But when you're just doing your thing, your own intellect is the only limitation.
Very often, concepts that were considered difficult or "advanced" came naturally to me. I don't think it was because I have an unusual amount of intelligence or talent, but I was always very interested and enthusiastic.
From this perspective, what's it like to understand advanced mathematics? You don't even notice, because you don't know it's advanced. By way of analogy:
Suppose you're in a class and the assignment is to bake cookies and bring them in on Monday. Everybody is handed out the same recipe: take 1 cup of flour, 2 eggs, etc. etc. Mix it all up. Portion out into teaspoon-sized servings. Bake in an oven at so-and-so temperature for such-and-such time. Let cool, then serve. Nothing fancy.
You come to class the next day, and everyone else's cookies are complete shit shows. One guy forgot to crack the eggs, and just threw whole eggs into a pile of flour and sugar. One guy didn't mix the ingredients, just poured separate piles of them on the cookie sheet. One guy put the raw cookie dough in the oven, then took it out, then turned the oven on. One guy, somehow or another, brought in a hamburger (?!). And of course, one guy brought in 100% immaculate cookies that are a wonder to behold, loudly talking about how easy the assignment was. But he brought his cookies in a box from the fanciest high-end cookie shop in town. (There's always that one cheater in class...)
You look around and think, "How the hell could people not follow that recipe?" That's (sometimes) what it's like to understand advanced mathematics. You don't feel like a genius, you just wonder how everyone else got it wrong.
I'll address this from several different perspectives:
Socially
First, I don't think there are any people that are "bad" at math. Most people think they are bad at math because they don't practice. I like to think of math as a muscle. The more you work out, the bigger gains you see. The bigger gains you see, the more you want to work out. It's an addiction.
When interacting with those who haven't studied math, you get a lot of "you're a math person" comments. Being a woman who loves math, I used to get a lot of nerd jokes growing up. This probably determined by emotional growth on some level. It would be interesting to examine just how much of my childhood math prowess affected my future adult development, or any mathematicians really. At the very least, I think my emotional intelligence became somewhat stunted because other people didn't understand/love math and science like I did. It get's a little lonely when you want to talk Moser's Worm and the other girls want to talk about Kim Kardashian's makeup. *sigh*
Intellectually
The world is your oyster.
You are able to see patterns more easily because you understand the underlying concepts behind several disciplines... physics especially. But also art, biology, sales... everything starts to fall into place and out of place at the same time. A lot of questions are answered, and more questions are brought up because you go down the math rabbit hole so to speak. You understand everything, but not really anything. You get comfortable with being in-between.
My favorite example of this is artwork. Several artists use mathematical concepts/ equations to create art... and you see those patterns mimicked in nature. Daina Taimina, adjunct professor at Cornell University, used crochet to create representations of non-euclidian math principles like hyperbolic geometry (below). As you can see, math can be truly beautiful.
Personally
Having a better understanding of math has made several things in a "non-math persons" life kind of tedious for me.
These things are:
Be it a combination of understanding physics and math at the same time (gasp) or whatever else... for most movies I find myself saying "that's not how that works... that's not how any of that works" because you know... physics.
OR
I find myself thinking how I could make that thing work.
A perfect example of this is the last Iron Man movie.
Can I tell you what happened during the movie?
No.
Can I tell you I spent most of the time during the movie thinking about how we could-maybe-one-day make his chest piece of perpetual power?
Yes.
Again...
"That's not how this works!!!!!!! That's not how any of this works!"
But most of all, understanding higher level math just makes life easy and hard at the same time. Who knows what the future holds for the world with all of the advanced mathematicians who are working in the tech industry now?
I can't wait to see.