There is something inherently beautiful about mathematics that becomes more and more profound the more I study it. I mean... what is a number? It seems to me that a number is just a human-invented label for something. Just a placeholder to describe something abstract.

All of mathematics stems from 2 simple assumptions:
1) Lets assume that the number 1 means what we all think it means. Let's assume that the idea of a 'single', 'solitary' thing is a practical thing to assume, and attach a symbol to this notion of 'unity'.
(Note how all I did there was write a bunch of synonyms for 1, without really defining it. Find me another way to do it, I challenge you. This, in itself is a beautiful thing, no? Doesn't it say something profound that we all understand this thing, but really can't define it without being redundant?)

2) Now let's have a generator function, such that whenever we give it input, it outputs a number that is greater by exactly 1.
f(x) = x + 1
(Note, we inadvertently invented addition in the process. Oops.)

I'm simplifying it a (shockingly tiny) bit, but that's the foundation of all mathematics. Everything else just spills out from those seemingly obviously trivial assumptions. So far, we've only defined 1,2,3,... but all of the decimals and fractions and rules of arithmetic and algebra just come spinning out of there, with a little application of logic.

Logic is the most powerful tool of math. Math and logic are somehow intrinsically linked. I find that beautiful as well.


Your friend may be referring to Goedel's incompleteness theorems:
"The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (e.g., a computer program, but it could be any sort of algorithm) is capable of proving all truths about the relations of the natural numbers (arithmetic). For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that such a system cannot demonstrate its own consistency."
-wikipedia

What that means is that it has been mathematically proven that no finite number of mathematical theories can yield a mathematically complete set of mathematical truths. There will always be more true statements to be made, no matter how robust our understanding of mathematics ever becomes.

I.e., it is absolutely proven that humans have invented/discovered something that can never be fully understood... not just by humans, but by any conceivable finite intelligence. I find this to be remarkably beautiful, too.