I have trouble understanding each point of view and I would like to receive help in order to understand each philosophical point of view correctly. I understand that there are many books and internet resources, but, for example, in MathSE I have found different opinions about the same question, so I am not sure if I understood answers correctly. I will try to be more specific in order to receive a unique answer (as much as it is possible).
As I understand, we would like to formalize what theory, axioms, formulas etc mean in mathematics. But in order to do that we need another language to express definition. Even more, in order to prove some results about theories from the logical point of view, we want this language to be as free from contradictions and have notions as intuitive as possible. To define what proof is, we would like to do it in a way that every proof could be checked at least in principle. Then the idea comes to make proof a finite sequence of expressions.
Question 1: do I understand correctly that both finitists and formalists agree that metamathematics is finitary - it uses only methods (for example, proof techniques) that are finite?
Then, I have read that finitists deny the existence of infinite sets such as real numbers. But, as I assume, Hilbert who was formalist accepted real numbers. What confuses me here is that real numbers are constructed and analyzed within a theory, so this is not metamathematics anymore. Therefore, all descriptions of real numbers are basically just some sequences of symbols. It is our choice to define within this theory what infinity or what a real number is, but it is not necessary. It is just some interpretation to make more connection to the real world, I guess. Why are finitists bothered with this then? I think that formalists do not say they exist in real life, they just do within theory which was constructed using finitary means.
Question 2: what is the difference between interpretations of real numbers of finitist and formalist?
I would appreciate any opinions and commentary.