Every time I ask why most people think that P ≠ NP, I get a response like "because theorem proving is harder than proof checking".
From this, I understand that proving theorems is hard. But why is this? In fact, it is known that a theorem, in mathematics, is a statement that has been proved on the basis of previously established statements. So a not proven theorem would not be called a theorem, would it? I mean, a theorem is easy to prove because it is defined to be so and people call it theorem because they know how to prove it, no? If we were talking about conjectures, I would have not asked this question.
When I was looking, I found this paper: Cook, Stephen (1971). "The complexity of theorem proving procedures". Proceedings of the Third Annual ACM Symposium on Theory of Computing. pp. 151–158. But this seems to confuse me furthermore.
I am sure I am missing something (maybe some obvious one). Would you please clarify this?