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.
I'm sure I could write more about why proofs are important, but I believe it all boils down to this. Proofs are important because some very intuitive results turn out to be false, and if we just accept results with no proof because something 'sounds correct', then we could be led down false pathways. Who knows what horrors that could reap?