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I'm interested to hear what very talented mathematicians and physicists have to say about "what it's like" to have an internalized sense of very advanced mathematical concepts... As someone who only completed college calculus and physics, and has some basic CS background, but who is very intrigued by mathematics, I've always been curious about this. Does it feel analogous to having mastery of another language like in programming or linguistics? Any honest, candid insights appreciated!
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Borislav AgapievBorislav Agapiev, Search Entrepreneur
59 upvotes by Kevin Lin (PhD Math), Anurag Bishnoi (Ph.D. student in Mathematics at Ghent University.), Jesse Rudolph, (more)
You can start appreciating and understanding the inner beauty of mathematics at a new, emotional level.

It is like art - having some knowledge and understanding of it will enable you to appreciate masterpieces even more.

Mathematics is, unfortunately, considered as dull and boring in (most of) the popular culture. Big part of that is due to the way it has been taught, almost exclusively as a seemingly endless sequence of rote memorizations of abstract formulae and objects most people have trouble relating to.

Understanding mathematics is really about understanding proofs, i.e. why theorems are true.  Instead of needing to remember bunch of statements, you know they are true because you know how to prove them.

Despite popular misconceptions, proofs of the most important theorems are often simple and elegant, things of beauty.

Consider Pythagora's theorem,  there are  many simple and elegant proofs of it -

Another example, a personal favorite, is Cantor's diagonalization proof there are more real numbers than natural numbers i.e. there is no bijection between them.

This fundamental proof, discovered surprisingly late in 1891, is the basis for such groundbreaking and seemingly unrelated works such as Turing's proof of undecidability of the halting problem and Goedel's incompleteness theorems.

Personally, I find it rather amazing that before 1891,  the world thought there was only one infinity (as I did, before understanding Cantor's proof for the first time).

Of course, the above examples are but a very small set showing the beauty of mathematics.

Having even the most basic understanding, and interest, in advanced mathematics will start uncovering this amazing world to you.