Thought experiments in mathematics: importance in research 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.
 A: I'm of the opinion that all of mathematics is a thought experiment. Unlike the empirical sciences which rely on observations mathematics is a creative discipline which only requires your imagination. Axioms are the beginning point of the experiment and theorems the result. For example, proof by contradiction makes this transparent as it uses thought experiment style reasoning to show something is impossible by imagining it was possible and deriving an absurdity.
A: I guess the answer (as everything in math) depends upon definitions of what "thought" and "experiment" is. When you're dealing, for example, with infinity... is that an instance of "thought experiment"? In case it is, the answer is obvious: math is built up from such thought experiments taking their place literally at each and every tiny place. In case it is not, then the essence of "thought experiment" remains to be an open question...
A: If I understand you correctly, you want to know how thought experiments, i.e. being aware of one's conscious experience, can guide research in mathematics. Interpreted this way, I would refer you to Mach's work, but also Husserl's phenomenology. Before dedicating himself fully to philosophy, Husserl obtained a Ph.D. in mathematics. In his first book, The Philosophy of Arithmetic, he aims to synthesize the psychology of impressions with the concept of number.
I don't know whether you'll find an algorithm for designing thought experiments there, but some awareness of what goes on in the mind when we assign numbers to things, or investigate their structure, might inspire you to arrive at your own.
