My professor of Epistemological Basis of Modern Science discipline was questioning about what we consider knowledge and what makes us believe or not in it's reliability.
To test us, he asked us to write down our justifications for why do we accept as true that 2 plus 2 is equal to 4. Everybody, including me, answered that we believe in it because we can prove it, like, I can take 2 beans and more 2 beans and in the end I will have 4 beans. Although the professor told us: "And if all the beans in the universe disappear", and of course he can extend it to any object we choose to make the proof. What he was trying to show us is that the logical-mathematical universe is independent of our universe.
Although I was pretty delighted with this question and I want to go deeper. I already searched about Peano axioms and Zermelo-Fraenkel axioms although I think the answer that I am looking for can't be explained by an axiom.
It is a complicated question for me, very confusing, but try to understand, what I want is the background process, the gears of addition, like, you can say that a+0=a and then say a+1 = a+S(0) = S(a+0) = S(a). Although it doesn't show what the addition operation itself is. Can addition be represented graphically? Like rows that rotates, or lines that join?
Summarizing, I think my question is: How can I understand addition, not only learn how to do it, not just reproduce what teachers had taught to me like a machine. How can I make a mental construct of this mathematical operation?