I do not understand the discipline in mathematics that is called "mathematical logic". For me, all of mathematics is based on logic and that is what makes it the exact science. Every theorem or lemma or result should be based on correct reasoning, and there should be some true logic behind it. Or am I mistaken?
Are there some results that cannot be treated with logic? What is the difference between the discipline of "mathematical logic" and the logic used in mathematics? Are there known results in mathematics that are not based on logic?
Find $x$ in $(1)$: $$(1)\quad x+1=0.$$ The logic is to find $x=-1$ by subtracting $-1$ and no one else can say otherwise.
For me, this is mathematics. Even though a lot of theorems are hard to understand (at least for me), there must be some kind of logic behind them. Am I right?