What is it like to understand complicated/advanced mathematics?

Question

Whenever I see very complex equations, they look, in a way, beautiful even though I don't understand them. This was directly taken from another question:

"- Definition 1 - Given an open subset $U \subset \mathbb{R}^n$, a smooth differential $p$-form on $U$ is a smooth function $\omega : U \mapsto \bigwedge^p(\mathbb{R}^n)^*$ such that $\omega = \sum_I{f_Ie_{i_1}^* \wedge \cdots \wedge e_{i_p}^*}$ for the smooth function $f_I$ on $U$ and the dual basis $\{e_{1}^*,\ldots,e_{n}^*\}$ of the basis $\{e_1,\ldots,e_n\}$ where $I = \{i_1,\ldots,i_p\} \subseteq \{1,\ldots,n\}$ with $i_1 < \cdots < i_p$. The vector space of all $p$-forms on $U$ is denoted $\Omega^p(U)$. The vector space $\Omega^*(U) = \bigoplus_{p \geq 0}{\Omega^p(U)}$ is the set of all differential forms on $U$."

When I see something like this I always wonder what it is like to understand all of the symbols and notation. Can someone describe this to me? Would understanding these mathematical notations make the beauty of it fade? Or would it grow stronger?

Answer 1

For me, it feels very simple and logical, and I am always slightly surprised others think it is hard. It gives me some degree of inner balance, slight feeling of fullfilness, satisfaction.

I believe your feelings would gradually change as your knowledge grows, however this is good, and is one of the most attractive aspects of learning.

You would go from "I believe" to "I know" - and the those are two very different levels.

The beauty won't fade, but will be transfigured. Also, new beauty will be revealed.

Answer 2

Specialization is the name of the game in mathematics today, just as is the case in medicine. If you ever have the misfortune of breaking both a shoulder and an ankle, you'll have to talk to twenty different specialists.

There is a joke that maybe you've heard before and you'll certainly hear again. Your local university once invited the world's most renowned expert in category theory to speak at the mathematics colloquium. The presentation was a success. Afterwards, the department chair and some professors went to a restaurant. The department chair picked up the tab. "What's a 20% tip on a \$205.37 bill?" he asked, turning to the category theorist. "Don't look at me," the category theorist said, "I don't do arithmetic."

If you want to lay it on even thicker, you can extend the joke to other practical applications of mathematics for which the category theorist is utterly clueless.

But another thing that you also have to understand is that understanding advanced math in real life is quite unlike what you see on TV. There is no bomb that will blow up if you fail to properly differentiate an integral in the next five minutes. Instead, you have plenty of time to go over the definitions and notations, consult various references, and take your time pondering what it all means. In some cases, you might even be able to contact the author of the article or book and ask for clarification.

In a nutshell, understanding advanced mathematics is outwardly quite boring. Andrew Wiles spent a lot of time alone trying to solve Fermat's last theorem. And when he presented it to the public and a mistake was discovered, it took him a year to fix it, plus help from another mathematician.

Of course I should give the disclaimer that I don't actually understand advanced mathematics, though in a pinch I can bluff my way through.

P.S. Maybe you love typography more than you love mathematics... so in closing, I say unto you: $$\left(\sum_{n = \pi(\phi)}^{\infty} \frac{\phi(\pi(\phi^\pi))}{n!}\right)^{\left(\sqrt{-(\phi(\pi(\phi^\pi)))}\right) \pi}.$$

Answer 3

A humble mathematician might admit he really does not understand any mathematics (I'm being very deliberate with my choice of pronoun here).

What are some mathematical concepts that you think you understand today which you did not understand ten years ago? Review these concepts and ask yourself how complicated and advanced these would have appeared to you ten years ago.

Another thing is that mathematicians, especially on this website, in their zeal to be both precise and concise, often wind up obfuscating simple concepts. For example, the sum of two odd numbers is an even number. This is a simple fact, but there are ways to state it that make it look very complicated and intimidating.