Gulnara wrote:1 + 1 means 1 and 1 and equals 2. What is 2? The mark called 2 was chosen and named so to be a shorthand when depicting, drawing or writing 1 and 1. Instead of writing 200 times 1 and 1 and 1..., we can shortly write 200. 1 is then shorthand for depicting actual object counted. Instead of drawing entire person, or building, or fruit, I can simply write the mark 1 which people agreed to go with something separate, individual, the thing of its own. It became anonymous mark that goes with anything, can be attached to anything. Same goes for every other number.
I think this answer is simplest and most direct. In essence, saying "1+1" is just another way of saying "2".
The idea of refuting "1 + 1 = 2" is then akin to refuting a definition. A definition may be useful or useless in given communicative context, or it can comport or not comport with common usage, but it can't properly be called correct or incorrect; hence it doesn't make sense to speak of a definition being proven or refuted.
However, to get a final satisfying answer on the overall issue requires a firm definition of each symbol. If, as someone mentioned already, we take the symbols as instructions on how to visualize various numbers of dots, we will regard the result as obvious: "Yes, if I imagine a dot on the left side of my visual field and I imagine another dot on the right side of my visual field at the same time, I am now clearly seeing two dots on my visual field."
In other words, I suggest that any uncertainty around "1 + 1 = 2" arises not from any empirical question, but from the fact that various people and various schools of mathematical thought will interpret those symbols differently. One person sees dots, while another envisions something much more complicated and abstract involving set theoretic concepts believed to be required for the foundations of mathematics. It is not so much that there is no clear answer to the OP's question as that there is no one agreed-upon way to specify the question, at least in a rigorous context.