I am curiosity about if thought experiments have importance in mathematics.
I know about famous thought experiments in physics and also some example in mathematics. Professors who followed that mental image obtained a great theory or theorem, or advance in their research, and seem as direct consequence of such mental image (what they caused) or thought experiment.
In physics I believe that these thought experiments are more popular and well known, but what about the importance of thought experiments in mathematics? In mathematics, the more important thing is rigor or examples that you can write on a sheet of paper. But is that all?
Question. What is the importance of thought experiments in research in mathematics? Have these any importance? Refers the literature if you need it, and I try to find and read these information. Many thanks.
I am asking about the importance of thought experiments that eventually can provide to a professor researching in some field a great advantage or a privileged point of view in his/her work. I am not asking about ingenious or artificious ideas, just about simple thought experiments that provide to the proffesor the idea for the solution of a remarkable theorem.