Question: I don't know if this is the right place to post this or not, but I will go ahead anyway (sorry if it ain't the right place)

Yesterday I was discussing a particular theorem of geometry with my brother which he just learnt in the school. I had asked him if he knew the proof for it, he replied saying his teacher has said that wouldn't be necessary.

Then, I asked him to sit and try to prove the theorem. He said that knowing the theorem counts for more than knowing the proof. How do I explain to him that knowing the proof is more important and how it can even help expand his thinking?

I know this question doesn't have a single pointed answer as is pre-requisite for questions posted here, but I would appreciate any replies

Answer: 
In my opinion, they're not important. But let me qualify that in a massive way.

Suppose we're interested in solving some particular real-life problem. We're going to make some "let" statements. Like maybe we've got $(x,y) \in \mathbb{R} \times \mathbb{R}$. And to solve the problem, we're going to let $(r,\theta)$ satisfy $x = r\cos \theta$ and $y = r\sin \theta$. Why are we allowed to do this? We're actually leveraging a theorem:

> 
> For all $x,y \in \mathbb{R}$, there exists $r \in \mathbb{R}\_+$ and $\theta \in \mathbb{R}$ such that $x = r\cos \theta$ and $y = r\sin \theta$.
> 
> 
> 

This legitimizes the "Let" statement. So theorems tell us the rules of the game; and therefore, the range of problems the human race is capable of solving is hard-limited by the theorems we know. So theorems are important.

Does that mean you *personally* ought to know the proof of a theorem? That we're not allowed to use it 'til we know the proof? Of course not.

That being said, if you want the ability to prove your *own* theorems and thereby move the human race forward, expanding the range of problems that we're capable of solving, well you'd better start going beyond the question: "What can I do with this theorem?"

You'd better start asking: "How do we even know its true?"

By the way, most theorems in mathematics aren't proved because of their direct applications to real world problems. This begs that we ask: "Why should we care about theorems that don't have direct applications to real-life problems?" You should post another question if this topic interests you.
One approach would be to give an example of a theorem that sounds so absurd that noone in his right mind would accept it without proof. For example, mention the Banach-Tarski result to the effect that the unit ball can be decomposed into 5 parts which can then be reassembled by rigid motions into... two unit balls.

Another example would be the equality of $0.999\ldots$ to $1$. Of course, this one he won't believe even after you give him a proof :-)