TITLE: What is the basis for a proof?
QUESTION [19 upvotes]: What is the basis for a proof? I am trying to understand what a proof is, what is the most simple example of a proof? like can you have a 'proof' for $1+1=2$?

REPLY [2 votes]: You are probably asking about formal mathematical proof or maybe you are confused about how trivial mathematical facts like 1+1=2 can be proven. In that sense you should remember that a proof is meaningful only relative to a formal proof system. A string $\pi$ is a proof for a statement $\varphi$ if the pair is accepted by the proof system (think of a proof system as an algorithm/computer program that gets $\pi$ and $\varphi$ and returns YES or NO. Normally a proof system should be sound, i.e. false statements should not have proofs).
If your question is more philosophical then you should first say what do you mean by $1$, $2$, $+$, and $=$ (and there are various opinions about these among philosophers of mathematics and mathematical philosophers). How can we answer the question if we haven't agreed on what these symbols mean? 
If you interpret them as symbols in the language of an arithmetic theory like PA or ZFC then your question is easy to answer. Here are two examples:
In the case of PA, $2$ is defined as $S(S(0))$ and $1$ is defined as $S(0)$ and $+$ is defined recursively as $x+0=x$ and $x+S(y)=S(x+y)$; and your question is to show that $S(S(0))=S(0)+S(0)$ which is not difficult. The symbol $S$ is the symbol for successor.
In case of ZFC, $0 = \emptyset, 1 = \{0\}$ and $2=\{0,1\}$ and etc and again the proof is not difficult.
To talk about a formal proof we should fix the meaning of the symbols and the theory, a formal proof outside any proof system is meaningless. Often people omit explicitly mentioning the formal system because it is clear from context, but it is there.
Many will interpret the question as asking about a formal proof inside a theory about symbols as above (somewhat formalist view of mathematics) but that is not the only way. There are other approaches where the linguistic/formal entities are not primary concern of mathematics but secondary means used for the purposes like communicating mathematical results to others, etc. Constructive mathematics and Platonism are two of the famous ones viewpoints. (And honestly, do you know anyone who believes in $1+1=2$ because of a formal proof in PA or ZFC?). Check for example the BHK interpretation of (constructive) proofs.
However, if you are asking more generally about what is a proof, then I think the following story is relevant:

In 1978, as an undergraduate, I attended Shimon's course Graph Algorithms. At some point, one student was annoyed at Shimon's "untraditional" way of analyzing algorithms, and asked whether Shimon's demonstrations constituted a proof and if so what is a proof. Shimon answer was immediate, short, and clear: A proof is whatever convinces me.
  [Stories about Shimon Even by Oded Goldreich]