What does it mean to prove something? What does it mean to prove something? I am constantly told that something is defined to work in some way (take as an example the truth table of p implies q) and that you don't need to prove it works right. But the problem is that I don't see the point in working with something which might turn out to be wrong. I am expected to take the risk?
Then proving something is just more or less about remembering all the arbitrary facts of a certain system and using this facts to explain something which happens inside of this system? Or should I assume it all doesn't matter, as long as I'll stay consequent then I'll be able to correct whatever facts are wrong automatically?
 A: You can define whatever you want. I say that an integer is floopy if it is even, prime and greater than 3. My definition is perfectly adequate, and I don't need to prove anything about it.
However, if I do just a bit of work, I can show that no integer is floopy. My definition is still absolutely fine; it's just that nothing satisfies it.
In that sense, I don't need to prove my definition to be true - what would that even mean? - but I should really justify that it's useful. (In this case, I can't justify it, because it describes non-existent objects.)
"Proving something is just about remembering the arbitrary facts of a system and using them to explain things inside the system" is basically spot on, yes. The art of maths is a) in finding the right system, b) spotting the facts inside the system, and c) finding the right way to combine what we know about the system to prove the facts we spot. We have settled on a certain system which seems to make sense and can be used to model reality in various useful ways - the integers, for instance, capture the notion of "number of objects" from the real world, so we study that system.
A: You can see :


*

*Peter Smith, An Introduction to Formal Logic (2003), page 1 :



By `argument' we mean, roughly, a chain of reasoning in support of a certain
  conclusion. So we must distinguish arguments from mere disagreements and disputes.

And page 37 :

When a chain of argument leads from initial premisses to a final conclusion
  via intermediate inferential steps, each one of which is clearly valid,
  then we will say that the argument constitutes a proof of the conclusion
  from those premisses.

This pattern is the core of mathematics : the conclusions derived by valid arguments from the axioms assumed as premises are the theorems of the mathematical theories "identified" by the said axioms.
